Lightning electromagnetic environment in the presence of a tall grounded strike object

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL.,, doi:.9/4jd555, 5 Lightning eletromagneti environment in the presene of a tall grounded strike objet Yoshihiro Baba Department of Eletrial Engineering, Doshisha University, Kyoto, Japan Vladimir A. Rakov Department of Eletrial and Computer Engineering, University of Florida, Gainesville, Florida, USA Reeived Otober 4; revised 5 January 5; aepted February 5; published May 5. [] We have analyzed and ompared distane dependenes of eletri and magneti fields due to a lightning strike to a tall objet and due to the same lightning strike to flat ground. In both ases, lightning was represented by a transmission line energized by a lumped voltage soure onneted at the hannel attahment point. The resultant total harge transfer to ground was the same regardless of the presene of strike objet. The eletri field for the strike-objet ase is redued relative to the flat-ground ase at loser distanes from the objet. If we assume, in an idealized ase, that the return stroke wave front speed is equal to the speed of light, v =, the urrent refletion oeffiient at the bottom of the strike objet r bot = (grounding impedane Z gr = ), and that at the top of the objet for upward-propagating waves r top = (harateristi impedane of the objet is equal to that of the hannel Z ob = Z h ), the ratio of the vertial eletri fields on ground for the strike-objet and flat-ground ases (eletri field attenuation fator) will be d/ p (d + h ), where h is the height of the strike objet and d is the horizontal distane from the objet. The orresponding ratio for the azimuthal magneti field is equal to unity. We show that the ratio for either eletri or magneti field inreases with dereasing r bot (r bot <), dereasing r top (r top < exept for the ase of r bot = ), and dereasing v (v < ), and at larger distanes an beome greater than unity. We additionally show that the ratio of the far fields for the strike-objet and flat-ground ases is given by ( r top )(/v + )/( + r gr ), where r gr is the urrent refletion oeffiient at the lightning hannel base when the hannel terminates diretly on ground. For realisti values of r top =.5, r gr =, and v =.5, this ratio (far field enhanement fator) is equal to.3. Citation: Baba, Y., and V. A. Rakov (5), Lightning eletromagneti environment in the presene of a tall grounded strike objet, J. Geophys. Res.,,, doi:.9/4jd555.. Introdution [] It is important to know the lightning eletromagneti environment in the viinity of a tall strike objet for studying lightning return-stroke proesses at early times and for optimizing lightning protetion means of nearby teleommuniation and power distribution lines. [3] Rahidi et al. [] have theoretially shown that the vertial eletri field and azimuthal magneti field at a distane of km from a 553-m-high objet struk by lightning are.6 times larger ompared to the ase when the same lightning attahes to flat ground. These alulations, based on the modified transmission line model with exponential urrent deay with height (MTLE) [Nui et al., 988], were performed for a urrent waveform were thought to be typial for negative subsequent return strokes. Rahidi et al. [] assumed that the urrent propagation speed along the strike objet was equal to the speed of light, Copyright 5 by the Amerian Geophysial Union. 48-7/5/4JD555$9. and the return stroke wave front speed was.63. They further assumed that the urrent refletion oeffiient at the bottom of the 553-m-high objet r bot =.48 and that at the top of the objet for upward-propagating waves r top =.5 (these urrent refletion oeffiients orresponded to, for example, a -W grounding impedane and a 9-W hannel impedane, if the harateristi impedane of the objet was assumed to be equal to 3 W). Note that Rahidi et al. [] employed a lumped urrent soure that injeted the same urrent into the hannel regardless of the presene of the strike objet, although the use of urrent soure (whih is haraterized by infinitely large impedane) is inonsistent with the speified urrent refletion oeffiient (r top =.5) at the top of the objet. We will show in this paper that the field enhanement effet at larger distanes is observed even when this inonsisteny is removed by replaing the urrent soure by an appropriate voltage soure. Rahidi et al. [] have interpreted the model-predited inrease in field magnitudes as being due to () the presene of two urrent wave fronts originating from the top of the tall objet and of8

2 propagating in opposite diretions and () the relatively high speed (v = ) of urrent waves propagating along the tall objet. It appears that the presene of a tall strike objet serves to enhane lightning eletri and magneti fields relative to the ase of strikes to flat ground. Enhanement of lightning eletri and magneti fields by a tall strike objet was also disussed by Diendorfer and Shulz [998], Rakov [], Kordi et al. [3], and Bermudez et al. [4]. [4] On the other hand, Fisher and Shnetzer [994] found that a strike objet appeared to redue eletri fields in its viinity. They examined the dependene of triggered lightning eletri fields on the height of strike objet at Fort MClellan, Alabama. The fields were measured at distanes of 9.3 and 9.3 m from the base of a metalli strike rod whose height was either 4.5 or m. They observed that the leader eletri fields (approximately equal to return stroke fields at suh lose distanes) tended to be redued as the strike objet height inreased. [5] Miyazaki and Ishii [4], using the Numerial Eletromagneti Code (NEC-) [Burke and Poggio, 98], examined the influene of the presene of tall strike objet (6 to 4 m in height) on the assoiated eletromagneti fields at ground level m to 5 km away from the hannel. They represented the lightning hannel by a vertial wire having distributed resistane ( W/m) and additional distributed indutane (3 mh/m), energized by a voltage soure onneted between the hannel and the strike objet represented by a vertial perfetly onduting wire. The voltage soure had internal resistane of 3 W. Grounding resistane of the strike objet was assumed to be 3 W, and ground ondutivity was set to.3 S/m. The ratio of the alulated vertial eletri field due to a lightning strike to a tall objet of 6 to 4 m in height to that due to the same strike to flat ground is smaller than unity at horizontal distanes of to 6 m from the hannel, but is larger than unity beyond 6 m. The ratio reahes its peak around several kilometers from the hannel, and then begins to derease with inreasing horizontal distane. Miyazaki and