Lightning strikes to tall objects: Currents inferred from far electromagnetic fields versus directly measured currents

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Lightning strikes to tall objects: Currents inferred from far electromagnetic fields versus directly measured currents
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  IX International Symposium on Lightning Protection 26  th -30 th  November 2007 – Foz do Iguaçu, Brazil   LIGHTNING STRIKES TO TALL TOWERS: CURRENTS INFERRED FROM ELECTROMAGNETIC FIELDS VERSUS DIRECTLY MEASURED CURRENTS   Yoshihiro Baba Vladimir A. Rakov Doshisha University   University of Florida ybaba@mail.doshisha.ac.jp   rakov@ece.ufl.edu 1-3 Miyakodani, Tatara, Kyotanabe, Kyoto 610-0321, Japan   Abstract - We have derived far-field-to-current conversion factors for lightning strikes to tall objects for (a) the initial peak current at the object top, (b) the largest peak current at the object top, and (c) the peak current at the object bottom. These far-field-to-current conversion factors are needed for proper interpretation of peak currents reported by lightning detection networks and are each expressed here as the product of (1) the far-magnetic- or electric-field-to-current conversion factor for lightning strikes to flat ground based on the transmission line model, F   H_flat  =2   cd/v  or F   E_flat  =2   0 cd/v , and (2) an appropriate correction factor,  f  , to account for the transient process in the strike object. In the above equations, c  is the speed of light, d   is the distance between the lightning channel and a far observation point, and v  is the wavefront propagation speed of lightning return stroke. The correction factors for the three considered cases are: (a)  f  tall_ini.top = v  /( v + c ), (b)  f  tall_top =[ 1+   bot  (1+                 top )] v  /( v + c ), and (c)  f  tall_bot  =(1+                 bot  )  v  /( v + c ), where bot   and top  are current reflection coefficients at the object bottom and at the object top for upward-propagating waves, respectively. The inverse of the correction factor for the initial peak current at the object top, ( v + c )/  v , differs from the far-field enhancement factor, k  tall  =[(1-   top )/(1+    gr  )]   ( v + c )/  v , in that the latter is influenced by current transmission coefficients, (1-   top ) and (1+    gr  ), at the object top (for downward-moving waves) and at the channel base for the flat-ground strike, respectively, while the former is not. 1 INTRODUCTION Modern lightning detection (or locating) systems provide lightning return stroke peak currents estimated from measured magnetic field peaks. Direct measurements of lightning currents on tall towers are used for testing the validity of field-to-current conversion equations (e.g., Diendorfer et al. , 1998 [1]; Diendorfer and Pichler, 2004 [2]; Lafkovici et al. , 2006 [3]). The field and current  peaks are usually assumed to be proportional to each other, with the proportionality coefficient (field-to-current conversion factor) being determined for strikes to flat ground. Therefore, these conversion factors, in general, are not applicable to strikes to tall (electrically long) objects. Further, in the case of strikes to tall objects, as a result of transient process in the object (e.g., Rakov, 2001 [4]), current waveforms can differ significantly at different heights along the object and can exhibit more than one peak (typically, secondary peak is larger than the initial one). In such cases, the field-to-current conversion factor depends on current measurement location (typically near the top or bottom of the tower) and on whether initial or largest current peak is used. Thus, for calibrating lightning locating systems (LLSs) using current measurements on towers, it is necessary to know appropriate far-field-to-current conversion factors. Lafkovici et al.  (2006) [3] have shown that largest current  peaks measured directly near the top (at a height of about 500 m) of the 553-m CN Tower in Toronto for 21 strokes in 7 presumably upward flashes initiated from the tower in 2005 are considerably (by a factor of two to three) smaller than the corresponding current peaks estimated by the North American Lightning Detection Network (NALDN). Thus, the NALDN, which is calibrated using triggered-lightning strokes terminating on small-height grounded objects (see, for example, Rakov, 2005 [5]), tended to significantly overestimate currents in strokes terminating on the CN Tower. In contrast, Diendorfer et al.  (1998) [1] reported that current peaks measured directly near the top of the 160-m Peissenberg tower in Germany for 44 return strokes and initial-stage pulses, all greater than about 3 kA, in 8 flashes terminating on the tower in 1997-1998 were on average 0.88 of the corresponding current peaks estimated by the Austrian lightning detection system (ALDIS). Also, Diendorfer and Pichler (2004) [2] reported that current peaks measured directly near the top of the 100-m Gaisberg tower in Austria for 334 return strokes terminating on the tower in 2000-2003 were on average 0.98 of the corresponding current peaks estimated by the European lightning detection network (EUCLID). Further, they found a good agreement  between directly measured and EUCLID-reported peak currents for 173 initial-stage pulses (larger than 2 kA). Essentially all Peissenberg and Gaisberg tower flashes were of upward type; that is, they were typically composed of the initial stage and one or more leader/return stroke sequences. In this paper, we will derive far-field-to-current conversion factors for lightning strikes to tall objects for (a) the initial peak current (occurring before the arrival of the first ground reflection at the top of strike object) at the object top, (b) the largest peak current (usually occurring  between the arrival of the first and second ground reflections at the object top) at the object top, and (c) the  peak current at the object bottom. We will use these conversion factors in examining the tower-measured and LLS-inferred currents. Additionally, we will discuss the relationship between the field-to-current conversion    factor for the initial peak at the object top considered here and the so-called far-field enhancement factor introduced  by Baba and Rakov (2005, Eq. B5) [6]. 2 ANALYSIS 2.1 Lightning Currents and Associated Far Magnetic Fields Fig. 1 (a) illustrates the case of lightning strike to a tall grounded object, in which two lossless uniform transmission lines represent the lightning channel (whose characteristic impedance is  Z  ch ) and the tall strike object of height h  (whose characteristic impedance is  Z  ob ). The current reflection coefficient at the bottom of the tall object and the current reflection coefficient at the top of the object for upward-propagating waves are given by  ρ  bot  = (  Z  ob -  Z  gr  )/(  Z  ob +  Z  gr  )   and  ρ  top =(  Z  ob -Z  ch )/(  Z  ob +  Z  ch ), respectively. In the former equation,  Z  gr   is the lumped grounding impedance. Current distributions,  I  (  z’ , t  ), along the tall object (0 ≤  z’ ≤   h ) and along the lightning channel (  z’   ≥  h ) for the configuration shown in Fig. 1 (a) are given by Baba and Rakov (2005) [7] and reproduced below 01 21()22for0(1a) '  ,','  ,'  n nbot top sctopn n nbot top sc h z nh I h t c c I z t h z nh I h t c c z h  ρ ρ  ρ  ρ ρ  ∞= + ⎡ ⎤−⎛ ⎞− −⎜ ⎟⎢ ⎥− ⎝ ⎠⎢ ⎥=⎢ ⎥+⎛ ⎞+ − −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦≤ ≤ ∑   ( ) 11 1()221for(1b) '  ,','  ,'  sctopn nbot top top scn  z h I h t v I z t  z h nh I h t v c z h  ρ  ρ ρ ρ  ∞−= ⎡ ⎤−⎛ ⎞−⎜ ⎟⎢ ⎥− ⎝ ⎠⎢ ⎥=⎢ ⎥−⎛ ⎞+ + − −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦≥ ∑   where  I  sc ( h , t  ) is the lightning short-circuit current, which is defined as the lightning current that would be measured at an ideally grounded object (  Z  gr  =0 or  Z  gr  <<  Z  ch ) of negligible