Why do some lightning return stroke models not reproduce the far-field zero crossing?

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2008jd011547, 2009 Why do some lightning return stroke models not reproduce the far-field zero crossing? A. Shoory, 1,2 F. Rachidi, 1 M. Rubinstein, 3 R. Moini, 2 and S. H. H. Sadeghi 2 Received 27 November 2008; revised 18 May 2009; accepted 27 May 2009; published 27 August [1] In this paper, we investigate the reasons why some return stroke models do not reproduce one of the characteristic features of the electromagnetic fields radiated by lightning, namely the far-field inversion of polarity with a zero crossing occurring in the tens of microseconds range. This study leads to physical insights into the behavior of the return stroke attenuation and speed as a function of height. Making reference to the properties of time domain far fields, we show that any one of the existing lightning return stroke channel models, including the electromagnetic models, sometimes known as Antenna Theory models, and the engineering models, should be able to predict a zero crossing in the far field as long as the duration of the return stroke current and the height of the channel are finite. However, the present versions of most of the available engineering and electromagnetic models predict the zero crossing to occur at times that fall well beyond those observed experimentally. We identify three mechanisms responsible for the time of occurrence of the reversal of polarity in the far fields: the current attenuation along the channel, the duration of the return stroke current, and the return stroke speed. The analysis of the MTLE and MTLL engineering models shows that the higher the attenuation of the current along the channel, the earlier the polarity reversal of the vertical electric field. Also, for a given value of the attenuation factor, higher propagation speeds correspond to earlier polarity reversal times. For the TCS model, in which the only adjustable parameter is the return-stroke speed, we show that the far-field zero crossing occurs considerably later than the values predicted by both the MTLE and the MTLL models. This is shown to be essentially due to the fact that the decrease of the current wave along the channel according to the TCS model is less pronounced than the current attenuation predicted by the MTLE and MTLL models. Furthermore, it is shown that, in the electromagnetic models, a uniformly distributed resistance along the channel does not lead to a zero crossing in the tens of microseconds range even for large resistance values. This is due to the significant increase of the current pulse duration, as it propagates along the channel as a result of the dispersion effect. The correct zero crossing time, however, can indeed be successfully reproduced if a nonlinear channel resistance is used, which prevents the significant increase of the current pulse duration as it propagates up the channel. Finally, based on the predictions of the MTLE, MTLL, and TCS models, it is shown that the zero crossing time decreases as the observation point moves farther away from the lightning channel, with the amount of variation of less than 10%. Citation: Shoory, A., F. Rachidi, M. Rubinstein, R. Moini, and S. H. H. Sadeghi (2009), Why do some lightning return stroke models not reproduce the far-field zero crossing?, J. Geophys. Res., 114,, doi: /2008jd Introduction [2] The electric and magnetic fields radiated from cloudto-ground lightning return strokes have been characterized 1 EPFL-STI-IEL, EMC Group, Swiss Federal Institute of Technology, Lausanne, Switzerland. 2 Electromagnetic Research Laboratory, Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran. 3 Institute for Information and Communication Technologies, University of Applied Sciences of Western Switzerland, Yverdon-les-bains, Switzerland. Copyright 2009 by the American Geophysical Union /09/2008JD as a function of the distance to the observation point for distances of 1 km and above [e.g., Lin et al., 1979; Rakov and Uman, 1998], and for distances within a few hundred meters [Rakov et al., 1998; Rubinstein et al., 1995]. The characteristic features identified in these studies include the flattening of the E-field after a few microseconds at very close range (100 m or less), the initial peak followed by a ramp increase for the E-field at distance ranges of 1 to 5 km, and, for the magnetic field, an initial peak followed by a hump at distances of 1 to 5 km. [3] Another characteristic feature of the return-stroke fields which will be dealt with in the present paper is the 1of9