Ishii noted that this derease was due to the propagation effets (attenuation of eletromagneti waves as they propagate over lossy ground). [6] In this paper, we will examine the distane dependenes of eletri and magneti fields due to lightning strikes to the top of a tall grounded objet and ompare those fields with their ounterparts for strikes to flat ground. In doing so, we will represent both the lightning hannel and the strike objet by lossless, uniform transmission lines energized at their juntion by a lumped voltage soure [Baba and Rakov, 5] (in fat, eah of the two vertial ondutors, lightning hannel or strike objet, an be viewed as the return path for the other [Thottappillil et al., ]). In speifying the soure, we will use the urrent waveform proposed by Nui et al. [99], whih is thought to be typial for lightning subsequent return strokes. [7] The struture of the paper is as follows. In setion, we present expressions for urrent along the tall strike objet and along the lightning hannel, to be used in alulating eletri and magneti fields in the viinity of the strike objet. In setion 3, using these expressions, we alulate vertial eletri fields and azimuthal magneti fields on a perfetly onduting ground due to a lightning strike to a -m-high objet under the following idealized onditions: There is perfet urrent refletion at the bottom of the objet (r bot = ), there is no refletion at the top of the objet for upward-propagating urrent waves (r top = ), and the return stroke wave front speed is equal to the speed of light (v = ). We ompare the resultant fields with those due to the same strike to flat ground. In setions 4, 5, and 6, we examine influenes on the distane dependenes of fields due to a lightning strike to the tall objet, relative to those due to the same strike to flat ground, of the urrent refletion oeffiient at the bottom of the strike objet (r bot =,.7, and ), urrent refletion oeffiient (for upward-propagating waves) at the top of the objet (r top =,.5, and ), and return stroke wave front speed (v = and.5), respetively. Additionally, we onsider the influene of objet height and lightning return stroke urrent risetime. In setion 7, we ompare the results obtained in this paper with experimental data, and in setion 8, with those obtained using other modeling methods. In Appendix A, we show that lightning strikes to a tall objet and to flat ground onsidered in this paper are assoiated with the same harge transfer to ground. In Appendix B, we derive an equation for the far field enhanement fator due to the presene of a tall strike objet using urrent expressions presented in setion.. Distribution of Current Along the Tall Strike Objet and Along the Lightning Channel [8] In this setion, we present expressions, derived by Baba and Rakov [5], for urrent along the tall strike objet and along the lightning hannel, to be used in the following setions in alulating eletri and magneti fields in the viinity of the strike objet. Figure a shows a transmission line representation of lightning strike to a tall grounded objet, omprising two lossless uniform transmission lines representing the lightning hannel (whose harateristi impedane is Z h ) and the tall strike objet of height h (whose harateristi impedane is Z ob ), a lumped grounding impedane (Z gr ), and a lumped voltage soure that generates a voltage waveform V (h, t) =Z h I s (h, t), where I s (h, t) is the lightning short-iruit urrent. The lightning short-iruit urrent, I s (h, t), is defined [Baba and Rakov, 5] as the lightning urrent that would be measured at an ideally grounded objet (Z gr =orz gr Z h )of negligible height (h ). The urrent propagation speed along the strike objet is assumed to be equal to the speed of light and that along the lightning hannel to be equal to v, the return stroke wave front speed. The urrent refletion oeffiient at the bottom of the tall objet (r bot ) and the urrent refletion oeffiient at the top of the objet for upward-propagating waves (r top ) are given by r bot ¼ Z ob Z gr Z ob þ Z gr : r top ¼ Z ob Z h Z ob þ Z h : Current distributions, I(z, t), along the tall objet ( z h) and along the lightning hannel (z h), for the ðþ of8

3 Figure. Lightning strikes (a) to a tall grounded objet of height h and (b) to flat ground, represented by lossless transmission lines onneted in series with a lumped voltage soure generating an arbitrary voltage waveform, V (h, t) = Z h I s (h, t) orv (, t) =Z h I s (, t), and a lumped grounding impedane (Z gr ). Z h is the harateristi impedane of the transmission line representing the lightning hannel, and Z ob is that representing the tall strike objet; r top is the urrent refletion oeffiient at the top of the tall objet for upwardpropagating waves, and r bot is the urrent refletion oeffiient at the bottom of the tall objet; r gr is the urrent refletion oeffiient at the hannel base (ground) in the absene of a strike objet. onfiguration shown in Figure a, are given by Along the strike objet Iz ð ; tþ ¼ r X top r n bot rn top I h z s h; t nh 3 6 n¼ þr nþ bot r n top I s h; t h þ z nh z h ðaþ Along the lightning hannel Iz ð ; t Þ ¼ r top I s h; t z h v 6 4 þ X þ r top n¼ z h; r n bot rn top 3 h; t z h nh 7 5 v I s ðbþ where n is an index representing the suessive multiple refletions ourring at the two ends of the strike objet. Equations (a) and (b) are the same as equations (a) and (b) of Baba and Rakov [5], exept v ref, the speed of urrent waves refleted from ground and then transmitted into the hannel, in equation (b) is replaed by v in equation (b). Rationale for replaing v ref with v is disussed by Baba and Rakov [5]. Equations (a) and (b) show that two urrent waves of the same magnitude, ( r top )I s (h, t)/, are initially injeted downward, into the tall objet, and upward, into the hannel. Note that Equation (a) is the same as equation (5) of Rahidi et al. [], who used a distributed-shunt-urrentsoure representation of the lightning hannel, and the struture of equation (b) is the same as that of equation (4) of Rahidi et al. [], although their equations are written in terms of the so-alled undisturbed (mathedonditions) urrent, I m (h, t) =I s (h, t)/, as disussed by Baba and Rakov [5]. [9] The urrent distribution, I(z, t), along the lightning hannel for the ase of strike to flat ground (see Figure b), is given by [Baba and Rakov, 5] Iz ð ; t Þ ¼ þ r gr I s ; t z ; ð3þ v where I s (, t) is the lightning short-iruit urrent (same as I s (h, t) in Equations (a) and (b) but injeted at z = instead of z = h), and r gr is the urrent refletion oeffiient at the hannel base (ground). Equation for r gr (although no downward-propagating urrent wave would be present along the uniform transmission line representing the lightning hannel shown in Figure b) is given by r gr ¼ Z h Z gr Z h þ Z gr : Note that equation (b) redues to equation (3) and equation (a) redues to equation (3) with z = when h approahes zero [Baba and Rakov, 5]. The total harge transfer to ground, alulated integrating urrent given by equation (a) at z =, is the same as that alulated integrating urrent given by equation (3) at z = (see Appendix A). Therefore urrent distributions for the ase of strikes to a tall objet (Equations (a) and (b)) and for the ase of strikes to flat ground (equation (3)) orrespond to the same lightning disharge, as required for examining the influene of the strike objet. On the other hand, urrents injeted into the lightning hannel in these two ases are generally not the same, as disussed next. [] It follows from equations (b) and (3) that urrents injeted into the lightning hannel from the soure for onfigurations shown in Figures a and b are given by I = ( r top )I s / and I = ( + r gr )I s /, respetively. These two urrents are different, unless r top = and r gr = (mathed onditions at the position of the soure) or r top = r gr (Z ob = Z gr ; i.e. r bot = ). Both situations are physially unrealisti, sine typially r gr =(Z gr Z h and Z gr Z ob ). If one fored the urrent injeted into the lightning hannel from the soure in onfiguration shown in Figure a to be the same as that in onfiguration shown in Figure b (by setting the magnitude of the voltage soure to V (h, t) = ( + r gr )/( r top )xz h I s (h, t) instead of Z h I s (h, t), while keeping V (, t) =Z h I s (, t)), then for realisti values of r gr = and r top =.5 the total harge transfer to ground for the tall-strike-objet ase would be.3 times larger than that for the flat-ground ase. [] Table shows the magnitudes of urrent waves injeted by the soure into the hannel, in terms of the lightning short-iruit urrent, for strikes to a tall objet (Figure a) and to flat ground (Figure b). The urrent magnitudes are alulated using equations (b) and (3), with the resultant harge transfer to ground being the same in ð4þ 3of8

4 Table. Magnitudes, I, of Current Waves Injeted Into the Lightning Channel From the Soure for the Configurations Shown in Figures a and b, as a Funtion of the Lightning Short-Ciruit Current, I s, for Different Sets of Current Refletion Coeffiients, r top, r bot, and r gr Current Refletion Coeffiients Strike to Tall Objet a Strike to Flat Ground b r top =,r bot =,r gr =(Z gr =W, Z ob = Z h ).5I s I s r top =,r bot =.9, r gr =.9 (Z gr =5W, Z ob = Z h = 9 W).5I s.95i s r top =.5, r bot =,r gr =(Z gr =W, Z ob = 3 W, Z h = 9 W).75I s I s r top =.5, r bot =,r gr = r top =.5 (Z gr = Z ob = 3 W, Z h =3Z ob = 9 W).75I s.75i s a Figure a: I =( r top )I s /. Figure b: I =(+r gr )I s /. both ases. As expeted, the injeted urrent magnitude depends on Z gr and Z h for strikes to flat ground (although usually Z gr Z h in whih ase the injeted urrent is equal to I s ) and on Z ob and Z h for strikes to the tall objet. 3. Basi Case (R bot = R gr =,R top =,v = ) 3.. Comparison of a Lightning Strike to a Tall Objet With That to Flat Ground [] In this setion, we ompare the vertial eletri field and azimuthal magneti field at ground level due to a lightning strike to a tall objet of height h = m with their ounterparts due to the same strike to flat ground. We onsider here an idealized situation in whih the urrent refletion oeffiient at the top of the tall objet is r top = (no refletion and perfet transmission; Z ob = Z h ) and the urrent refletion oeffiient at the bottom of the objet is r bot = (perfet refletion; Z gr = W). In the ase of lightning strike to flat ground, we assume that the urrent refletion oeffiient at the hannel base (ground) is r gr = (perfet refletion; Z gr =W). Also, we assume here that the return-stroke speed is equal to the speed of light, v =, whih will greatly simplify our analysis. This is our basi ase. We will examine the influenes of variation in r bot (and r gr ), r top, and v on omputed fields in setions 4, 5, and 6, respetively. [3] Figure a shows vertial eletri fields on perfetly onduting ground for a lightning strike to the -m-high objet, at horizontal distanes of d = 3, 6,, and 3 m from the objet. The eletri fields (inluding the eletrostati, indution, and radiation omponents) were alulated using the expression for the eletri field due to an infinitesimal urrent dipole [Uman et al., 975; Thottappillil et al., 998] that was integrated over the radiating setions of the hannel and the strike objet. The presene of ground was aounted for using the image theory. Figure b shows the orresponding eletri fields alulated for the same lightning strike to flat ground. Figures 3a and 3b are similar to Figures a and b, respetively, but for azimuthal magneti fields (inluding the indution and radiation omponents). Note that vertial sales in Figures a and b are different, while in Figures 3a and 3b they are the same. Current distributions along the -m-high objet and along the lightning hannel, used in alulating fields shown in Figures a and 3a, are given by Equations (a) and (b), and urrent distribution along the lightning hannel used in alulating fields shown in Figures b and 3b is given by equation (3). Sine r top is assumed to be equal to (Z ob = Z h ), and r bot to be equal to (Z gr = ), the magnitude of urrent waves injeted initially into the tall objet and into the hannel in onfiguration of Figure a is.5i s (h, t), and that of the urrent wave injeted into the hannel in onfiguration of Figure b is I s (, t). As noted in setion, we used a urrent waveform proposed by Nui et al. [99] as the lightning short-iruit urrent, I s (h, t) ori s (, t). Note that the eletri and magneti field waveforms in Figures and 3 have idential shapes that are the same as the urrent waveshape. This is a result of our assumptions (r top =,r bot =, and v = ), under whih two spherial TEM waves are formed, as further disussed in setion 3.. [4] As seen in Figures a and b, at distanes ranging from 3 to 3 m the magnitude of the vertial eletri field due to a lightning strike to the top of the -m-high objet is smaller than that due to the same strike to flat ground. Although no field waveforms are shown here, as the distane inreases beyond 3 m, the ratio of eletri field magnitudes for these two ases approahes unity. The redution of lightning eletri field in lose proximity of Figure. Basi ase (r bot = r gr =,r top =,v = ). Vertial eletri field waveforms (a) due to a lightning strike to -m-high objet and (b) due to the same lightning strike to flat ground, at horizontal distanes of d = 3, 6,, and 3 m from the lightning hannel. 4of8