height ( h ≈ 0),  n  is an index representing the successive multiple reflections occurring at the two ends of the strike object. The current distribution,  I  (  z’,t  ), along the lightning channel for the case of the same strike to flat ground [see Fig. 1 (b)] , can be obtained from (1b) by setting h =0 to yield Reference ground  Z   gr   Voltage (  Z  ch ) TL representing channel TL representing tall object Grounding impedance (  Z  ob ) source top   bot    h   V  0 (0, t  ) V  0 ( h , t  ) Reference groundGround surface  gr    V  0 =Z  ch  I  sc   (a) (b) Fig. 1. Transmission line representation of lightning strikes (a) to a tall grounded object of height h  and (b) to flat ground. Adapted from Baba and Rakov (2005) [7]. ( )  ⎟ ⎠ ⎞⎜⎝ ⎛ −+= v zt  I t  z I  scgr  ',021,'  ρ   (2) where  I  sc (0, t  ) is the lightning short-circuit current [same as  I  sc ( h,t  ) in (1a) and (1b) but injected at  z’ =0 instead of  z’=h ], and  ρ  gr    is the current reflection coefficient at the channel base (ground), which is given by  ρ  gr  =(  Z  ch -  Z  gr  )/(  Z  ch +  Z  gr  ). Fig. 2 (a) shows waveforms of current at the top (  z’=h ) and bottom (  z’ =0) of strike object of height h =100 m, for 0510152001234Time [ μ s ]    C  u  r  r  e  n   t   [   k   A   ] 5 h  =100 m,  ρ   top  = -0.5,  ρ   bot  =1Top of strike object ( z'=h  )Bottom of strike object ( z'=  0)  I   bot.peak   I   top.peak   I   top.ini.peak   (a) 0510152001234Time [ μ s ]    C  u  r  r  e  n   t   [   k   A   ] 5Channel base ( z'=  0) h  =0 (strike to flat ground),  ρ   gr  =1  I  bas.peak   (b) Fig. 2. (a) Waveforms of current for a lightning strike to a 100-m object at the top (  z’ =100 m) and bottom (  z’ =0) of the object, and (b) waveform of current at the channel base (  z’ =0) for the same lightning strike to flat ground. Nucci et al.’s  (1990) [8] current waveform was used to specify the lightning short-circuit current,  I  sc ( h , t  ) or  I  sc (0, t  ). 01020304050012345Time [ μ s ]    A  z   i  m  u   t   h  a   l  m  a  g  n  e   t   i  c   f   i  e   l   d   [  m   A   /  m    ] d=  50 kmStrike to tall object h  =100 m,  ρ   top  = -0.5,  ρ   bot  =1, v  =0.5 c  Strike to flat ground:39.2 mA/m17.4 mA/m  ρ   gr  =1, v  =0.5 c  39.2 / 17.4 = 2.25   Fig. 3. Azimuthal magnetic field waveforms calculated on  perfectly conducting ground at distance d  =50 km for a lightning strike to a grounded object of height h =100 m and for the same strike to flat ground. Note that the largest magnetic field peak corresponds to the initial (not the largest) current peak at the object top [see Fig. 2 (a)].     ρ  top =-0.5,  ρ  bot  =1. Fig. 2 (b) shows the waveform of current at the channel base (  z’ =0) for the same lightning strike to flat ground, for  ρ  gr  =1. In these calculations, current waveform proposed by Nucci et al . (1990) [8] was used to specify the lightning short-circuit current,  I  sc . This waveform is characterized by peak of  I  sc.peak  =11 kA and 10-to-90% risetime of  RT  =0.15 μ s, which are thought to be typical for subsequent strokes. In Fig. 2 (a), the initial peak of current at the object top is  I  top.ini.peak  =8.2 kA, the largest peak current at the object top is  I  top.peak  =12 kA, and the peak current at the object bottom is  I  bot.peak  =16 kA. In Fig. 2 (b), the peak current at the channel base is  I  bas.peak  = I  sc.peak  =11 kA. Fig. 3 shows waveforms of azimuthal magnetic field, calculated using the expression for the radiation component of magnetic field due to an infinitesimal dipole (Uman et al.,  1975) [9] that was integrated over the radiating sections of the channel and the strike object, for a lightning strike to a tall grounded object of height h =100 m on perfectly conducting ground at horizontal distance d  =50 km. Also shown in Fig. 3 is the azimuthal magnetic field waveform for the same lightning strike to flat ground. In these field calculations, the return-stroke wavefront speed was set to v =0.5 c . 