2 bipolarity of both electric and magnetic fields at distances of about 50 km or beyond, with a zero crossing occurring in the tens of microseconds range [note that A. Pavlick et al. (Characteristics of distant lightning electric fields, paper presented at International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), Naples, Italy, 2002) reported that some 4% of distant electromagnetic field waveforms do not exhibit a pronounced zero crossing within 400 microseconds of the initial peak]. Typically, the zero crossing time is about 50 microseconds for the first return-stroke and about 35 microseconds for subsequent return-strokes [Rakov and Uman, 2003, chapter 4]. [4] The above characteristic features have been used to test the ability of return-stroke models to reproduce the observed electromagnetic fields (for engineering models, see, for instance, Gomes and Cooray [2000], Nucci et al. [1990], Rakov and Uman [1998], Thottappillil et al. [1997], and Thottappillil and Uman [1993] and for electromagnetic models, sometimes known as Antenna Theory or AT models, see Baba and Rakov [2008]). The procedure in these studies was to assume a typical return-stroke current at the channel base, to compute the electromagnetic fields adopting a given model, and to examine how well the model reproduces the significant features of the observed fields. [5] In particular, it has been shown that, while some of the engineering models, namely the Bruce-Golde (BG) model [Bruce and Golde, 1941], the Transmission Line (TL) model [Uman and McLain, 1969], the Traveling Current Source (TCS) model (F. Heidler, Traveling current source model for LEMP calculation, paper presented at 6th Symposium and Technical Exhibition on Electromagnetic Compatibility, Zurich, Switzerland, 1985) and the Diendorfer-Uman (DU) model [Diendorfer and Uman, 1990] are not able to reproduce the correct far-field zero crossing, others, such as the Modified Transmission Line models (MTLE [Rachidi and Nucci, 1990; C. A. Nucci et al., On lightning return stroke models for LEMP calculations, paper presented at 19th International Conference on Lightning Protection, ICLP, Graz, Austria, May, 1988] and MTLL [V. A. Rakov and A. A. Dulzon, A modified transmission line model for lightning return stroke field calculations, paper presented at International Zurich Symposium on Electromagnetic Compatibility, Zurich, Switzerland, 1991]) are successful in reproducing this feature [Nucci et al., 1990; Thottappillil et al., 1997]. [6] On the other hand, electromagnetic or antenna theory (AT) models in their original form appear to be unsuccessful at reproducing the far-field zero crossing [e.g., Moini et al., 2000] within the experimentally observed range although, recently, the feature was indeed reproduced through the introduction of a nonlinear distributed resistance along the channel (R. Alaee et al., A non-linear antenna theory model of lightning return stroke channel, paper presented at 18th International Zurich Symposium on EMC, Munich, Germany, 2007). [7] Why do some return-stroke models reproduce the far field zero crossing while others do not? This paper attempts to answer this question. [8] The paper is organized as follows. In section 2, we discuss the mathematical foundation requiring the far field to experience a polarity reversal. Sections 3 and 4 deal, respectively, with the ability of engineering and AT models to reproduce the far field zero crossing. Finally, a summary and conclusions are given in section Theoretical Background [9] The characteristics of far fields in the time domain have been recently investigated thoroughly by Yaghjian and Hansen [1996], who derived several universal properties for them. Yaghjian and Hansen showed, in particular, that the time integral of the far electric and magnetic fields radiated by a source of finite spatial extent equals zero if the source turns on and off in a finite time interval. This implies that the far electric and magnetic fields radiated by such a source should experience at least one polarity reversal or zero crossing to satisfy a vanishing time integral. Thus, if a return stroke model uses a finite-duration channel base current and a finite-height channel, the far fields will necessarily exhibit at least one zero crossing. [10] Both the engineering and electromagnetic returnstroke models represent the lightning channel as either a single source or a spatially finite array of sources (This is the