**4.2 Method of images**

70 Fourier Transform Applications

Fig. 14. Normalized FFT moduli for NCBC and Heidler's functions representing standard

**4. Electromagnetic field calculation using antenna model of the lightning** 

Based on experimentally measured characteristics of natural lightning (Berger et al., 1975; Lin et al. 1979; Anderson & Eriksson 1980) and also artificially initiated lightning discharges, LEMF can be estimated using some of the models from literature (Rakov & Uman, 1998, 2006), and lightning currents are assumed to propagate with attenuation and distortion while distributing charge along the channel. In the case of perfectly conducting ground calculations are simple in time domain, but full-wave approach in the case of a lossy ground is complex to implement, so Fourier transform is applied. After calculations are done in frequency domain, the conversion to time domain is performed by Inverse Fast Fourier transform (IFFT). A solution of the Sommerfeld's problem is required for the lossy ground.

The problem of LEMF calculation can be easily solved directly in time domain if the ground is treated as perfectly conductive. In such a case there exist vertical electric and azimuthal magnetic field components in the observed point at the ground surface. Horizontal component of electric field is zero at the perfectly conducting ground surface, but non-zero above the ground surface. However, it exists above, under, and at the surface of the lossy ground. Vertical component of the lightning electric and azimuthal component of the magnetic field can be easily (but in that case approximately) determined at the distances greater than a kilometer under the assumption of perfectly conducting ground. For smaller

0.25/100μs as the function of frequency

A few alternatives are proposed in literature.

**4.1 Alternatives to full-wave time domain computation** 

**channel at a lossy ground** 

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 calculate electromagnetic field in both near and far zone.

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,

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 73

One approximate solution is obtained by use of approximation of the spectral reflection coefficient of a plane wave with exponential series, having one or more terms which is named in literature as the method of images. For a horizontal electric dipole one complex image is the simplest approximation, which was first introduced by Wait, 1955, and later justified in (Wait & Spies, 1969). An important contribution was given by Bannister, (Bannister, 1966), who also showed that techniques of the theory of images in the case of finitely conductive ground are not restricted just to a quasi-static extent of the problem (Bannister, 1978). He later published a review paper "Summary of image theory expressions for the quasi-static fields of antennas at or above the earth's surface" (Bannister, 1979). Bannister extended the validity of the approximation to high-frequency problems by introducing exponential function exp(-*u*0*d*) where *u*0=(α2+γ02)1/2 and γ0 is the complex propagation constant in the air. Thus the effects of propagation are included, the same solution obtained, and for both approximations there is the assumption that the refraction coefficient of ground-to-air is much greater than 1 (*n*10=γ1/γ0>>1). Mahmoud & Metwally, 1981, used discrete, so as the combination of discrete and continuous images. Mahmoud, 1984, extended the theory to multiple images and to a layered ground. Distributed images were also used and named "the exact theory of continuous images" in (Lindell & Alanen, 1984). By using Prony's method and nonlinear optimization technique Chow et al., 1991, analytically expressed Sommerfeld's integrals in closed form of spatial complex images. Shubair & Chow, 1993, discussed the impact of complex images on the problem of vertical antennas in the presence of lossy media and took advantage of the spatial Green's function in closed form in order to obtain a superposition of the sources influence, quasi-dynamic image and three complex images. Arand et al., 2003, used the method of discrete complex images and Generalized pencil of function technique to obtain the locations and impact of current sources. A procedure to approximate Sommerfeld's integrals was carried out by Popovic & Petrovic, 1993, named "a simple near-exact solution" which uses a few images determined so to approximately satisfy the boundary conditions in a limited area of the interface and to yield accurate field values in the domain of the antenna. Electromagnetic models including influence of the lossy ground on LEMF calculation often use approximations of Sommerfeld's integral kernel on the basis of theory of images in frequency domain. Takashima et al., 1980, proposed a modified theory of images for smaller distances from the point of observation to the vertical electric dipole. As in lightning research is necessary to treat also larger distances from the point of observation to the vertical electric dipole, this approximation can not satisfy the needs of computation in the range of distances of interest. For larger distances from the point of observation to the vertical electric dipole complete expressions, derived by King, 1990, were used for EM field of a vertical electric dipole over the lossy medium, if the condition |γ2|2>>|γ1|2 or |γ2|>>3|γ1| is satisfied. King & Sandler, 1994, confirmed with their results the validity of