2.2 Far-Field-to-Current Conversion Factors 2.2.1 Lightning Strikes to Flat Ground The peak of azimuthal magnetic field (radiation-component)  H   flat.peak   on perfectly conducting ground plane at far distance d   from the lightning channel terminating on ground, according to the transmission line (TL) model (Uman et al. , 1975) [9], is given by 111(3)222  . . . gr  flat peak bas peak sc peak   H v I v I cd cd   ρ π π  += =  so that the far-magnetic-field-to-current conversion factor is given by 2(4)  . _ . bas peak  H flat  flat peak   I  cd F  H v π  = =  where  I  bas.peak   is the peak current at the channel base (  z’ =0), and  I  sc.peak   is the peak of the lightning short-circuit current.  Note that the far-magnetic-field-to-current conversion factor employed in modern lightning detection systems (0.185 in the NALDN and 0.23 in the ALDIS and EUCLID, where the magnetic field is expressed in so-called LLP units) has been determined empirically comparing system’s response to triggered-lightning strokes terminating on small-height grounded objects and directly measured channel-base currents for these same strokes. This empirical conversion factor implies a constant value of return-stroke speed. 2.2.2 Lightning Strikes to Tall Objects Since currents measured on tall objects can differ at different heights along the object and can exhibit more than one peak, we will consider three cases: (a) the initial  peak current at the object top, (b) the largest peak current at the object top, and (c) the peak current at the object  bottom. Far-field-to-current conversion factors for lightning strikes to tall objects are defined here as the  product of (i) the far-field-to-current conversion factor for lightning strikes to flat ground, F   H_flat   given by (4), and (ii) an appropriate correction factor,  f  , to account for the transient process in the strike object. The peak of azimuthal magnetic field at far distance d   on  perfectly conducting ground (Uman et al. , 1975) [9] due to current waves propagating from the junction point  between the strike object and vertical lightning channel  both upward and downward at speeds v  and c , respectively, is given by ( )( ) 1211(5)22  . . . . tall peak top ini peak topsc peak   H v c I cd v c I cd  π  ρ π  = +−= +   Note that (5) is valid only for 0 ≤   t    ≤  2 h/c  when  ρ  bot  = 1 and for 0 ≤   t    ≤  h/c  when 0 <  ρ  bot   < 1 (Baba and Rakov, 2005) [7]. Using (4) and (5), one can express the far-magnetic-field-to-current conversion factor for lightning strikes to tall objects for the initial peak current at the object top as 2(6a)where(6b)  . . _ _ . _ . _ . _ . top ini peak  H tall ini top tall ini top H flat tall peak tall ini top  I  cd F f F  H v cv f v c π  = = =+=+  In (6a), F   H_flat   is the far-field-to-current conversion factor for lightning strikes to flat ground given by (4), and  f  tall_ini.top   is the correction factor to account for the transient process in the strike object. Note that  f  tall_ini.top  can be also derived and is the same for the case of far vertical electric field. Clearly,  F   H_tall_ini.top  is smaller than F   H_flat   due to the presence of two waves moving in opposite directions from the top of the strike object. Both F   H_tall_ini.top  and  f  tall_ini.top  increase with increasing v . The maximum value of   f  tall_ini.top  corresponds to v=c  and is equal to 0.5. When v =0.5 c ,  f  tall_ini.top = 0.33 (see Table 1). When the risetime,  RT  , of  I  sc ( h, t  ) is less than 2 h/c , and its overall duration is much longer than 2 h/c , the largest peak of current at the object top (  z’=h ),  I  top.peak  , occurs between t  =2 h/c  and 4 h/c  and can be approximated from (1b) by  I  top.peak    ≈   (1-  ρ  top )/2 × [1+  ρ  bot    (1+  ρ  top )]  I  sc.peak = [1+  ρ  bot (1+  ρ  top )]  I  top.ini.peak  . The peak current at the object bottom (  z’ =0),  I  bot.peak  ,  occurs