**4.6 Frequency domain results for lightning electric field**

Results for vertical and radial electric field as functions of the normalized radial distance β0ρ for the normalized antenna height *h*/λ0=0.25, normalized radius of the antenna *a*/λ0=0.0005 at the ground of specific conductivity σ1=0.01S/m and for different values of electric permittivity ε*r*1 as parameter, are presented in Fig. 19, and for ε*r*1=10 and for different values of specific conductivity σ1 as parameter in Fig. 20. Vertical and radial component of the electric field at radial distances *r* from 50 to 250m (0.5λ0≤*r≤*2.5λ0) from 78 Fourier Transform Applications

*h*=2600m, *a*=5cm, *Z*'=0.1 Ω/m, ε*r*1=10 and σ1=0.01S/m, *N*FFT=1024 points for FFT and Δ*f*=6.425kHz. Fig. 18 shows the results for input conductance and input susceptance. For different frequencies, different number of segments along the VMA in the range 1≤*N≤*200

Fig. 17. Input resistance and reactance of the vertical antenna, for *h*=2600m, *a*=0.05m, and *Z*'=0.1Ω/m, at the lossy ground of parameters ε*r*1=10 and σ1=0.01S/m, versus frequency

Fig. 18. Input conductance and susceptance of the vertical antenna, for *h*=2600m, *a*=0.05m, and *Z*'=0.1Ω/m, at the lossy ground of parameters ε*r*1=10 and σ1=0.01S/m, versus frequency

Results for vertical and radial electric field as functions of the normalized radial distance β0ρ for the normalized antenna height *h*/λ0=0.25, normalized radius of the antenna *a*/λ0=0.0005 at the ground of specific conductivity σ1=0.01S/m and for different values of electric permittivity ε*r*1 as parameter, are presented in Fig. 19, and for ε*r*1=10 and for different values of specific conductivity σ1 as parameter in Fig. 20. Vertical and radial component of the electric field at radial distances *r* from 50 to 250m (0.5λ0≤*r≤*2.5λ0) from

**4.6 Frequency domain results for lightning electric field** 

was chosen, depending on the observed frequency.

Fig. 19. Vertical and radial electric field of the quarter-wavelength monopole antenna at the lossy ground, for σ1=0.01S/m and ε*r*1 as parameter, as functions of normalized distance

Fig. 20. Vertical and radial electric field of the quarter-wavelength monopole antenna at the lossy ground, for ε*r*1=10 and σ1 as parameter, as the functions of normalized distance

the base of the antenna having height *h*=300m, circular cross section of radius *a*=5cm at the ground surface of parameters ε*r*1=10 and σ1=0.01S/m, for the frequency *f*=3MHz, the polynomial degree of the current distribution approximation along each of the segments *nk*=3, and the number of segments 20, 30 and 50, is presented in Fig. 21. The results obtained for the electric field in the points at a height *z*=1.5m above the ground surface, for the radial distances 0.5λ0≤*r≤*2.5λ0, differ a little from results for the field at the ground surface (*z*=0).

#### **4.7 Time domain results for lightning electromagnetic field**

Results for vertical electric and azimuthal magnetic field components are presented in Figs. 22-24 for different distances from the channel-base: *r*=500m, 5km and 100km. For CBC

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 81

approximation), concept of current source, NEC2 computer program (Burke & Poggio, 1981) and FFT/IFFT in 8192 points, i.e. calculations for 4092 positive frequencies. It is interesting that results very similar to (Shoory et al., 2005) would be obtained with NCBC

Fig. 23. Vertical electric and azimuthal magnetic field at the ground surface (*z*=0) for *r*=5km

Fourier transform proved to be very successful in lightning research. It enables calculations in frequency domain which are more suitable for including lossy ground effects than in time domain. It also provides information about frequency spectra of the quantities of interest in lightning research. For antenna modeling of a lightning stroke in frequency domain, Sommerfeld's integral is calculated efficiently using Two-image approximation.

Fig. 24. Vertical electric and azimuthal magnetic field at the ground surface (*z*=0) for

& MTLE, but for λ=6000m.

*r*=100km

**5. Conclusion** 

Fig. 21. Vertical and radial electric field of the antenna *h*=300m at the lossy ground, for ε*r*1=10 and σ1=0.01S/m for *f*=3MHz, as the functions of distance

Fig. 22. Vertical electric and azimuthal magnetic field at the ground surface (*z*=0) for *r*=500m

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, and two different decaying constants λ=2000m and λ=4500m, the results are compared to the results from (Nucci, 1990) calculated for perfectly conducting ground using the same Modified Transmission Line Model with Exponential Decay (MTLE) with the decaying constant λ=2000m (Nucci et al., 1990), the same return-stroke speed *v*=1.3. 108m/s and the channel having height *H*=2600m and radius *a*=0.05m. For the same *v*, *H* and *a*, for the distributed resistance *R*'=0.1Ω/m along the antenna, driven by a Dirac delta current source connected across a 3.25m gap, the results are presented also for the perfectly conducting ground (Shoory et al., 2005). Shoory et al., 2005, used the electromagnetic model and a similar procedure to here presented: EFIE type equation, Method of Moments (MoM), method of images for the approximate solution of Sommerfeld's integrals (another approximation), concept of current source, NEC2 computer program (Burke & Poggio, 1981) and FFT/IFFT in 8192 points, i.e. calculations for 4092 positive frequencies. It is interesting that results very similar to (Shoory et al., 2005) would be obtained with NCBC & MTLE, but for λ=6000m.

Fig. 23. Vertical electric and azimuthal magnetic field at the ground surface (*z*=0) for *r*=5km

Fig. 24. Vertical electric and azimuthal magnetic field at the ground surface (*z*=0) for *r*=100km
