**3. Fourier transform of NCBC function**

Whether a function is given analytically, graphically or numerically, its Fourier transform can be obtained analytically, numerically or using some of the commercial programs. Faster or slower rising/decaying of the function and its waveshape at the end of the covered impulse duration time determine the number *N* of needed points for Fast Fourier Transform.

For NCBC function Fourier and Laplace transforms are calculated analytically. It should be noted that we obtain Fourier transform for any Fourier-transformable function which is zero for *t*<0 if we substitute variable *s* with j2π*f* in its Laplace transform.

The analytical expression for the unilateral Laplace transform of NCBC function is:

$$I(s) = I\_m t\_m \left[ \frac{\exp(a)}{\left(a + st\_m\right)^{a+1}} \gamma(a+1, a+st\_m) + \sum\_{i=1}^n \frac{c\_i \exp(b\_i)}{\left(b\_i + st\_m\right)^{b\_i+1}} \Gamma(b\_i+1, b\_i + st\_m) \right] \tag{4}$$

for 0 ( 1, ) exp( )d *x <sup>a</sup>* γ+ = − *a x t tt* ∫ and ( 1, ) exp( )d *<sup>a</sup> x* Γ *a x t tt* ∞ += − ∫ the incomplete Gamma functions, as defined in (Abramowitz, Stegun, 1970).

Fourier transform of NCBC function is obtained from (4) for *s*= j2π*f*, so:

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 65

the real functions, this makes the time for computing twice shorter. This is very useful if the number of FFT points is great and relevant calculations in frequency domain are timeconsuming. For smaller number of FFT points the total computing time is shortened, but the

*f*<sup>k</sup> *f*(Hz) Re{*I*(*f*)} Im{*I*(*f*)} |*I*(*f*)/*I*(0)| *f*1 0.0000E+00 0.6232E-04 0.0000E+00 0.1000E+01 *f*2 0.6403E+04 0.6892E-05 -0.2172E-04 0.3657E+00 *f*3 0.1281E+05 0.8713E-06 -0.1179E-04 0.1896E+00 *f*4 0.1921E+05 -0.2839E-06 -0.7867E-05 0.1263E+00 *f*5 0.2561E+05 -0.6610E-06 -0.5842E-05 0.9433E-01 *f*35 0.2177E+06 -0.6334E-06 -0.6164E-07 0.1021E-01 *f*108 0.6851E+06 0.4510E-07 0.4597E-07 0.1033E-02 *f*352 0.2248E+07 0.4931E-09 0.6259E-08 0.1007E-03 *f*2478 0.1586E+08 0.8158E-12 0.6237E-09 0.1001E-04 *f*3912 0.2503E+08 -0.2014E-12 0.6261E-10 0.1005E-05 *f*4078 0.2611E+08 -0.1417E-12 0.6350E-11 0.1019E-06 *f*4095 0.2621E+08 -0.1377E-12 0.6689E-12 0.1096E-07 *f*4097 0.2623E+08 -1.3765E-13 0.0000E+00 2.2088E-09

sampling interval in time domain is critical in that case.

Table 1. FFT of the HVP function in 8196 points

Fig. 7. Normalized FFT modulus of HVP as the function of frequency

times greater step in time domain ∆*T*≈76.256ns than if *N*=8092.

For example, for *N*=2048 and the same time interval *T*=*N*∆*T*≈156.172μs the sampling frequency is *f*g=1/∆*T*≈13.114MHz, which gives the same frequency step ∆*f*≈6.4kHz, but four

$$I(f) = I\_m t\_m \left| \frac{\exp(a)}{\left(a + j \mathbf{2} \pi f \mathbf{t}\_m\right)^{a+1}} \gamma(a+1, a+j \mathbf{2} \pi f \mathbf{t}\_m) + \sum\_{i=1}^n \frac{c\_i \exp(b\_i)}{\left(b\_i + j \mathbf{2} \pi f \mathbf{t}\_m\right)^{b\_i+1}} \Gamma(b\_i + 1, b\_i + j \mathbf{2} \pi f \mathbf{t}\_m) \right|. \tag{5}$$