between t=h/c  and 3 h/c  and can be approximated from (1b) by  I  bot.peak    ≈   (1-  ρ  top )/2 × (1+  ρ  bot  )  I  sc.peak = (1+  ρ  bot  )  I  top.ini.peak  . Therefore, far-magnetic-field-to-current conversion factors for the largest peak current at the object top and for the peak current at the object bottom can be expressed as ( )( ) 11(7a)where11(7b)  . . . _ _ . . _ _ _ bot top top ini peak top peak  H tall toptall peak tall peak tall top H flat tall top bot top  I  I F  H H  f F v f v c  ρ ρ  ρ ρ  ⎡ ⎤+ +⎣ ⎦= ==⎡ ⎤= + +⎣ ⎦ +      ( )( ) 1(8a)where1(8b)  . . . _ _ . . _ _ _ bot peak bot top ini peak  H tall bot tall peak tall peak tall bot H flat tall bot bot   I I F  H H  f F v f v c  ρ  ρ  += === ++   Expressions for all the correction factors considered here are summarized in Table 1. 3 DISCUSSION 3.1 Comparison with Experimental Data In this section, we use the far-field-to-current conversion factors derived in Section 2.2 in examining the tower- measured and LLS-inferred currents. Lafkovici et al . (2006) [3] reported that the peak currents,  I  CNT  , measured directly near the top of the 553-m CN Tower for 21 strokes in 7 presumably upward flashes initiated from the tower in 2005 were a factor of 0.38, on average, smaller than the corresponding NALDN- reported peak currents,  I   NALDN  . This ratio of  I  CNT   and  I   NALDN   was estimated as the inverse of the slope of linear regression equation,  I   NALDN  =2.61  I  CNT   - 1.83, reported by Lafkovici et al. , with the intercept (-1.83) being neglected.  I  CNT   corresponds to the largest peak current at the object top,  I  top.peak  , and the corresponding correction factor for  ρ  top =-0.5,  ρ  bot  =1, and v =0.5 c  is equal to  f  tall_top =0.5 (see Table 1), which is not much different from the observed ratio  I  CNT   /  I   NALDN   =0.38. If     ρ  bot    were changed to 0.8 (Janischewskyj et al ., 1996 [11]),  f  tall_top  would be 0.47.  Note that for the 553-m CN Tower, 2 h/c =3.7 μ s, which is larger than current risetimes (0.18 to 2.0 μ s; A. Hussein,  personal communication) in Lafkovici et al.’s  study. It is worth mentioning that Lafkovici et al . expressed caution regarding their directly measured currents, stating that the CN Tower current measuring system is in need of more accurate calibration. In contrast to CN Tower observations, Diendorfer   et al . (1998) [1] reported that largest current peaks measured directly near the top of the 160-m Peissenberg tower in Germany for 44 return strokes and initial-stage pulses, all Table 1. Correction factors,  f  , to be used in far-field-to-current conversion equations,  I=f  (2 π cd/v )  H   or  I=f  (2 π ε  0     c 2 d/v )  E  , to account for transient process in the strike object. Peak current of interest Correction factor equation Expected value*   Peak current at the channel base for strikes to flat ground,  I  bas.peak    1  flat   f   =  1 Initial peak current at the object top,  I  top.ini.peak     _ . tall ini top v f v c =+  0.33 Largest peak current at the object top,  I  top.peak    ( ) 11  _ tall top bot top v f v c  ρ ρ  ⎡ ⎤= + +⎣ ⎦ + 0.5 Peak current at the object bottom,  I  bot.peak    ( ) 1  _ tall bot bot  v f v c  ρ  = ++  0.67  f  tall_ini.top ,  f  tall_top , and  f  tall_bot   are valid only for  RT    ≤  2 h/c  when  ρ  bot    = 1 and for  RT    ≤   h/c when 0 <  ρ  bot   < 1. * Expected values are calculated for  ρ  top =-0.5,  ρ  bot  =1, and v =0.5 c  (Rakov, 2001 [4], 2007 [10]). greater than about 3 kA, in 8 flashes that terminated on the tower in 1997-1998 were on average 0.88 of the corresponding current peaks estimated by the ALDIS. Further, Diendorfer and Pichler (2004) [2] reported that the current peaks measured directly near the top of the 100-m Gaisberg tower in Austria for 334 strokes that terminated on the tower in 2000-2003 were, on average, 0.98 of the corresponding current peaks estimated by the EUCLID. We now discuss possible