The new approximation TIA can be classified into two-image approximations. The basic idea for obtaining a new approximation of the spectral reflection coefficient is matching the approximation of reflection coefficient as a two-terms function (with three unknown constants) at two points, but also matching its first derivative at one point, which resulted in a very efficient approximation of Sommerfeld's integral. TIA gives good results for modeling in both near and far zone, as well as physically conceivable arrangement of images and representation of the problem (as the distances of the images are real values and

these expressions over certain types of ground.

boundary conditions are fulfilled at the wire antenna surface, and a voltage or current source is assumed at the channel-base. Unknown current distribution along the antenna is determined by solving some system of integro-differential equations of EFIE or MFIE type satisfying boundary condition for electric or magnetic field components, respectively. One of those is e.g. System of integral equations of two potentials (SIE-TP), (Rancic, 1995), which is of EFIE type. Current distribution along the antenna can be approximated e.g. by polynomials (Popovic, 1970) with unknown complex coefficients. These can be determined by using Method of Moments (MoM) (Harrington, 1968) i.e. by matching in a sufficient number of points along the antenna. Based on that current distribution LEMF is calculated using electromagnetic theory relations. Thus, the problem is solved in full (without mentioned approximations of some LEMF components using others) on the basis of approximate solution of Sommerfeld's problem of a dipole radiating at the arbitrary height above the lossy half-space, which is a classical problem in electromagnetics.

The problem of a dipole radiation in the presence of a lossy medium was noticed yet in 1909, by Sommerfeld, who determined the solution in the form of superposition of inhomogeneous plane waves. His work had such a great influence on later theoretical research in this area that the solution of more complex problems which shows the influence of a lossy medium on the properties of linear antennas is marked as Sommerfeld's solution. Sommerfeld was the first who treated all four cases of elementary Hertz's dipoles in the air applying his formulation through the cylindrical waves. Van der Pol, 1931, proposed approximate method for solving Sommerfeld's integrals. Class of integrals obtained by Sommerfeld's formulation which strictly defines boundary conditions on the surface of discontinuity is known in literature as "integrals of Sommerfeld's type". These integrals have limits of integration 0 and infinity, and the integrand is a product of Bessel's function (*J*0 or *J*1), exponential, and one more function, so it is of very complex shape. Depending on the complexity of the model, this function has different forms so that different problems appear in its numerical integration. Since these integrals are highly oscillatory and slowly decreasing functions they don't have solutions in the closed form. In the analysis of antennas above lossy media, there is an inevitable and major problem to determine Sommerfeld's integral as accurate as possible in a wide range of values of electrical parameters of the medium, for different positions of current elements, different frequencies and positions of the observed point in the field. There are many different ways to approximate Sommerfeld's integral which can be found in literature. The model that includes complex mathematical functions in the case of lossy ground gives result equivalent to the field of infinite number of current images. Simpler approximations have limitations as for the values of electrical parameters of the medium, the position of the point in the field (near or far zone) and positions of matching points near or at the interface of two half-spaces. In order to include the influence of electrical parameters of a real ground on antenna characteristics, direct Sommerfeld's approach to the problem can be also used, but it is very complex for solving from both the analytical and numerical point of view. In addition to the direct approach to this problem, it is possible to modify the path of integration of Sommerfeld's integral, or to use interpolation and speed up the calculation, or to use tables of Sommerfeld's integrals, but there were also some attempts to solve the problem by obtaining approximations in closed form for some special cases of Sommerfeld's integrals. A suitable solution would be the one to determine efficiently Sommerfeld's integrals for arbitrary positions and orientations of dipoles, arbitrary positions of the points in LEMF and a wide range of frequencies.