It can be obtained also numerically, or by using some computer program such as FAS (Walker, 1996) with the application of FFT (Fast Fourier Transform). Any frequency domain calculations can be done just for the positive frequencies, as for FFT of real functions the following relations are valid:

$$\operatorname{Re}\left\{\underline{I}\_k(f)\right\} = \operatorname{Re}\left\{\underline{I}\_{N-k}(f)\right\} \tag{6}$$

and

$$\operatorname{Im}\left\{\underline{I}\_k(f)\right\} = -\operatorname{Im}\left\{\underline{I}\_{N-k}(f)\right\}.\tag{7}$$

This feature makes the time for computing FFT twice shorter for real functions, which is very useful if the number of FFT points is great so that calculations are time-consuming. For smaller number of FFT points the computing time is shortened. FFT results given for NCBC function representing lightning current indicate that the major part of its power is in the lower frequency range. The modulus of FFT decays fast as the function of frequency, so it is better to use some other than linear scaling of frequencies in order to cover the lower part of the frequency range. For very high frequencies, the values of real and imaginary parts of FFT of the pulse functions near the end of the frequency interval can be taken as zeros, as they are relatively very small comparing to the values at lower frequencies. That doesn't make much influence on computation results if IFFT (Inverse Fast Fourier Transform) is done for obtaining time domain results based on frequency domain calculations. The sufficient number of FFT points for lightning research studies should be 1024, but better results are obtained with 2048 or 4096 points. This depends on the time interval to cover and the current waveshape.

Fourier transform of Heidler's function is calculated approximately (Andreotti et al., 2005). Some FFT results for Heidler's function are also given in (Vujevic & Lovric, 2010). NCBC/CBC have analytical solutions for Fourier transform which enables obtaining analytical solution for LEMF results in the case of perfectly conducting ground.

The modulus of Fourier transform of HVP function 1.2/50μs is presented in Fig. 7, for *I*(*f*) normalized to *I*(0), as the function of log10*f*. For *f*2≈6.4kHz≈103.8Hz the normalized modulus of FFT is |*I*(*f*2)/*I*(0)|≈0.3657. The results are obtained for FFT calculated in *N*=8192 points, for the time step ∆*T*≈19.064ns, frequency step ∆*f*≈6.4kHz, and limit frequency *f*g≈52.455 MHz. From Fig. 7 is obvious that using frequencies higher than a few MHz is not necessary.

For some frequencies up to the chosen limit frequency the results are presented in Table 1 for FFT of HVP function 1.2/50μs calculated in *N*=8192 points, for ∆*T*≈19.064ns, ∆*f*≈6.4kHz, and *f*g≈52.455MHz. For the chosen number of points for FFT, the corresponding time interval is *T*=*N*∆*T*≈156.17μs and the frequency step is ∆*f*=(*N*∆*T*)-1= *T*-1. For such values there are 8192 frequencies in the chosen frequency interval [-0.5*f*g,+0.5 *f*g], and the highest positive frequency is *f*4097≈26.2275MHz. FFT results for |*I*(*f*)|≈10-*n*|*I*(0)| for *n*=1, 2, 3, …, 8, are also given in Table 1, corresponding approximately to frequencies *f*5, *f*35, *f*108, *f*352, *f*2478, *f*3912, *f*4078, *f*4095, and they also point out to the fast decaying of FFT modulus. As (6) and (7) are valid for 64 Fourier Transform Applications

*<sup>n</sup> i i m m a b <sup>m</sup> ii m*

<sup>=</sup> <sup>⎢</sup> γ+ + + Γ+ + <sup>⎥</sup> <sup>⎢</sup> + + <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

+ + <sup>=</sup>

It can be obtained also numerically, or by using some computer program such as FAS (Walker, 1996) with the application of FFT (Fast Fourier Transform). Any frequency domain calculations can be done just for the positive frequencies, as for FFT of real functions the