case for both versions of the engineering models, namely the lumped source models and distributed current source models [Rachidi and Nucci, 1990; Cooray, 2002; Maslowski and Rakov, 2007].) which are turned on only for a period of time given by the duration of the current, which is of the order of 100 microseconds. The far fields integrated over all time can be nonzero only when the current approaches a static value as time tends to infinity since, in this case, the source does not satisfy Yaghjian and Hansen s [1996] source turnoff condition. [11] The above considerations suggest that, in principle, any one of the existing models of the lightning return stroke channel, including the electromagnetic and engineering models, should predict a zero crossing in the far field. Note that one fundamental phenomenon responsible for the far field zero crossing is the boundary conditions at the top of the finite-length channel. Indeed, in electromagnetic models, the negative reflection appearing when the current reaches the top of the channel would result in an abrupt inversion of polarity in the far field, as confirmed by numerical simulations. Correspondingly, in engineering models, it is generally assumed that the current pulse vanishes abruptly when reaching top of the channel (perfectly matched termination). If the current reaches the top of the channel with a non negligible amplitude, such as in the TL model, the abrupt vanishing of the current at the channel top results in the so-called mirror image effect [Uman et al., 1975] which, in turn, will result in an abrupt reversal of polarity of the far field. [12] However, in order to reproduce the smooth inversion of polarity observed in experimental data and with zerocrossing times as low as 30 microseconds for subsequent strokes, other mechanisms must be present to force the far field reversal of polarity prior to the arrival of the current pulse at the channel top. We hypothesize here that these mechanisms are (1) the current decay rate and its dispersion as it propagates along the channel, (2) the duration of the channel base current, and (3) the speed of the return stroke as it travels up the channel. In the next sections, we will study these factors. 2of9

3 Figure 1. The lightning return-stroke channel-base current adopted from Nucci et al. [1990]. [13] It is worth noting that other mechanisms might also play a role in the far field reversal of polarity, namely the channel geometry, as discussed by Cooray et al. [2008]. [14] For each of the considered models, we will study the variation of the zero-crossing time as a function of the attenuation, if this parameter is adjustable in that particular model, of the return stroke speed, and of the duration of the channel base current. 3. Engineering Models [15] Analyses of various engineering models namely the BG, DU, TCS, TL, MTLL, and MTLE have shown that a zero crossing in the distant electric field compatible with experimental measurements can only be reproduced by the MTLE and the MTLL models [Nucci et al., 1990; Rakov and Uman, 1998]. Note that the electromagnetic field produced using the other models also exhibits a zero crossing, as a result of the Yaghjian and Hansen [1996] theorem discussed above. However, this zero crossing is either a result of the mirror image effect (for the TL model), or it occurs well beyond the typical time windows observed in experimental waveforms. [16] To the best of our knowledge, no direct algorithm relating the parameters of the engineering models with the radiation fields zero crossing time is available. In this study, we will adopt an iterative approach to obtain curves of the zero crossing times as a function of the parameters of three engineering models, namely, TCS (Heidler, presented paper, 1985), MTLL (Rakov and Dulzon, presented paper, 1991) and MTLE [Rachidi and Nucci, 1990; Nucci et al., presented paper, 1988]. To this end, we will use the exact formulation of the models to calculate, numerically, the zero-crossing times. Table 1. Spatial-Temporal Relationships of the Current Along the Channel for the TCS, MTLL, and MTLE Models Model Expression TCS i(z 0 ið0; t þ z, t) = =cþ z 0 vt 0 z 0 > vt MTLL i(z 0 ð1 z, t) = =HÞið0; t z 0 =vþ z 0 vt 0 z 0 > vt MTLE i(z 0 expð z, t) = =lþið0; t z 0 =vþ z 0 vt 0 z 0 > vt Figure 2. Current wave attenuation along the channel for (a) the MTLE model (l = 2 km), (b) the MTLL model (H = 7.5 km), and (c) the TCS model. [17] To investigate the capabilities of the engineering models in predicting the zero