5 Figure 3. Same as Figure, but for the azimuthal magneti field. grounded strike objet might be regarded as the eletri field shielding effet of the objet. [5] As seen in Figures 3a and 3b, at any distane the azimuthal magneti field due to a lightning strike to the -m-high objet is idential to that in the absene of the objet (due to a strike to flat ground). [6] We will further disuss distane dependenes of eletri and magneti fields for the strike-objet and flatground ases, as well as of field ratios (eletri field attenuation fators), in setion Analysis of Distane Dependenes of the Ratios of Eletri and Magneti Fields for Tall-Objet and Flat-Ground Cases [7] In this setion, we utilize the ideal transmission line theory developed by Thottappillil et al. [], whih will allow us to obtain easy-to-analyze analytial expressions for lightning eletri and magneti fields. These expressions are valid for the ase of () ideal grounding (Z gr = ), () no refletion at the juntion between the lightning hannel and tall strike objet (Z h = Z ob ), and (3) propagation of all urrent waves, both along the strike objet and along the hannel, at the speed of light. Clearly, these are the same assumptions we made in formulating our basi ase. In setions 4, 5, and 6, we will examine the influene of eah of these three assumptions on the inferenes regarding the lightning eletromagneti environment in the viinity of a tall strike objet made in this setion. [8] Additionally, the analytial field expressions used in this setion require that both the lightning hannel and the strike objet are approximated by ondutors of vanishing radius [Thottappillil et al., ]. This latter idealization is disussed by Kordi et al. [], Thottappillil and Uman [], and Baba and Rakov [3]. Sine we assumed Z gr = (ideal grounding), we an use the method of images to replae the onfiguration shown in Figure a by a vertial wire of infinite extent that is energized by two voltage soures, as shown in Figure 4a. The onfiguration shown in Figure 4a in turn an be replaed by its equivalent involving two infinitely long vertial wires, eah energized by a single soure, as shown in Figure 4b. Note that the two vertial wires shown in Figure 4b are atually olloated and shown separated for illustrative purpose only. Eah soure generates two urrent waves propagating without attenuation or distortion in the upward and downward diretions. Sine the urrent wave speed is assumed to be equal to the speed of light, the resultant eletromagneti field struture is spherial TEM [e.g., Thottappillil et al., ; Kordi et al., ; Baba and Rakov, 3], and we an apply the analytial expressions for TEM-wave eletri and magneti fields derived by Thottappillil et al. []. Total fields due to both wires shown in Figure 4b, orresponding to the onfiguration shown in Figure a (with Z gr = and Z h = Z ob ), will be obtained using the priniple of superposition. [9] The vertial eletri field, E z, on the referene ground plane at horizontal distane d from the wire energized at height h (left wire in Figure 4b) is given by [Thottappillil et al., ] E z ðd; t Þ ¼ E q ðd; tþsin q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ¼ pe ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d p :5I s h; t þ h ; ð5þ d þ h where E q (d, t) istheq-omponent of the eletri field, e is the permittivity of vauum, p (d + h ) is the radial distane Figure 4. Approximation of the onfiguration shown in Figure a in the ase of r bot = and r top =(Z gr =W, Z ob = Z h ). (a) Vertial wire of zero radius and infinite longitudinal extent, energized by two voltage soures. The position of the imaginary referene ground plane is indiated by a horizontal dotted line. (b) Superposition of two wires, eah energized by a single, zero-impedane soure. Eah wire produes a spherial TEM wave. Geometrial parameters used in deriving eletri and magneti field equations are shown. 5of8

6 from the soure (at the top of the strike objet) to the observation point, and q is the angle between the vertial wire and the straight line passing through both the soure and the observation point. Current injeted into the wire in this ase is.5i s, as disussed above (see Table ). Note that equation (5) gives the total eletri field whih is the sum of the eletrostati, indution, and radiation omponents [Thottappillil et al., ]. [] The wire whose exitation point is below the referene ground plane (right wire in Figure 4b) produes, due to symmetry, the same vertial eletri field on the referene ground plane as the other wire whose exitation point is above the referene ground plane. Hene the total vertial eletri field, E z_tall, on ground at horizontal distane d from the strike objet of height h is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! E z tall ðd; tþ ¼ p pe ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I s h; t d þ h d þ h : ð6þ Equation (6) shows that the vertial eletri field in the viinity of a tall objet is inversely proportional to the radial distane, p (d + h ), from the soure at the top of the tall objet to the observation point. The inverse dependene of the total eletri field on the radial distane, p (d + h ), from the soure at height h, not expeted for a vertial lightning hannel, is due to the assumption v =. [] For the ase of strike to flat ground, equation for the vertial eletri field an be obtained by setting h =in equation (6) and is given by E z f lat ðd; tþ ¼ pe d I s ; t d : ð7þ Equation (7) shows that the vertial eletri field (inluding its eletrostati, indution, and radiation omponents) on ground, due to a lightning strike to flat ground is inversely proportional to horizontal distane d from the hannel, as expeted for a spherial TEM wave whose soure is loated on the ground plane. [] The ratio E z_tall to E z_flat given by equations (6) and (7), respetively, is E z tall ðd; tþ E z f lat ðd; tþ ¼ d d=h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð8þ d þ h ðd=hþ þ As expeted, the ratio, whih an be viewed as the eletri field attenuation fator due to the presene of the strike objet, is equal to unity when h =ord h. Figure 5 shows the ratio E z_tall /E z_flat as a funtion of d/h, alulated using equation (8). E z_tall is muh less than E z_flat at d h, one half of E z_flat at d = h/ p 3 (= 6 m for h = m), and nearly equal to E z_flat beyond d =3h to 4h. [3] The azimuthal magneti field, H j, at the referene ground plane at horizontal distane d from the left wire energized at height h (see Figure 4b) is given by H j