This feature makes the time for computing FFT twice shorter for real functions, which is very useful if the number of FFT points is great so that calculations are time-consuming. For smaller number of FFT points the computing time is shortened. FFT results given for NCBC function representing lightning current indicate that the major part of its power is in the lower frequency range. The modulus of FFT decays fast as the function of frequency, so it is better to use some other than linear scaling of frequencies in order to cover the lower part of the frequency range. For very high frequencies, the values of real and imaginary parts of FFT of the pulse functions near the end of the frequency interval can be taken as zeros, as they are relatively very small comparing to the values at lower frequencies. That doesn't make much influence on computation results if IFFT (Inverse Fast Fourier Transform) is done for obtaining time domain results based on frequency domain calculations. The sufficient number of FFT points for lightning research studies should be 1024, but better results are obtained with 2048 or 4096 points. This depends on the time interval to cover and

Fourier transform of Heidler's function is calculated approximately (Andreotti et al., 2005). Some FFT results for Heidler's function are also given in (Vujevic & Lovric, 2010). NCBC/CBC have analytical solutions for Fourier transform which enables obtaining

The modulus of Fourier transform of HVP function 1.2/50μs is presented in Fig. 7, for *I*(*f*) normalized to *I*(0), as the function of log10*f*. For *f*2≈6.4kHz≈103.8Hz the normalized modulus of FFT is |*I*(*f*2)/*I*(0)|≈0.3657. The results are obtained for FFT calculated in *N*=8192 points, for the time step ∆*T*≈19.064ns, frequency step ∆*f*≈6.4kHz, and limit frequency *f*g≈52.455 MHz. From Fig. 7 is obvious that using frequencies higher than a few MHz is not necessary. For some frequencies up to the chosen limit frequency the results are presented in Table 1 for FFT of HVP function 1.2/50μs calculated in *N*=8192 points, for ∆*T*≈19.064ns, ∆*f*≈6.4kHz, and *f*g≈52.455MHz. For the chosen number of points for FFT, the corresponding time interval is *T*=*N*∆*T*≈156.17μs and the frequency step is ∆*f*=(*N*∆*T*)-1= *T*-1. For such values there are 8192 frequencies in the chosen frequency interval [-0.5*f*g,+0.5 *f*g], and the highest positive frequency is *f*4097≈26.2275MHz. FFT results for |*I*(*f*)|≈10-*n*|*I*(0)| for *n*=1, 2, 3, …, 8, are also given in Table 1, corresponding approximately to frequencies *f*5, *f*35, *f*108, *f*352, *f*2478, *f*3912, *f*4078, *f*4095, and they also point out to the fast decaying of FFT modulus. As (6) and (7) are valid for

analytical solution for LEMF results in the case of perfectly conducting ground.

exp( ) exp( ) ( ) ( 1, j2 ) ( 1, j2 ) j2 j2 *<sup>i</sup>*

*m i i m <sup>a</sup> c b If It a a ft b b ft*

1

⎡ ⎤

 π

Re ( ) Re ( ) {*I k N f* } = {*I* <sup>−</sup>*<sup>k</sup> f* } (6)

Im ( ) Im ( ) {*I k N f* } = − {*I* <sup>−</sup>*<sup>k</sup> f* } . (7)

 π

∑ . (5)

( ) ( ) 1 1

*a ft b ft* π

π

following relations are valid:

the current waveshape.

and

the real functions, this makes the time for computing twice shorter. This is very useful if the number of FFT points is great and relevant calculations in frequency domain are timeconsuming. For smaller number of FFT points the total computing time is shortened, but the sampling interval in time domain is critical in that case.