crossing of the far radiated electric field, we will consider a straight, vertical lightning channel with the attachment point at ground level. We will use a channel with a height of H = 9.5 km (except, as we will see, for the TCS model) and with a channel-base current initially equal to the one used by Nucci et al. [1990], shown in Figure 1. We will use the general expressions for the electromagnetic fields [Uman et al., 1975], assuming the ground as a perfectly conducting half-space. [18] For each model, the spatial-temporal relations of the current at a time t along the channel and at a height of z 0 as a function of channel base current are shown in Table 1, where H is the channel height, c is the speed of light in free space, v is the current wave propagation speed, and l is the decay height constant for the MTLE model. The 3of9

4 Figure 3. Variation of the zero crossing time versus attenuation constant l in the MTLE model for different current wave propagation speeds. corresponding current distributions at selected heights along the channel are shown in Figures 2a 2c. For Figure 2, the channel height was assumed to be H = 7.5 km and the propagation speed was assumed to be v = m/s. The decay height constant for the MTLE model was set to l = 2 km. [19] Figure 3 shows, for the MTLE model, a plot of the zero-crossing time of the electric field at a distance of r = 100 km, as a function of the exponential decay height constant l. The curves in the plot were obtained for three different values of the current wave propagation speed, i.e., v = m/s, v = m/s, and v = m/s. As can be seen from Figure 2, increasing the value of l results in an increase in the time of zero crossing. In other words, the higher the attenuation of the current along the channel, the earlier the polarity reversal of the vertical electric field. Also, for a given value of the attenuation factor, higher propagation speeds correspond to earlier occurrence of the field zero crossing as the more attenuated currents contribute earlier to the far radiated electric field. This result is consistent with experimental observations which indicate, on the one hand, greater average zero crossing times for first strokes compared to subsequent strokes [Cooray and Lundquist, 1985; Lin et al., 1979] and, on the other hand, smaller average return-stroke speeds for first strokes compared to subsequent strokes [e.g., Idone and Orville, 1982; Mach and Rust, 1989]. [20] Regarding the MTLL model, only the channel height and the propagation speed are adjustable parameters. Since Figure 4. Variation of the zero crossing time versus channel height H in the MTLL model for different current wave propagation speeds. Figure 5. Variation of the zero crossing time versus current wave propagation speed in the TCS model (channel height of H = 17.5 km is assumed in this case). the current in this model decays linearly from its maximum at the channel base to the zero value at the channel top, the height of the channel is also a measure of the current wave attenuation along the channel. In this case the plot of the zero crossing time versus channel height for the same values of the propagation speeds as for the MTLE model is shown in Figure 4. It can be seen that, for a given propagation speed, earlier zero crossing times correspond to shorter channel heights. In fact, shorter channels impose more attenuation on the current wave, resulting in earlier zero crossing times. Also, the effect of the propagation speed appears to be qualitatively the same as that in the MTLE model. [21] For the third and last engineering model considered here, the TCS model, Figure 5 shows the zero crossing time as a function of the propagation speed. Note that the returnstroke speed is the only adjustable parameter of this model. Due to very late zero crossing times in this case, an unrealistic channel height of H = 17.5 km was adopted to produce the plots in Figure 5. It can be seen that the far field zero crossing for the TCS model occurs considerably later than the values predicted by both the MTLE and the MTLL models. This is essentially due to the fact that the decrease of the current wave along the channel is less pronounced in the TCS model than the current attenuation according to the MTLE and MTLL models as can be inferred from the curves of Figures 2a 2c. [22] Note that, assuming that the zero crossing time occurs prior to the arrival of the current at the top of the channel, the results presented in Figures 3 and 5 are not affected by the value for the channel