ðd; tþ ¼ pd :5I s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! d h; t þ h : ð9þ The wire whose exitation point is below the referene ground plane (right wire in Figure 4b) produes the same azimuthal magneti field on the referene ground plane as the other wire whose exitation point is above the referene ground plane. Hene the total azimuthal magneti field, H j_tall, on ground at horizontal distane d from the strike objet is given by H j tall ðd; tþ ¼ pd I s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! d h; t þ h : ðþ Equation () shows that the azimuthal magneti field in the viinity of a tall objet is inversely proportional to the horizontal distane d from the objet. [4] Setting h = in equation () we obtain the orresponding equation for the ase of strike to flat ground, H j f lat ðd; tþ ¼ pd I s ; t d : ðþ The ratio H j_tall to H j_flat given by equations () and (), respetively, is H j tall ðd; tþ ¼ : ðþ H j f lat ðd; tþ Equation (), plotted as a funtion of d/h in Figure 5, shows that H j_tall is the same as H j_flat regardless of distane or strike objet height. [5] Note that one an alulate the same eletri field waveforms as shown in Figures a and b, using equations (6) and (7), respetively, and the same magneti field waveforms as shown in Figures 3a and 3b using equations () and (), respetively. This onfirms that equations (6), (7), (), and () are exat, provided that the assumptions made in deriving these analytial equations are valid. We will examine these assumptions in setions 4, 5, and 6. [6] In summary, the vertial eletri field is strongly attenuated at small distanes from the strike objet (relative to the fields due to the same strike to flat ground). This is onsistent with the boundary ondition at the juntion between eletrially long strike objet and perfetly onduting ground, whih requires that the harge density vanishes (while urrent doubles) at z =. On the other hand, at large distanes, the vertial eletri field is essentially not influened by the presene of the strike objet. The azimuthal magneti field is the same regardless of the presene of the strike objet. We will show in setions 4, 5, and 6 that both eletri and magneti fields an be enhaned (eletri fields at larger distanes only) by the presene of the tall strike objet, when the assumptions made in this setion are relaxed. 4. Influene of Imperfet Current Refletion From Ground [7] Here, we examine the influene of the assumption r bot = r gr = made in the basi ase (see setion 3), assuming r bot = r gr =.7 in setion 4. and onsidering three values of r bot,,.7, and, in setion Comparison of a Lightning Strike to a Tall Objet With That to Flat Ground [8] In this setion, we ompare the vertial eletri field and azimuthal magneti field at ground level due to a 6of8

7 Figure 5. Ratios E z_tall /E z_flat and H j _ tall /H j_ flat, eah as a funtion of d/h, alulated using equations (8) and (), respetively. lightning strike to -m-high objet with those due to the same strike to flat ground, assuming that r bot = r gr =.7 (orresponding, for example, to Z gr =5W and Z ob = Z h = 3 W). Note that Janishewskyj et al. [996], from their analysis of five urrent waveforms measured 474 m above ground on the 553-m CN tower, inferred r bot to vary from.34 to.43, and Fuhs [998], from 3 simultaneous urrent measurements at the top and bottom of the 6-m Figure 7. Same as Figure 6, but for the azimuthal magneti field. Figure 6. Imperfet ground refletion ase (r bot = r gr =.7, r top =,v = ). Vertial eletri field waveforms (a) due to a lightning strike to -m high objet and (b) due to the same lightning strike to flat ground, at horizontal distanes of d = 3, 6,, and 3 m from the lightning hannel. Peissenberg tower, found r bot to vary from.64 to.8. All other assumptions remain the same as in setion 3. [9] Figure 6a shows vertial eletri fields on perfetly onduting ground for a lightning strike to the -m-high objet at horizontal distanes of d = 3, 6,, and 3 m from the objet. Figure 6b shows the orresponding eletri fields for the same lightning strike to flat ground. Figures 7a and 7b are similar to Figures 6a and 6b, respetively, but for azimuthal magneti fields. The fields were alulated in the same manner as in setion 3.. Note that vertial sales in Figures 6a and 6b are different, while in Figures 7a and 7b they are the same. [3] As seen in Figures 6a and 6b, in the ase of r bot = r gr =.7, the peak of the vertial eletri field due to a lightning strike to the -m-high objet is smaller than that due to the same strike to flat ground at d =3to m, but larger at d = 3 m. This indiates that imperfet ground refletion serves to enhane eletri fields at larger distanes from the strike objet, the effet not observed when r bot = r gr = (see Figure 5). As seen in Figure 6a, the peak of vertial eletri field alulated for r bot =.7 inreases and then dereases with inreasing horizontal distane from the strike objet: 4.3, 4.9, 4.3, and. kv/m at d = 3, 6,, and 3 m, respetively. Also, the eletri field peak in the ase of r bot =.7 is smaller than in the ase of r bot = (see Figure a), by 3, 4, 7, and % at d = 3, 6,, and 3 m, respetively. The latter result indiates that the influene of imperfet urrent refletion from ground is more signifiant at loser distanes. 7of8

8 [3] As seen in Figures 7a and 7b, in the ase of r bot = r gr =.7, the peak of the azimuthal magneti field due to a lightning strike to the -m-high objet is larger than that due to the same strike to flat ground at d = 6 to 3 m, while being the same at d = 3 m. Thus, similar to eletri fields, imperfet ground refletion serves to enhane magneti fields at larger distanes from the strike objet. The peak of azimuthal magneti field alulated for r bot =.7 monotonially dereases with inreasing the horizontal distane from the strike objet: 49, 6, 6, and 5.7 A/m at d = 3, 6,, and 3 m, respetively. Similar to the eletri field peak, the magneti field peak is smaller in the ase of r bot =.7 (see Figure 7a) than in the ase of r bot = (see Figure 3a), by 5,, 7, and % at d = 3, 6,, and 3 m, respetively. 