Table 1. FFT of the HVP function in 8196 points

Fig. 7. Normalized FFT modulus of HVP as the function of frequency

For example, for *N*=2048 and the same time interval *T*=*N*∆*T*≈156.172μs the sampling frequency is *f*g=1/∆*T*≈13.114MHz, which gives the same frequency step ∆*f*≈6.4kHz, but four times greater step in time domain ∆*T*≈76.256ns than if *N*=8092.

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 67

As the value |*I*(*f*2)/*I*(0)|≈0.3657 is obtained already for the frequency *f*2≈6.4kHz (Fig. 7), it is better to use some other then linear scaling of frequencies in order to cover better the lower part of the frequency range. For very high frequencies, the values of real and imaginary part of FFT near the end of the frequency interval can be taken as zeros, as they are relatively very small comparing to their values at low frequencies. That doesn't make much influence on the computation results if IFFT (Inverse Fast Fourier Transform) is done for obtaining time domain results based on frequency domain calculations. Both FFT and IFFT can be done using program FAS (Walker, 1996) in 128, 256, 512, 1024, 2048, 4096 or 8192 points. The input data can be given as analytical functions (as in the case of NCBC or other channel-base current functions) or discrete values in the corresponding set of values for that quantity (as

In order to analyze lightning electromagnetic field in a far zone, as an external excitation of some receiving antenna structure, vertical electric field and azimuthal magnetic field can be represented with the waveshape of NCBC function according to the measured data and its Fourier transform should be known. Fourier transform of the channel-base current is also necessary if calculating equivalent voltage source as a product of the input impedance and the channel-base current in frequency domain (Moini et al. 2000, Shoory et al. 2005), instead of the current source itself at the channel-base. A few computer programs for antenna analysis are used for calculations (Richmond, 1974, 1992; Bewensee, 1978; Burke & Poggio, 1980, 1981; Djordjevic et al., 2002), and most of these use the voltage generator as excitation

Fig. 10. Normalized FFT modulus for NCBC, CBC and HVP as a function of frequency

For CBC function with parameters *Im*=11kA, *tm*=0.5826μs, *a*=1.5 and *b*=0.02 the results of FFT are presented in Figs. 10-12. This function is chosen adequate to the DEXP, except for the parameter *a* = 1.5 instead of *a* = 0.5 in the rising part, in order to satisfy the mentioned

in the case of experimentally measured data).

(Grcev et al., 2003).

Fig. 8 shows real part of FFT for HVP function in the considered frequency range [-4096∆*f*, +4096∆*f*], and enlarged rectangle area for the range of frequencies around zero *f*є[-200∆*f,* +200∆*f*]*=*[-1280.638kHz,+1280.638kHz]. It can be seen that Eq. (6) is valid.

Fig. 8. Real part of FFT for HVP in the frequency range [-0.5*f*g,+0.5*f*g]=[-4096∆*f*,+4096∆*f*] and enlarged rectangle area for frequencies *f*є[-200∆*f* ,+200∆*f*]

Fig. 9 shows imaginary part of FFT for HVP function in the same considered frequency range [-0.5*f*g,+0.5*f*g], and enlarged rectangle area for *f*є[-200∆*f,* +200∆*f*]. It can be seen that Eq. (7) is valid.

FFT results given in Table 1 for this pulse function indicate that the major part of its power is in the lower frequency range. The modulus of FFT decays rapid as the function of frequency which can be noticed both in Table 1 and Fig. 7.

Fig. 9. Imaginary part of FFT for HVP in the range [-0.5*f*g,+0.5*f*g]=[-4096∆*f*,+4096∆*f*] and enlarged rectangle area for frequencies *f*є[-200∆*f* ,+200∆*f*]

66 Fourier Transform Applications

Fig. 8 shows real part of FFT for HVP function in the considered frequency range [-4096∆*f*, +4096∆*f*], and enlarged rectangle area for the range of frequencies around zero

Fig. 8. Real part of FFT for HVP in the frequency range [-0.5*f*g,+0.5*f*g]=[-4096∆*f*,+4096∆*f*] and

Fig. 9 shows imaginary part of FFT for HVP function in the same considered frequency range [-0.5*f*g,+0.5*f*g], and enlarged rectangle area for *f*є[-200∆*f,* +200∆*f*]. It can be seen that Eq.