reasons why the flat-ground conversion factor appears to be applicable to Peissenberg and Gaisberg tower strikes. For strikes to a 160- or 100-m tower, if current risetimes are larger than 1.1 or 0.67      μ s, respectively, the  RT   < 2 h/c  condition (for  ρ  bot  =1) assumed in deriving equations for far-field-to-current conversion factors is not satisfied. Figs. 4 (a) and (b) show correction factors to account for transient process in the strike object as a function of  RT/  (2 h/c ) for  ρ  top =-0.5,  ρ  bot  =1 (Fig. 4a) and 0.7 (Fig. 4b), and v =0.5 c . Values of correction factors were calculated from peak currents at the top and bottom of a 100-m-high strike object and associated azimuthal magnetic field peaks at distance d= 50 km. A ramp-like current waveform was used to specify the lightning short-circuit current. Note that, since the initial peak current at the object top is identical to the largest peak current for  RT   > 2 h/c , the correction factor for the initial peak current at the object top,  f  tall_ini.top  is shown only for 0 ≤    RT    ≤ 2 h/c . In the case of  RT   > 2 h/c ,  f  tall_top  and  f  tall_bot   increase relative to the values given in Table 1 (Baba and Rakov, 2005) [6]. For example, the correction factor for the largest peak at the object top in Fig. 4 (a) is 0.79 for  RT   =4 h/c  and 0.90 and for  RT  =12 h/c , compared to 0.5 for  RT   < 2 h/c . For the case of  RT   >>2 h/c , the tower acts as a lumped circuit and each of the correction factors is equal to unity. Note that  f  tall_ini.top ,  f  tall_top , and  f  tall_bot   for  RT    ≤  2 h/c  in Fig. 4 (a) (0.33, 0.50, and 0.67, respectively) and   RT    ≤   h/c  in Fig. 4 (b) (0.33, 0.45, and 0.56, respectively) are equal to theoretically derived values given by (6b), (7b), and (8b), respectively. We estimated from Fig. 4 (a) the expected equivalent  RTs  for the 160-m Peissenberg tower and the 100-m Gaisberg tower (  f  tall_top =0.88 and 0.98, respectively) to be about 5(2 h/c )=5.5  μ s and >8(2 h/c )=5.3 μ s, respectively. Interestingly, Diendorfer and Pichler (2004) [2] found that essentially no correction factor was needed for either return strokes or initial-stage pulses. For the latter, the geometric mean 10-to-90%  RT   was estimated by Miki et al . (2005) [12] to be 61 μ s for the Peissenberg tower and 110 μ s for the Gaisberg tower (both>>2 h/c ), although Diendorfer and Pichler and Diendorfer et al . considered only larger (>2 kA and >3 kA, respectively) pulses, as opposed to Miki et al . who included in their analysis all measurable pulses. Another factor possibly contributing to the discrepancy  between the CN-Tower data on the one hand and Peissenberg and Gaisberg tower data on the other hand is the propagation model used in the NALDN, but not in the ALDIS or EUCLID. This model partially compensates the measured field peak for its attenuation due to     propagation over finitely-conducting ground. Uncompensated field propagation effects should lead to an overestimation of ratio of measured and inferred currents as well as of the expected  RTs . When only sensors corresponding to relatively small propagation effects are considered, ratios of measured and inferred currents for the Peissenberg and Gaisberg towers become smaller, 0.71 (Diendorfer et al. , 1998 [1]; Fig. 3) and 0.67 (Schulz and Diendorfer, 2004 [13]; Fig. 3B), respectively, and the corresponding expected equivalent  RTs  become 1.7(2 h/c )=1.8  μ s and 1.5(2 h/c )=1.0  μ s. These latter values of  RT   appear to be reasonable. 