4.. Analysis of Distane Dependenes of the Ratios of Eletri and Magneti Fields for Tall-Objet and Flat-Ground Cases [3] In this setion, we further disuss the influene of the urrent refletion oeffiient at ground on distane dependenes of eletri and magneti fields for the strikeobjet and flat-ground ases. In doing so, we will use a onfiguration that involves four appropriately energized vertial wires generating TEM waves. The total eletri and magneti fields will be obtained as a superposition of these TEM waves. [33] Rakov et al. [995] onsidered the lightning M-omponent eletri field as a superposition of field ontributions from a downward-progressing inident urrent wave and an upward-progressing urrent wave refleted from ground. We employ their approah below in deriving equations for eletri and magneti fields due to a lightning strike to the -m-high objet, assuming no refletion at the top of the strike objet (r top = ) and imperfet urrent refletion at the bottom of the objet (r bot < ). We refer to urrent waves, propagating upward and downward from the top of the objet,.5i s (h, t (z h)/v) and.5i s (h, t (h z )/), respetively, as inident urrent waves, and to an upward-propagating urrent wave refleted from the bottom of the objet (from the ground),.5r bot I s (h, t (h + z )/), as a refleted urrent wave. Equations (6) and () give total eletri and magneti fields, respetively, inluding their inident and refleted omponents for the ase of r bot =. In order to employ an arbitrary value of r bot, we first eliminate the refleted-wave ontributions from equations (6) and () and then add a ontribution from the refleted wave orresponding to the new value of r bot. This an be aomplished by modifying the two-wire onfiguration shown in Figure 4b to inlude two additional wires as shown in Figure 8. Note that these four wires are olloated and shown separated for illustrative purpose only. Wires and in Figure 8 are the same as the two wires in Figure 4b, while wires 3 and 4, energized at the referene ground plane, anel the perfet ground refletion and add an imperfet ground refletion, respetively. Indeed, wire 3 injets urrent waves, I (, t) =.5I s (h, t h/), whih anel the perfetly refleted urrent waves produed by wires and. Thus wires,, and 3 produe inident urrent waves, whih orrespond to the ase of r bot = (mathed onditions at the bottom of the strike objet). Wire 4 produes urrent waves refleted (imperfetly) from ground, I ref (, t) =.5r bot I s (h, t h/). Note that wires 3 and 4 produe waves that anel eah other if r bot = (perfet ground refletion ase). [34] As stated above, wires,, and 3 in Figure 8 represent inident urrent waves that are absorbed at ground level, and wire 4 represents refleted urrent waves. Thus, from equation (6) and equation (5) with h = and noting that I s (, t d/)=i s (h, t h/ d/), the vertial eletri field at the referene ground plane is given by E z tall ðd; tþ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pe ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d p I s h; t þ h d þ h pe d :5I s h; t h d þ pe d :5r boti s h; t h d ; ð3þ where the first term whih varies as / p (d + h ) is the field due to wires and in Figure 8, the seond and third terms whih vary as /d are the fields due to wires 3 and 4, respetively. If r bot =, the total field is given by the first term of equation (3), as in the basi ase onsidered in setion 3. If h = (the ase of strike to flat ground), equation (3) redues to E z f lat ðd; tþ ¼ pe d I s ; t d ð pe d r botþ:5i s ; t d pe d I s ; t d : ð4þ ¼ þ r bot If r bot =, equation (4) redues to equation (7). [35] Similarly, the azimuthal magneti fields due to strikes to a tall objet and to flat ground are given by H j tall ðd; tþ ¼ pd I s pd :5I s þ pd :5r gri s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! d h; t þ h h; t h d h; t h d ð5þ H j f lat ðd; tþ ¼ pd I s ; t d ð pd r botþ:5i s ; t d ¼ þ r bot pd I s ; t d : ð6þ If r bot =, equation (6) redues to equation (). [36] Figures 9a and 9b show vertial eletri field waveforms, alulated using equation (3) for a lightning strike to a -m-high objet, on a perfetly onduting ground at horizontal distanes of d = 3 and 3 m from the objet, respetively. Figure 9 shows azimuthal magneti field waveforms, alulated using equation (5) on a perfetly onduting ground at a horizontal distane of d =3m from the objet. Azimuthal magneti field waveforms at 8of8

9 magneti field due to the inident urrent wave remains unipolar. [38] The information about onditions at ground (about urrent waves refleted from ground or about the absorption of the inident urrent wave at ground) arrives at the Figure 8. Approximation of the onfiguration shown in Figure a in the ase of r top =(Z ob = Z h ) and r bot < (Z gr > ), omprising four olloated wires of vanishing radii and infinite longitudinal extent. Wires and are the same as the two wires shown in Figure 4b. They aount for both inident urrent waves and urrent waves refleted perfetly from ground (r bot = ). Wires 3 and 4, energized at the referene ground plane, anel the perfet ground refletion and add an imperfet ground refletion, respetively. Note that eah wire supports unattenuated urrent waves propagating outward from the soure and produes a spherial TEM wave. d = 3 m are not shown in this paper, but their shapes are almost idential to those of vertial eletri field waveforms at the same distane, shown in Figure 9b. In these alulations, we onsidered two values of r bot,.7 and. The solid-line urves in Figure 9 are the total fields (total fields for r bot = are the same as the fields due to the inident urrent wave), dashed-line urves are the fields due to the inident urrent wave and dotted-line urves due to the refleted urrent wave. [37] In Figure 9, the field waveforms due to inident urrent waves (dashed-line urves) begin to deay abruptly at time t =(h + d)/:.43 ms ford = 3 m, and.3 ms ford = 3 m. At d = 3 m, the vertial eletri field due to the inident urrent wave hanges from about 4 kv/m to 5 kv/m, while the azimuthal magneti field due to the same inident urrent wave deays from about 4 A/m to 3 A/m. At d = 3 m, the vertial eletri field and azimuthal magneti field (not shown here) due to the inident urrent wave exhibit similar waveshapes: They begin to deay abruptly after their peaks, at.3 ms, and then maintain their magnitudes at about 5% of the initial peak. This abrupt deay of inident fields is due to the seond term, having the sign opposite to that of the first term, in equations (3) and (5), and signifies that the inident urrent wave is absorbed at the referene ground plane. In equation (3), the first term is a funtion of / p (d + h ), while the seond (negative) term is a funtion of /d. Therefore, at a very lose horizontal distane, suh as at d = 3 m, the magnitude of the seond (negative) term beomes larger than that of the first term. This is the reason for the hange of polarity of the vertial eletri field due to the inident urrent wave at d = 3 m. In ontrast, in equation (5), both the first term and the seond (negative) term are eah a funtion of /d. Thus the azimuthal Figure 9. (a, b) Vertial eletri field waveforms alulated using equation (3) for a lightning strike to a -mhigh objet on a perfetly onduting ground at horizontal distanes of d = 3 m and d = 3 m from the objet, respetively. () Azimuthal magneti field waveforms alulated using equation (5) at a horizontal distane of d = 3 m from the objet. Azimuthal magneti field waveforms at d = 3 m, not shown in this paper, exhibit essentially the same shape as those of the vertial eletri field waveforms shown in Figure 9b. The solid-line urves represent the total field (total field for r bot = is the same as the field due to the inident urrent wave), dashed-line urves represent the field due to the inident urrent wave and dotted-line urves due to the refleted urrent wave. 9of8