FFT results given in Table 1 for this pulse function indicate that the major part of its power is in the lower frequency range. The modulus of FFT decays rapid as the function of

Fig. 9. Imaginary part of FFT for HVP in the range [-0.5*f*g,+0.5*f*g]=[-4096∆*f*,+4096∆*f*] and

enlarged rectangle area for frequencies *f*є[-200∆*f* ,+200∆*f*]

frequency which can be noticed both in Table 1 and Fig. 7.

enlarged rectangle area for frequencies *f*є[-200∆*f* ,+200∆*f*]

(7) is valid.

*f*є[-200∆*f,* +200∆*f*]*=*[-1280.638kHz,+1280.638kHz]. It can be seen that Eq. (6) is valid.

As the value |*I*(*f*2)/*I*(0)|≈0.3657 is obtained already for the frequency *f*2≈6.4kHz (Fig. 7), it is better to use some other then linear scaling of frequencies in order to cover better the lower part of the frequency range. For very high frequencies, the values of real and imaginary part of FFT near the end of the frequency interval can be taken as zeros, as they are relatively very small comparing to their values at low frequencies. That doesn't make much influence on the computation results if IFFT (Inverse Fast Fourier Transform) is done for obtaining time domain results based on frequency domain calculations. Both FFT and IFFT can be done using program FAS (Walker, 1996) in 128, 256, 512, 1024, 2048, 4096 or 8192 points. The input data can be given as analytical functions (as in the case of NCBC or other channel-base current functions) or discrete values in the corresponding set of values for that quantity (as in the case of experimentally measured data).

In order to analyze lightning electromagnetic field in a far zone, as an external excitation of some receiving antenna structure, vertical electric field and azimuthal magnetic field can be represented with the waveshape of NCBC function according to the measured data and its Fourier transform should be known. Fourier transform of the channel-base current is also necessary if calculating equivalent voltage source as a product of the input impedance and the channel-base current in frequency domain (Moini et al. 2000, Shoory et al. 2005), instead of the current source itself at the channel-base. A few computer programs for antenna analysis are used for calculations (Richmond, 1974, 1992; Bewensee, 1978; Burke & Poggio, 1980, 1981; Djordjevic et al., 2002), and most of these use the voltage generator as excitation (Grcev et al., 2003).

Fig. 10. Normalized FFT modulus for NCBC, CBC and HVP as a function of frequency

For CBC function with parameters *Im*=11kA, *tm*=0.5826μs, *a*=1.5 and *b*=0.02 the results of FFT are presented in Figs. 10-12. This function is chosen adequate to the DEXP, except for the parameter *a* = 1.5 instead of *a* = 0.5 in the rising part, in order to satisfy the mentioned

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 69

the results are given in these figures also for the high voltage pulse (HVP) function 1.2/50μs, although the function with those parameters is not adequate for lightning channel-base currents modeling. All the results are presented for the functions normalized to the maximum values. The results for moduli of FFT for *I*(*f*) normalized to *I*(0), as the function of frequency, are presented in Fig. 10. For *f*2≈6.4kHz the normalized modulus of FFT for NCBC function is |*I*(*f*2)/*I*(0)|≈0.2426, for CBC function |*I*(*f*2)/*I*(0)|≈0.6242, and for HVP function

Real and imaginary parts of FFT in the frequency range *f*є[-200∆*f* ,+200∆*f*] are presented in

Log-log dependence of FFT modulus on frequency is presented in Fig. 13 for CBC function with parameters *Im*=11kA, *tm*=0.5826μs, *a*=1.5 and *b*=0.02 and for NCBC function with

Normalized FFT modulus as the function of frequency for NCBC with parameters *Im*=25.176kA, *tm*=0.65μs, *a*=20, *b*=0.00467, and for Heidler's function with parameters *I*0=25kA, η=0.993, *τ*1=0.454μs, *τ*2=143μs, both representing IEC 62305 standard negative first stroke lightning current 0.25/100μs for lightning protection level (LPL) III-IV, is given in Fig. 14.

