**4.5 Input impedance of a vertical antenna modeling the lightning discharge channel**

Input impedance, *Zul*, is important for the antenna analysis in frequency domain and presents integral characteristic of the antenna structure, which also allows checking the accuracy of TIA for Sommerfeld's integral kernel. Satisfactory agreement for the polynomial degree *n*>2 was confirmed. It is enough to choose a polynomial degree *n*=2 if the length of the antenna is not greater than 0.6λ0, for *N*=1 segment of the antenna. The polynomial degree should not take values *n*>8 as the polynomial approximation of the antenna current is not appropriate for those. For σ1λ0<10-1 the obtained values for the input impedance/admittance are not dependent on the normalized conductivity, but approximately equal to the values of the input impedance/admittance in the case of a perfect dielectric of relative dielectric constant ε*r*1.

For the antenna of length *h*=2600m the results for input impedance are obtained for the frequency step Δ*f* =6.425kHz and the selected maximum frequency *f*max/2=3.2896MHz for FFT transforming time interval [0,*t*] into the frequency interval [-*f*max/2, *f*max/2]. For an arbitrary overall height *h* the segmentation should be done depending on the frequency i.e. wavelength into the segments of length *lk*≤ λ0/2 for a selected polynomial degree *nk*=3. For frequencies *f* <500kHz the antenna can be treated as one segment, so that *N*=1 is enough for calculations, whereas for higher frequencies is necessary to divide the antenna into segments. E.g. about 20 segments are required for frequencies around 1MHz, and about 200 segments for frequencies around 10MHz, if the chosen polynomial degree of the current approximation is *nk*=3 along each of the segments. Fig. 17 shows the results for input resistance and input reactance of the antenna modeling lightning discharge channel, for

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 79

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

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

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

surface (*z*=0).

*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 was chosen, depending on the observed frequency.

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