10 Figure. Illustration of the influene of imperfet urrent refletion from ground. Shown are ratios E z_tall / E z_flat (solid irles) and H j _ tall /H j_flat (open irles), eah as a funtion of d/h, in the ase of v =, r top =(Z ob = Z h ) and r bot = r gr =,.7, and (Z gr =, 5, and 3 W if Z ob = Z h = 3 W). observation point (h + d p (d + h ))/ later than the information about the inident urrent wave injeted at the top of the strike objet. For example, this time delay is about. ms ifd = 3 m and h = m, and about.3 ms if d = 3 m and h = m. As a onsequene, the influene of ground refletion is smaller for more distant observation points and taller strike objets. Also, the first term in equation (3) varies approximately as /d if d h. Thus the distane dependene of the vertial eletri field is similar to that of the azimuthal magneti field at a distant observation point. [39] We now disuss distane dependenes of the ratio of fields due to a lightning strike to a tall objet and those due to the same strike to flat ground, in the ase of v =, r top = (orresponding to Z ob = Z h ), and three values of r bot =,.7, and (orresponding to Z gr =, 5, and 3 W, respetively, if Z ob = Z h = 3 W). For r gr =.7, the magnitude of urrent waves injeted initially into the hannel and into the objet from the soure in the onfiguration shown in Figure a (strike to tall objet) is ( r top )I s (h, t)/ =.5I s (h, t), and that of the urrent wave injeted into the hannel from the soure in the onfiguration shown in Figure b (strike to flat ground) is ( + r gr )I s (, t)/ =.85I s (, t). [4] The omputed ratios E z_tall /E z_flat and H j _ tall /H j_flat, eah as a funtion of d/h, are shown in Figure. It is lear from Figure that the ratios E z_tall /E z_flat and H j_tall /H j_flat inrease with dereasing r bot. The former ratio exeeds unity when d/h is about.7 and for r bot = r gr = and.7, respetively. As d/h inreases, both the eletri and magneti field ratios approah the far field enhanement fator given by ( r top )(/v + )/( + r gr ) (see equation (B5) in Appendix B), whih is equal to for r top =,r gr =, and v =, and.7 for r top =,r gr =.7, and v =. 5. Influene of Current Refletion at the Top of the Tall Strike Objet [4] Here, we examine the influene of the assumption r top = made in the basi ase (see setion 3), assuming r top =.5 in setion 5. and onsidering three values of r top,,.5, and, in setion Comparison of a Lightning Strike to a Tall Objet With That to Flat Ground [4] In this setion we assume that r top =.5 (orresponding, for example, to Z ob = 3 W and Z h = 9 W), with all other onditions being the same as in the basi ase, presented in setion 3. Note that Janishewskyj et al. [996], from their analysis of five urrent waveforms measured 474 m above ground on the CN tower, inferred r top to vary from.7 to.49, and Fuhs [998], from 3 simultaneous urrent measurements at the top and bottom of the Peissenberg tower, found r top to vary from.39 to.68. [43] Figures and, to be ompared with Figures a and 3a, show waveforms of vertial eletri field and azimuthal magneti field, respetively. The orresponding field waveforms alulated for the same lightning strike to flat ground are the same as those shown in Figures b and 3b, respetively. The fields were alulated in the same manner as in setion 3.. Note that the magnitude of urrent waves injeted into the hannel and into the objet from the soure at the top of the objet (see Figure a) is ( r top )I s (h, t)/ =.75I s (h, t) and that of urrent wave injeted into the hannel from the soure at ground level (see Figure b) is ( + r gr )I s (, t)/ = I s (, t). [44] As seen in Figures and, the influene of urrent waves refleted from the top of the -m-high objet first appears in the field waveforms at t =h/ + p (d + h )/: for example, at. ms atd = 3 m and at.7 ms atd = 3 m. At eah distane, the field reahes its peak before the information about urrent waves refleted from the objet top arrives at the observation point. If the strike objet is higher than m, this information arrives at the observation point even later. Thus the urrent refletion itself at the top of -m-high objet does not influene the peak values of eletri and magneti fields. However, sine the magnitude of urrent waves injeted into the hannel and into the objet from the soure, whih is given by ( r top )I s (h, t)/, inreases with dereasing Figure. Refletion from the objet top ase (r bot = r gr =,r top =.5, v = ). Vertial eletri field waveforms due to a lightning strike to -m-high objet at horizontal distanes of d = 3, 6,, and 3 m from the lightning hannel. of 8

11 respetively. The magneti field ratio is independent of d/h and equal to,.5, and for r top =,.5, and, respetively. The distant eletri field ratios and magneti field ratios are equal to the far field enhanement fator given by ( r top )(/v + )/( + r gr ) (see equation (B5) in Appendix B). Figure. Same as Figure, but for the azimuthal magneti field. Azimuthal magneti field waveforms due to the same lightning to flat ground are the same as those shown in Figure 3b. r top (r top < ), the field magnitudes inrease with dereasing r top. Note that in the ase of r bot =(Z ob = Z gr ), r gr beomes equal to r top (see equations () and (4)), and thereby ( r top )I s / beomes equal to ( + r gr )I s /, regardless of the value of r top. In this speial ase (r bot =), the ratios of E z_tall /E z_flat and H j _ tall /H j_flat are independent of the value of r top. [45] In the ase of r top =.5, the peak of the vertial eletri field due to a lightning strike to the -m-high objet is smaller than that due to the same strike to flat ground at d = 3 and 6 m, but is larger at d = and 3 m (ompare Figures and b). This indiates that the presene of strike objet with r top < serves to attenuate relatively lose eletri fields and enhane relatively distant eletri fields. [46] The peak of the azimuthal magneti field due to a lightning strike to the -m-high objet in the ase of r top =.5 is.5 times larger than that due to the same strike to flat ground at any horizontal distane (ompare Figures and 3b). Reall that when r top =, the former is idential to the latter. This indiates that the presene of strike objet with r top < serves to enhane magneti fields. 