Fig. 13. Normalized FFT moduli for NCBC and CBC functions as the function of frequency

The results show that for frequencies higher than 10MHz FFT modulus of these channelbase current functions is less than 1o*/*oo of its value at 10kHz, so for frequencies higher than

FFT results for NCBC function given in Fig. 14 are also in good agreement with the results given in (Vujevic & Lovric, 2010) for the corresponding Heidler's function with parameters

10MHz calculations of LEMF are not needed in frequency domain.

*I*0=25kA, η=0.993, *τ*1=0.454μs, *τ*2=143μs, and for LPL III-IV.

parameters *Im*=11kA, *tm*=0.472μs, *a*=1.1, *b*1=0.16, *c*1=0.34, *b*2=0.0047, and *c*2=0.66.


Figs. 11 and 12.

condition for lightning discharge channel-base current functions (having the first derivative equal to zero at *t*=0+ and the concave to convex shape in the rising part). In the same figures there are the results for the new channel base current function (NCBC) with parameters *Im*=11kA, *tm*=0.472μs, *a*=1.1, *b*1=0.16, *c*1=0.34, *b*2=0.0047, and *c*2=0.66. Just for the comparison,

Fig. 11. Real part of the normalized FFT for NCBC, CBC and HVP functions in the frequency range [-200∆*f* ,+200∆*f*]

Fig. 12. Imaginary part of the normalized FFT for NCBC, CBC and HVP functions in the frequency range [-200∆*f* ,+200∆*f*]

68 Fourier Transform Applications

condition for lightning discharge channel-base current functions (having the first derivative equal to zero at *t*=0+ and the concave to convex shape in the rising part). In the same figures there are the results for the new channel base current function (NCBC) with parameters *Im*=11kA, *tm*=0.472μs, *a*=1.1, *b*1=0.16, *c*1=0.34, *b*2=0.0047, and *c*2=0.66. Just for the comparison,

Fig. 11. Real part of the normalized FFT for NCBC, CBC and HVP functions in the frequency

Fig. 12. Imaginary part of the normalized FFT for NCBC, CBC and HVP functions in the

range [-200∆*f* ,+200∆*f*]

frequency range [-200∆*f* ,+200∆*f*]

the results are given in these figures also for the high voltage pulse (HVP) function 1.2/50μs, although the function with those parameters is not adequate for lightning channel-base currents modeling. All the results are presented for the functions normalized to the maximum values. The results for moduli of FFT for *I*(*f*) normalized to *I*(0), as the function of frequency, are presented in Fig. 10. For *f*2≈6.4kHz the normalized modulus of FFT for NCBC function is |*I*(*f*2)/*I*(0)|≈0.2426, for CBC function |*I*(*f*2)/*I*(0)|≈0.6242, and for HVP function |*I*(*f*2)/*I*(0)|≈0.3657.

Real and imaginary parts of FFT in the frequency range *f*є[-200∆*f* ,+200∆*f*] are presented in Figs. 11 and 12.

Log-log dependence of FFT modulus on frequency is presented in Fig. 13 for CBC function with parameters *Im*=11kA, *tm*=0.5826μs, *a*=1.5 and *b*=0.02 and for NCBC function with parameters *Im*=11kA, *tm*=0.472μs, *a*=1.1, *b*1=0.16, *c*1=0.34, *b*2=0.0047, and *c*2=0.66.

Normalized FFT modulus as the function of frequency for NCBC with parameters *Im*=25.176kA, *tm*=0.65μs, *a*=20, *b*=0.00467, and for Heidler's function with parameters *I*0=25kA, η=0.993, *τ*1=0.454μs, *τ*2=143μs, both representing IEC 62305 standard negative first stroke lightning current 0.25/100μs for lightning protection level (LPL) III-IV, is given in Fig. 14.