72 Fourier Transform Applications

boundary conditions are fulfilled at the wire antenna surface, and a voltage or current source is assumed at the channel-base. Unknown current distribution along the antenna is determined by solving some system of integro-differential equations of EFIE or MFIE type satisfying boundary condition for electric or magnetic field components, respectively. One of those is e.g. System of integral equations of two potentials (SIE-TP), (Rancic, 1995), which is of EFIE type. Current distribution along the antenna can be approximated e.g. by polynomials (Popovic, 1970) with unknown complex coefficients. These can be determined by using Method of Moments (MoM) (Harrington, 1968) i.e. by matching in a sufficient number of points along the antenna. Based on that current distribution LEMF is calculated using electromagnetic theory relations. Thus, the problem is solved in full (without mentioned approximations of some LEMF components using others) on the basis of approximate solution of Sommerfeld's problem of a dipole radiating at the arbitrary height

The problem of a dipole radiation in the presence of a lossy medium was noticed yet in 1909, by Sommerfeld, who determined the solution in the form of superposition of inhomogeneous plane waves. His work had such a great influence on later theoretical research in this area that the solution of more complex problems which shows the influence of a lossy medium on the properties of linear antennas is marked as Sommerfeld's solution. Sommerfeld was the first who treated all four cases of elementary Hertz's dipoles in the air applying his formulation through the cylindrical waves. Van der Pol, 1931, proposed approximate method for solving Sommerfeld's integrals. Class of integrals obtained by Sommerfeld's formulation which strictly defines boundary conditions on the surface of discontinuity is known in literature as "integrals of Sommerfeld's type". These integrals have limits of integration 0 and infinity, and the integrand is a product of Bessel's function (*J*0 or *J*1), exponential, and one more function, so it is of very complex shape. Depending on the complexity of the model, this function has different forms so that different problems appear in its numerical integration. Since these integrals are highly oscillatory and slowly decreasing functions they don't have solutions in the closed form. In the analysis of antennas above lossy media, there is an inevitable and major problem to determine Sommerfeld's integral as accurate as possible in a wide range of values of electrical parameters of the medium, for different positions of current elements, different frequencies and positions of the observed point in the field. There are many different ways to approximate Sommerfeld's integral which can be found in literature. The model that includes complex mathematical functions in the case of lossy ground gives result equivalent to the field of infinite number of current images. Simpler approximations have limitations as for the values of electrical parameters of the medium, the position of the point in the field (near or far zone) and positions of matching points near or at the interface of two half-spaces. In order to include the influence of electrical parameters of a real ground on antenna characteristics, direct Sommerfeld's approach to the problem can be also used, but it is very complex for solving from both the analytical and numerical point of view. In addition to the direct approach to this problem, it is possible to modify the path of integration of Sommerfeld's integral, or to use interpolation and speed up the calculation, or to use tables of Sommerfeld's integrals, but there were also some attempts to solve the problem by obtaining approximations in closed form for some special cases of Sommerfeld's integrals. A suitable solution would be the one to determine efficiently Sommerfeld's integrals for arbitrary positions and orientations of dipoles,

above the lossy half-space, which is a classical problem in electromagnetics.

arbitrary positions of the points in LEMF and a wide range of frequencies.