height and the boundary conditions at the top of the channel. [23] The variation of the zero crossing time as a function of the distance of the observation point is also shown in Figure 6 for the MTLE, the MTLL and the TCS models. It can be seen that farther observation points correspond to earlier zero crossing times. This effect can, however, be compensated by the propagation effects above a lossy ground, which results in the broadening of the field waveforms and therefore an increase of the zero crossing time [Rakov and Uman, 2003]. [24] The above analysis supports our hypothesis that current attenuation along the channel influences the far field zero crossing time [see also Rakov and Dulzon, presented paper, 1991, Figures 1 and 4]. This statement is correct for a 4of9

5 The pulse duration is defined here as the time window over which the current amplitude is greater than 5% of its peak. An interesting feature is that for very short pulse durations of the order of 50 ms the TCS model predicts earlier zero crossing times compared with the MTLL model. Also, the sensitivity of the zero crossing time to the pulse duration in the TCS model is more pronounced than those observed in the MTLE and MTLL models. It is worth noting that other studies have used different waveforms for the lightning return stroke channel-base current [e.g., Rachidi et al., 2001; G. Diendorfer, Effect of an elevated strike object on the lightning electromagnetic fields, paper presented at 9th International Symposium on Electromagnetic Compatibility, Zurich, Switzerland, March, 1991] which are based on Figure 6. Variation of the zero crossing time versus the distance of the observation point from the channel base for (a) the MTLE model with l = 2 km, (b) the MTLL model with H = 7.5 km, and (c) the TCS model with H = 17.5 km. v = m/s is assumed for the three models. given channel-base current waveshape, and assuming a constant return-stroke speed along the channel. It is important to note that, regardless of the return-stroke model, a larger current pulse duration corresponds to a larger value of the zero crossing time. This is reflected in Figure 7, where the zero crossing time is plotted for different pulse durations for the three models. The obtained results are in agreement with the conclusions of R. Thottappillil et al. (Influence of channel base current and varying return stroke speed on the calculated fields of three important return stroke models, paper presented at International Conference on Lightning and Static Electricity, Cocoa Beach, Florida, USA, April, 1991) based on the use of two specific current pulse waveforms. The parameters used are the same as in Figure 6. Figure 7. Variation of the zero crossing time versus channel-base current pulse duration for (a) the MTLE with l = 2 km, (b) the MTLL model with H = 7.5 km, and (c) the TCS model with H = 17.5 km. v = m/s is assumed for the three models. 5of9

6 assuming an exponentially decreasing speed with height, with an initial value of m/s and an exponential decay constant of 2 km. They found that the zero crossing time of the MTLE model is not affected much by the variable return-stroke speed. However, it was found that the zero crossing times for the TCS and DU models occurred earlier by about 15 ms. 4. Antenna Theory (AT) Models [26] In electromagnetic or AT models, the lightning channel is modeled as a vertical wire antenna and the lightning return-stroke current is injected by a voltage or Figure 8. Current waves along the channel at different heights, namely 1, 2, 3, and 4 km for the antenna theory model with L D =6mH/m and (a) R D =1W/m, (b) R D =2W/m, and (c) R D =3W/m. measured current waveforms obtained using either natural or triggered lightning. These waveforms are characterized by different pulse durations and therefore will result in different zero crossing times. [25] Optical observations have shown that the returnstroke speed usually decreases with height [Idone and Orville, 1982]. Measurements carried out by Mach and Rust [Mach and Rust, 1989] on short channels (less than 500 m) show a mean value of m/s, whereas for channel heights exceeding 500 m ( long channel ), the mean value drops to about m/s. Thottappillil et al. (presented paper, 1991) presented distant electric field calculations (using the MTLE, TCS and DU models) Figure 9. Vertical electric field at a distance of r = 100 km from the lightning channel at ground level for the antenna theory model with L D =6mH/m and (a) R D =1W/m, (b) R D = 2 W/m, and (c) R D =3W/m. 6of9