0.00.20.40.60.81.002468 RT   /2 h  / c      C  o  r  r  e  c   t   i  o  n   f  a  c   t  o  r  s  ρ   top  =-0.5,  ρ   bot  =1, v  =0.5 c RT _ 2 h   /  c  0.5 f  tall_top  0.67 f  tall_bot  f  tall_ini.top  0.33  (a) 0.00.20.40.60.81.002468 RT   /(2 h   /  c  )    C  o  r  r  e  c   t   i  o  n   f  a  c   t  o  r  s  ρ   top  =-0.5,  ρ     bot  =0.7, v  =0.5 c RT _h   /  c  0.45 f  tall_top  0.56 f  tall_bot  f  tall_ini.top  0.33  (b) Fig. 4. Correction factors,  f  , in far-field-to-current conversion equations,  I  =  f (2 π cd  / v )  H   or  I  =  f (2 πε 0 cd  / v )  E  , to account for transient process in the strike object as a function of  RT  /(2 h / c ). Values of  f   were calculated from peak currents at the top and  bottom of a 100-m-high strike object and associated azimuthal magnetic field peaks at distance d  =50 km for (a)  ρ  top =-0.5,  ρ  bot  =1, and v =0.5 c , and (b)  ρ  top =-0.5,  ρ  bot  =0.7, and v =0.5 c . A ramp-like current waveform was used to specify the lightning short-circuit current. When  RT   > 2 h / c , the initial peak coincides with the largest peak, so that  f  tall_ini.top =f  tall_top . 3.2 Relation Between the Far-Field-to-Current Conversion Factor and the Far-Field Enhancement Factor In this section, we discuss the relation between the far-field-to-current conversion factor derived in this paper for the initial peak at the object top and the so-called far-field enhancement factor due to the presence of tall strike object.  Baba and  Rakov  (2005) [6] have defined the far-field enhancement factor due to the presence of tall strike object as the ratio in magnitudes of far vertical electric or azimuthal magnetic field on perfectly conducting ground due to lightning strikes to a tall object and that due to the same strike to flat ground. In doing so, they used the TL representation of both the lightning channel and strike object shown in Fig. 1. The same lightning strike means that the short-circuit current and channel impedance, as well as the total charge transfer to ground are the same regardless of the presence of strike object. Under these conditions, the initial peak of lightning current for the strike-object case is (1-  ρ  top )  I  sc.peak  /2, and the peak of lightning current for the flat-ground case is (1+  ρ  gr  )  I  sc.peak  /2. Note that both the far-field-to-current conversion factors derived in this paper and the far-field enhancement factor derived by Baba and Rakov (2005) [6] are valid only for  RT    ≤  2 h/c  when  ρ  bot    = 1 and for  RT    ≤   h/c when 0 <  ρ  bot   < 1. The far-field enhancement factor of Baba and Rakov is reproduced below ( ) ( ) ( ) ( ) 11 top toptallgr gr  v c v ck vv  ρ  τ τ  ρ  − + += =+  (9) From comparison of (6b) and (9),  _ . toptall ini topgr tall v f v c k  τ τ  = =+  (10)   so that 2  _ _ . top H tall ini topgr tall cd F k v τ  π τ  =  (11) Clearly, k  tall  is influenced by current transmission coefficients, τ  top =(1-  ρ  top ) and τ  gr  =(1+  ρ  gr  ), at the object top (for downward-moving waves) and at the channel base for the flat-ground strike, respectively, while  f  tall_ini.top  is not. Pavanello (2007) [14] has derived the far-field enhancement factor (which he calls the “tower enhancement factor”) using a distributed-shunt current source representation of lightning strikes to a tall object and flat ground (Rachidi  et al. , 2002 [15]), as opposed to the lumped-source representation considered above. Pavanello’s far-field enhancement factor (also given by  Baba and Rakov , 2005, Eq. B8 [6]) is reproduced below ( ) 111  _ top toptall Pavgr  c cv vk cv  ρ ρ  ρ  ⎛ ⎞− + −⎜ ⎟⎝ ⎠=+  (12) In the latter model, shunt current sources distributed along the lightning channel are activated progressively when the return stroke wavefront propagating upward at speed v  arrives at their altitudes. The resultant partial current waves are assumed to propagate downward at speed c , and the upward waves reflected from ground or the object top are assumed to propagate along the channel also at speed c . The upward-moving wave was assumed to be absorbed at the front, but the field contribution from the front in deriving (12) was ignored. If, in order to avoid upward-wave absorbtion at the front, one changes the speed of reflected current wave propagating upward
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