One approximate solution is obtained by use of approximation of the spectral reflection coefficient of a plane wave with exponential series, having one or more terms which is named in literature as the method of images. For a horizontal electric dipole one complex image is the simplest approximation, which was first introduced by Wait, 1955, and later justified in (Wait & Spies, 1969). An important contribution was given by Bannister, (Bannister, 1966), who also showed that techniques of the theory of images in the case of finitely conductive ground are not restricted just to a quasi-static extent of the problem (Bannister, 1978). He later published a review paper "Summary of image theory expressions for the quasi-static fields of antennas at or above the earth's surface" (Bannister, 1979). Bannister extended the validity of the approximation to high-frequency problems by introducing exponential function exp(-*u*0*d*) where *u*0=(α2+γ02)1/2 and γ0 is the complex propagation constant in the air. Thus the effects of propagation are included, the same solution obtained, and for both approximations there is the assumption that the refraction coefficient of ground-to-air is much greater than 1 (*n*10=γ1/γ0>>1). Mahmoud & Metwally, 1981, used discrete, so as the combination of discrete and continuous images. Mahmoud, 1984, extended the theory to multiple images and to a layered ground. Distributed images were also used and named "the exact theory of continuous images" in (Lindell & Alanen, 1984). By using Prony's method and nonlinear optimization technique Chow et al., 1991, analytically expressed Sommerfeld's integrals in closed form of spatial complex images. Shubair & Chow, 1993, discussed the impact of complex images on the problem of vertical antennas in the presence of lossy media and took advantage of the spatial Green's function in closed form in order to obtain a superposition of the sources influence, quasi-dynamic image and three complex images. Arand et al., 2003, used the method of discrete complex images and Generalized pencil of function technique to obtain the locations and impact of current sources. A procedure to approximate Sommerfeld's integrals was carried out by Popovic & Petrovic, 1993, named "a simple near-exact solution" which uses a few images determined so to approximately satisfy the boundary conditions in a limited area of the interface and to yield accurate field values in the domain of the antenna. Electromagnetic models including influence of the lossy ground on LEMF calculation often use approximations of Sommerfeld's integral kernel on the basis of theory of images in frequency domain. Takashima et al., 1980, proposed a modified theory of images for smaller distances from the point of observation to the vertical electric dipole. As in lightning research is necessary to treat also larger distances from the point of observation to the vertical electric dipole, this approximation can not satisfy the needs of computation in the range of distances of interest. For larger distances from the point of observation to the vertical electric dipole complete expressions, derived by King, 1990, were used for EM field of a vertical electric dipole over the lossy medium, if the condition |γ2|2>>|γ1|2 or |γ2|>>3|γ1| is satisfied. King & Sandler, 1994, confirmed with their results the validity of these expressions over certain types of ground.

The new approximation TIA can be classified into two-image approximations. The basic idea for obtaining a new approximation of the spectral reflection coefficient is matching the approximation of reflection coefficient as a two-terms function (with three unknown constants) at two points, but also matching its first derivative at one point, which resulted in a very efficient approximation of Sommerfeld's integral. TIA gives good results for modeling in both near and far zone, as well as physically conceivable arrangement of images and representation of the problem (as the distances of the images are real values and

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 75

Electric field is determined based on the current along the antenna from the expression

2 <sup>0</sup> grad ˆ ˆ *E E* =− ϕ−γ Π= ρ+ <sup>ρ</sup> *E zz*

Polynomial approximation used to represent the current distribution along *k*-th segment

, , 0 ( ) ( / ), *nk*

*k k mk k k m Is B s l* =

0 0 0 0 0

0 00 0 0 0 0 0

*n n*

*s s*

*s s*

= = ϕ +γ ϕ γ − +γ Π γ − ∫ ∫

γ ϕ γ − +γ ⎡ ⎤ γ − ∫ ∫ ⎣ ⎦

0 0

*n n*

*s s*

*s s*

= =

where *nk* is the polynomial degree and *Bmk* are unknown complex coefficients to be determined from the system of equations SIE-TP. By satisfying boundary condition for the

( )ch ( ) d ( )sh ( ) d

'( ) ( )sh ( ) d sh( ),

*nn n n n*

*Z sI s s s s C s*

( ) ( )sh ( ) d ( )ch ( ) d

*s s sss s sss*

*n n sn*

*s ss s s sss* Π

*n sn*

[ ] 0 2 <sup>0</sup>

[ ] 0 2 <sup>0</sup>

'( ) ( )ch ( ) d ch( )

*nn n n n*

*Z sI s s s s C s*

Polynomial approximation is used to represent the current distribution along *k*-th segment