7 Figure 10. Current waves along the channel at different heights of namely 1, 2, 3, and 4 km adopted from Alaee et al. (presented paper, 2007): (a) the original AT model and (b) the nonlinear AT model. current source at the base of the channel. The current distribution along the channel is found, e.g., by solving an electric field integral equation [e.g., Moini et al., 2000]. In order to control the current attenuation rate along the channel and the propagation speed, distributed resistances and inductances are usually included in the model [e.g., Baba and Ishii, 2003; Bonyadi-Ram et al., 2008; Moini et al., 2000]. [27] Figures 8a 8c presents the spatial and temporal distribution of the current along the channel predicted by the AT model starting from the channel-base current of Figure 1 and considering three different values for the distributed resistance, namely 1, 2, and 3 Ohms/m. The results for the AT model are obtained using the thin wire time domain code (TWTD [Miller et al., 1973]). In this model we used a distributed inductance of L D =6mH/m according to Baba and Ishii [2003] to obtain a propagation speed of about v = m/s. Figures 9a 9c present the corresponding electric fields computed at a distance of 100 km. Note that in the AT model, the effective attenuation is a function of both the distributed resistance and distributed inductance along the channel [Bonyadi-Ram et al., 2008]. In fact, both parameters are responsible for the propagation speed and current wave attenuation. [28] The obtained results reveal that even large distributed resistance values of up to 10 Ohms/m along the channel do not produce a zero crossing in the tens of microseconds range. To explain this result, it is worth noting that the propagation of the current along the channel as a result of the distributed resistances and inductances is very different in nature from the propagation of the current in the engineering models. In the engineering models, the current at a given height is simply obtained in terms of an attenuated and delayed function of the channel-base current, without suffering from the accumulation of the attenuation and dispersion caused by the propagation [Rakov and Uman, 1998]. Note that in the engineering models (other than the TL model), the current attenuation along the channel is not viewed as being due to the losses, but rather to the neutralization of the charges initially stored in the leader channel and in the corona sheath around it [e.g., Maslowski and Rakov, 2006; Nucci et al., 1990]. [29] In the AT models, however, the attenuation and dispersion of the current at a given height z are the result of cumulative effects of the distributed resistances and inductances from 0 to z. The dispersion of the current results in a significant increase of the current pulse duration as it propagates along the channel, as can be seen in Figure 8. It is this increase of the current pulse duration that prevents the far field from exhibiting a zero crossing in the tens of microseconds range. [30] Recently, the effect of a nonlinear channel resistance on the resulting electromagnetic field was discussed by Alaee et al. (presented paper, 2007) and De Conti et al. [2008]. It was shown, in particular, that the inclusion of a time-varying channel resistance might reproduce the far-field polarity reversal in the ms range [De Conti et al., 2008; Alaee et al., presented paper, 2007]. Figure 10 presents the current profiles computed using the AT model with constant resistance (Figure 10a) and the recent model by Alaee et al. (Figure 10b) in which a time-varying channel resistance is included (Alaee et al., presented paper, 2007). In the model of Alaee et al., the current propagation speed was controlled using a distributed inductance of H/m. They used a time-variable nonlinear resistance decaying with the temporal-radial expansion of the channel as 1/psr 2 (t) where the channel radius is r(t) = i(t) 1/3 t 1/2 m[braginskii, 1958]. The parameter s is the channel conductivity and is assumed to be ohm 1 m 1. [31] From Figure 10, it can be seen that the inclusion of a time-varying channel resistance limits, to some extent, the Figure 11. Vertical electric field at a distance of r = 100 km from the lightning channel at ground level adopted from Alaee et al. (presented paper, 2007) for the original AT model and the nonlinear AT model. 7of9