Fig. 13. Normalized FFT moduli for NCBC and CBC functions as the function of frequency

The results show that for frequencies higher than 10MHz FFT modulus of these channelbase current functions is less than 1o*/*oo of its value at 10kHz, so for frequencies higher than 10MHz calculations of LEMF are not needed in frequency domain.

FFT results for NCBC function given in Fig. 14 are also in good agreement with the results given in (Vujevic & Lovric, 2010) for the corresponding Heidler's function with parameters *I*0=25kA, η=0.993, *τ*1=0.454μs, *τ*2=143μs, and for LPL III-IV.

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 71

distances, propagation above ground of finite conductivity results in the noticeable attenuation of the high-frequency components of electric and magnetic field, and thus in appearance of the horizontal electric field at the surface. Finite conductivity has greater impact on horizontal than on vertical electric field, so calculation of horizontal component

Approximate formulas in frequency domain are often used for calculation of the horizontal electric field in air, up to heights of tens of meters above the ground surface. These formulas can be integrated in the calculation of LEMF in time domain, but the obtained expressions are much more complex. There are simple approximations: the assumption of perfectly conducting ground, "wavetilt" formula, Cooray's approach and Rubinstein's approach. Cooray, 1992, proposed the calculation of horizontal electric field at the surface of finitely conductive ground using azimuthal magnetic induction and the expression for ground surface impedance. He showed that this simple formula provides very accurate results at the distances of about 200m. Rubinstein, 1996, proposed expression for the horizontal electric field with two terms: 1) horizontal electric field calculated under the assumption of perfectly conducting ground, and 2) the correction factor, given as a product of the magnetic field calculated under the same assumption and the function similar as in "wavetilt" formula which represents the effect of finite conductivity. The basic assumption of Rubinstein's approximation is σ1>>ε0εr1, and that finite ground conductivity does not affect the horizontal magnetic field at the surface. If this is not the case, then more general formula can be written, known in literature as the Cooray-Rubinstein's formula (Cooray, 2002). Wait gave generalization of Cooray-Rubinstein's formula and the exact evaluation of horizontal electric field, showing under which circumstances this general expression reduces to Cooray-Rubinstein's formula (Wait, 1997). Cooray and Lindquist, 1983, and Cooray, 1987, using the attenuation function in time domain proposed by Wait, 1956, included effects of the finite conductivity, and obtained results for the electric field that are in better agreement with experiments. Terms for approximate formulas in time domain are complex, so the

Method of images gives an approximate solution to the Sommerfeld's integral. Complex image technique often uses one or more terms of exponential series to approximate plane wave reflection coefficient. Thus, multiple discrete and/or continuous image sources are introduced, and this technique also proved not to be limited to a quasi-static range alone. Using this technique, different authors (Shubair & Chow, 1993; Yang & Zhou, 2004; Popovic & Petrovic, 1993; and Petrovic, 2005) obtained results for Sommerfeld's integral kernel for vertical dipoles above a lossy half-space which were used for the comparison with the new Two-image approximation (TIA). Approximate formulas are often valid for a limited range of ground electrical parameters, field point distances, or heights of dipoles above ground, but TIA approximation of Sommerfeld's integral kernel proposed in (Rancic & Javor, 2006 & 2007) has the advantage of being valid in a wide range of lossy ground electrical parameters, for various heights of vertical electric dipoles above the ground, and for possibility to

LEMF can be calculated using thin wire antenna modeling of a lightning channel assumed as vertical, without branches and reflections from the end. If using electromagnetic model,

requires rigorous computation or, at least, acceptable approximations.

approach in frequency domain is preferred.

calculate electromagnetic field in both near and far zone.

**4.2 Method of images** 

Fig. 14. Normalized FFT moduli for NCBC and Heidler's functions representing standard 0.25/100μs as the function of frequency