JJG

JG JJG . (11)

<sup>=</sup> ∑ (12)

*m*

[ ] <sup>2</sup>

+ γ −= γ ∫ (13)

*n*

+ γ −= γ ∫ . (14)

0=-∂П/∂*z*0 represents the scalar potential at

*n*

[] [] <sup>2</sup>

Fig. 15. Lightning channel model at a lossy ground

tangential component of electric field, SIE-TP is obtained as:

0

0

*s*

with axis *sk*', so that in (13) and (14), φ(s)=-div Π

=

*ns*

*s*

=

*ns*

with axis *sk*' is given with:

not complex as in other approximations). TIA is similar to two-image approximations of Sommerfeld's integral, and therefore is their abbreviation retained, but the way of deriving the corresponding expressions is different. Sommerfeld's integral kernel results are compared to the results from literature for different values of lossy ground electrical parameters and various heights of vertical dipoles above the ground (Javor & Rancic, 2009). For the spectral reflection coefficient the following approximation is used

$$
\tilde{R}\_{z10}(u\_0) \equiv B + A \operatorname{e}^{-(u\_0 - u\_{0c})d\_0} \,, \tag{8}
$$

where *A* and *B* are unknown complex coefficients, *d*0 is the distance from the source to the second image, and *u*0*<sup>c</sup>* is the characteristic value chosen so that *u*0*<sup>c</sup>*=γ0. Unknown constants are determined from equations obtained by matching the value of spectral reflection coefficient in two points (*u*0→∞ and *u*0=γ0), and its first derivative at *u*0=γ0. This results in *B*=*R*∞, *A*=*R*0-*R*∞ and *d*0=γ0-1(1+*n*10-2), for *R*∞ and *R*0 the quasi-stationary reflection coefficients

$$R\_{\text{cr}} = (n\_{10}{}^2 - 1) / (n\_{10}{}^2 + 1) \tag{9}$$

and

$$R\_0 = (n\_{10} - 1) \;/\; (n\_{10} + 1) \tag{10}$$

Instead of a complex distance of the second image the value is selected as *dim*=|γ0-1(1+*n*10-2)|, based on numerical experiments.

#### **4.3 Antenna model of a lightning discharge at a lossy ground**

The simplest approximation of a LD channel is a vertical transmitting antenna, with an excitation at its base, positioned at a lossy ground surface as in Fig. 15. For the application of an electromagnetic model and equations of antenna theory, it is necessary to divide this vertical antenna into segments of the length not greater than approximately half of the wavelength corresponding to the frequency for which the analysis is done. Vertical rod antenna with an excitation by an ideal Dirac's voltage source in the channel-base, having voltage *u*(*t*)=*U*δ(*t*) and frequency *f*, is presented in Fig. 15. The total height *h* of the antenna models the height of a LD channel, which may be as high as several thousands of meters in natural conditions. In such a model is assumed that the antenna is of a circular cross section with the radius *a*, so that *a*<<λ0, where λ 0 is the wave-length in the air corresponding to the frequency *f*. The corresponding angular frequency is ω=2π*f*. The antenna is divided into *N* segments, so that *h*=*l*1+ *l*2+...+*lN*, and the length of each segment is *lk*>>*ak*, whereas radius *ak*<<λ0. It can be simply taken that *ak* =*a* for all segments, for *k*=1, 2,..., *N*. The division into segments is necessary because of the wide range of frequencies of interest and for some of those the antenna should be divided into hundreds of segments. The lossy ground is treated as homogeneous, linear and isotropic half-space of electrical parameters: specific conductivity σ1, electric permittivity ε1=ε0εr1, and magnetic permeability μ1=μ0. Other parameters of two half-spaces are defined as: the complex conductivity σ*i*=σ*i*+jωε*i*, complex propagation constant γ*i*=(jωμ*i* σ*i*)1/2, for *i*=0 denoting air and *i*=1 denoting ground of the relative complex permittivity εr1=εr1-jεi1=εr1-j60σ1λ0, and ground to the air refraction index *n*10=γ1/γ0=(εr1)1/2.