8 dispersion effect and prevents the increase of the current pulse duration as it propagates upward. As a result, the corresponding electric field at 100 km, shown in Figure 11, exhibits an inversion of polarity at about 45 ms. 5. Summary and Conclusions [32] We analyzed in this paper one of the characteristic features of the electromagnetic fields radiated by lightning, namely the far-field inversion of polarity with a zero crossing occurring in the tens of microseconds range. [33] Based on the properties of time domain far fields derived by Yaghjian and Hansen, we showed that, in principle, any model of the lightning return stroke channel, including the electromagnetic or AT models and engineering models, should predict a zero crossing in the far field. However, most of the engineering and AT models fail to reproduce the inversion of polarity at the experimentally observed zero crossing times, which are as low as 30 microseconds for subsequent strokes. [34] We showed that at least three factors influence the zero-crossing of the distant electric and magnetic fields from lightning return strokes: (1) the rate of attenuation and the dispersion of the return stroke current as it propagates along the channel, (2) the duration of the channel base current pulse, and (3) the speed of the return stroke as it travels up the channel. [35] An analysis of the MTLE and MTLL engineering models showed that the higher the attenuation of the current along the channel, the earlier the polarity reversal of the vertical electric field. Also, for a given value of the attenuation factor, higher propagation speeds correspond to earlier polarity reversal times. This result is consistent with experimental observations which indicate, on the one hand, larger average zero crossing times for first strokes compared to subsequent strokes and, on the other hand, smaller average return-stroke speeds for first strokes compared to subsequent strokes. [36] For the TCS model, in which the only adjustable parameter is the return-stroke speed, we showed that the far field zero crossing occurs considerably later than for both the MTLE and the MTLL models. This was found to be essentially due to the fact that the decrease of the current wave along the channel according to the TCS model is less pronounced than the current attenuation according to MTLE and MTLL models. [37] Finally, we showed that, in the AT models, a zero crossing in the tens of microsecond range cannot be obtained using a uniformly distributed resistance even if large resistance values are used. This is essentially due to the significant increase of the current pulse duration due to dispersion as it propagates along the channel. A zero crossing at experimentally observed times, however, can be successfully reproduced if a nonlinear channel resistance is used, which prevents the significant increase of the current pulse duration as it propagates upward the channel. [38] Work is in progress to derive analytical formulas for the zero crossing times and the ratio of the opposite polarity overshoot to the initial field peak. [39] Acknowledgments. This work is partially supported by Swiss National Science Foundation (Project ), State Secretariat for Education and Research (Swiss Scholarship for the University Studies), and European COST Action P18 The Physics of Lightning Flash and Its Effects. We thank Yoshihiro Baba, Fridolin Heidler, Vladimir A. Rakov, and Martin A. Uman, as well as three anonymous reviewers, for their valuable comments. References Baba, Y., and M. Ishii (2003), Characteristics of electromagnetic returnstroke models, IEEE Trans. Electromagn. Compat., 45, Baba, Y., and V. A. Rakov (2008), Applications of electromagnetic models of the lightning return stroke, IEEE Trans. Power Delivery, 23, Bonyadi-Ram, S., et al. (2008), On representation of lightning return stroke as a lossy monopole antenna with inductive loading, IEEE Trans. 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9 Uman, M. A., et al. (1975), The electromagnetic radiation from a finite antenna, Am. J. Phys., 43, Yaghjian, A. D., and T. B. Hansen (1996), Time domain far fields, J. Appl. Phys., 79, R. Moini and S. H. H. Sadeghi, Electromagnetic Research Laboratory, Department of Electrical Engineering, Amirkabir University of Technology, 424 Hafez Avenue, Tehran, Iran. F. Rachidi and A. Shoory, EPFL-STI-IEL, EMC Group, Swiss Federal Institute of Technology, EPFL-STI-LRE, Station 11, Lausanne CH-1015, Switzerland. (farhad.rachidi@epfl.ch) M. Rubinstein, Institute for Information and Communication Technologies, University of Applied Sciences of Western Switzerland, Route de Cheseaux 1, CH-1400 Yverdon-les-bains, Switzerland. 9of9