5.. Analysis of Distane Dependenes of the Ratios of Eletri and Magneti Fields for Tall-Objet and Flat-Ground Cases [47] The ratios E z_tall /E z_flat and H j _ tall /H j_flat, eah as a funtion of d/h, in the ase of r bot = r gr = (orresponding to Z gr = ) and r top =,.5, and (orresponding to Z h = Z ob, Z h =3Z ob, and Z h Z ob ) are shown in Figure 3. The magnitudes of urrent waves injeted into the hannel and into the objet from the soure at the top of the objet (see Figure a), given by ( r top )I s (h, t)/, are.5i s (h, t),.75i s (h, t), and I s (h, t) for r top =,.5, and, respetively. The magnitude of urrent wave injeted into the hannel from the soure at ground level (see Figure b), given by ( + r gr )I s (, t)/, is I s (, t). [48] It is lear from Figure 3 that the ratios E z_tall /E z_flat and H j _ tall /H j_flat inrease with dereasing r top. The former exeeds when d/h is about and.6 for r top =.5 and, respetively. At larger distanes, the eletri field ratio approahes,.5, and for r top =,.5, and, 6. Influene of Return-Stroke Speed Being Less Than the Speed of Light [49] Here we examine the influene of the assumption v = made in the basi ase (see setion 3), onsidering two values of return-stroke speed, v = and v = Comparison of a Lightning Strike to a Tall Objet With That to Flat Ground [5] In this setion, we assume that v =.5, with all other onditions being the same as in the basi ase, presented in setion 3. Note that typial values of return stroke wave front speed are one third to two thirds of [e.g., Rakov, 4]. Also note that sine any urrent wave is assumed to propagate along the lightning hannel at speed v (see equation (b)), ground-refleted urrent waves are unable to ath up with the return-stroke front, and hene there is no need to deal with refletions at the front. [5] Figure 4a (to be ompared with Figure a) shows vertial eletri field waveforms for a lightning strike to the -m-high objet, and Figure 4b (to be ompared with Figure b) shows the orresponding eletri field waveforms for the same lightning strike to flat ground. Figures 5a and 5b are the same as Figures 4a and 4b, respetively, but for azimuthal magneti fields. The fields were alulated in the same manner as in setion 3.. Note that vertial sales in Figures 4a and 4b are different, while in Figures 5a and 5b they are the same. [5] As seen in Figures 4a and 4b, in the ase of v =.5, the peak of the vertial eletri field due to a lightning strike to the -m-high objet is smaller than that due to the same strike to flat ground for all distanes onsidered (d = 3 to 3 m). The vertial eletri fields within d = 3 m reah their peaks, whih are shown in the upper Figure 3. Illustration of the influene of urrent refletion at the top of the tall strike objet. Shown are ratios E z_tall /E z_flat (solid irles) and H j _ tall /H j_flat (open irles), eah as a funtion of d/h, in the ase of v =, r bot = r gr = (Z gr = ) and r top =,.5, and (Z h = Z ob, Z h =3Z ob, and Z h Z ob ). of 8

12 Figure 4. Less than the speed of light ase (r bot = r gr =, r top =v =.5). Vertial eletri field waveforms (a) due to a lightning strike to -m-high objet and (b) due to the same lightning strike to flat ground, at horizontal distanes of d = 3, 6,, and 3 m from the lightning hannel. H j _ tall /H j_flat for v =.5 are almost the same as those for v =. The abrupt inrease in the ratio E z_tall /E z_flat between d =hand 3h (see Figure 6b) is due to the fat that for d h both E z_tall and E z_flat rise to their peaks in several miroseonds or more while for d >3h the fields rise to their peaks within ms (beause the radiation field omponent beomes dominant at larger distanes). Beyond d = 3h, ratios E z_tall /E z_flat and H j_tall /H j_flat attain the value of.5, whih is equal to the far field enhanement fator given by ( r top )(/v + )/( + r gr )=.5 (see equation (B5) in Appendix B). [56] In the following, we will estimate the influene of strike objet height (see Figure 7) and return-stroke urrent risetime (see Figure 8). As seen in Figure 7, the overall variation of the ratio E z_tall /E z_flat for the ase h = m is quite similar to that for h = m (see Figure 6). When a urrent waveform whose risetime is 3 times longer than that of the urrent waveform proposed by Nui et al. [99] is used, the ratio E z_tall /E z_flat inreases abruptly between d = 3h and 4h, as seen in Figure 8b and in ontrast with Figure 6b. The modified, longer urrent risetime is longer than h/ =.33 ms. As a result, the ratios E z_tall /E z_flat and H j _ tall /H j_flat assymptotially approah.34, whih is smaller than the far field enhanement fator given by ( r top )(/v + )/( + r gr ) =.5. [57] It appears from Figure 6 that at shorter distanes, d < h, equations (8) and () derived assuming r bot = r gr =,r top =, and v = also reasonably represent the ase of r bot = r gr =,r top =, and v =.5. On the other hand, sine the ratios are dependent on the values of r bot, right orner of Figures 4a and 4b, within 7 ms, although the field waveforms are shown only up to 3 ms. [53] As seen in Figures 5a and 5b, the peak of the azimuthal magneti field due to a lightning strike to the -m-high objet is larger than that due to the same strike to flat ground for all distanes onsidered (d = 3 to 3 m). This is in ontrast with the ase of v = (see Figures 3a and 3b) for whih the magneti fields are independent of the presene of the strike objet. The azimuthal magneti field peaks are shown in the upper right orner of Figures 5a and 5b. At d = 3 m, the peak ours at about 4 ms in Figure 5b, although the field waveforms are shown only up to 3 ms. 6.. Analysis of Distane Dependenes of the Ratios of Eletri and Magneti Fields for Tall-Objet and Flat-Ground Cases [54] The ratios E z_tall /E z_flat and H j _ tall /H j_flat, eah as a funtion of d/h, in the ase of r bot = r gr =,r top =, and v =.5 and, are shown in Figures 6a and 6b. The magnitude of urrent waves injeted into the hannel and into the objet from the soure at the top of the strike objet (see Figure a) is.5i s (h, t), and that of urrent wave injeted into the hannel from the soure at ground level (see Figure b) is I s (, t). [55] It is lear from Figure 6a that in the viinity of strike objet (d < 3h), the ratios E z_tall /E z_flat and Figure 5. Same as Figure 4, but for the azimuthal magneti field. of 8