Fig. 15. Lightning channel model at a lossy ground

0

*s*

=

74 Fourier Transform Applications

not complex as in other approximations). TIA is similar to two-image approximations of Sommerfeld's integral, and therefore is their abbreviation retained, but the way of deriving the corresponding expressions is different. Sommerfeld's integral kernel results are compared to the results from literature for different values of lossy ground electrical parameters and various heights of vertical dipoles above the ground (Javor & Rancic, 2009).

10 0 () e , *uu d <sup>c</sup> R u BA <sup>z</sup>*

where *A* and *B* are unknown complex coefficients, *d*0 is the distance from the source to the second image, and *u*0*<sup>c</sup>* is the characteristic value chosen so that *u*0*<sup>c</sup>*=γ0. Unknown constants are determined from equations obtained by matching the value of spectral reflection coefficient in two points (*u*0→∞ and *u*0=γ0), and its first derivative at *u*0=γ0. This results in *B*=*R*∞, *A*=*R*0-*R*∞ and *d*0=γ0-1(1+*n*10-2), for *R*∞ and *R*0 the quasi-stationary reflection

2 2

Instead of a complex distance of the second image the value is selected as *dim*=|γ0-1(1+*n*10-2)|,

The simplest approximation of a LD channel is a vertical transmitting antenna, with an excitation at its base, positioned at a lossy ground surface as in Fig. 15. For the application of an electromagnetic model and equations of antenna theory, it is necessary to divide this vertical antenna into segments of the length not greater than approximately half of the wavelength corresponding to the frequency for which the analysis is done. Vertical rod antenna with an excitation by an ideal Dirac's voltage source in the channel-base, having voltage *u*(*t*)=*U*δ(*t*) and frequency *f*, is presented in Fig. 15. The total height *h* of the antenna models the height of a LD channel, which may be as high as several thousands of meters in natural conditions. In such a model is assumed that the antenna is of a circular cross section with the radius *a*, so that *a*<<λ0, where λ 0 is the wave-length in the air corresponding to the frequency *f*. The corresponding angular frequency is ω=2π*f*. The antenna is divided into *N* segments, so that *h*=*l*1+ *l*2+...+*lN*, and the length of each segment is *lk*>>*ak*, whereas radius *ak*<<λ0. It can be simply taken that *ak* =*a* for all segments, for *k*=1, 2,..., *N*. The division into segments is necessary because of the wide range of frequencies of interest and for some of those the antenna should be divided into hundreds of segments. The lossy ground is treated as homogeneous, linear and isotropic half-space of electrical parameters: specific conductivity σ1, electric permittivity ε1=ε0εr1, and magnetic permeability μ1=μ0. Other parameters of two half-spaces are defined as: the complex conductivity σ*i*=σ*i*+jωε*i*, complex propagation constant γ*i*=(jωμ*i* σ*i*)1/2, for *i*=0 denoting air and *i*=1 denoting ground of the relative complex permittivity εr1=εr1-jεi1=εr1-j60σ1λ0, and

<sup>000</sup> ( )

− − ≅ + (8)

<sup>10</sup> <sup>10</sup> *Rn n* ( 1)/( 1) <sup>∞</sup> = − + (9)

0 10 <sup>10</sup> *Rn n* = ( 1) /( 1) − + (10)

For the spectral reflection coefficient the following approximation is used

**4.3 Antenna model of a lightning discharge at a lossy ground** 

ground to the air refraction index *n*10=γ1/γ0=(εr1)1/2.

coefficients

based on numerical experiments.

and

Electric field is determined based on the current along the antenna from the expression

$$
\overrightarrow{E} = -\mathbf{grad}\,\mathbf{op} - \chi\_0 \,^2 \overrightarrow{\Pi} = E\_{\mathbf{p}} \hat{\mathbf{p}} + E\_z \hat{\mathbf{z}}\,.\tag{11}
$$

Polynomial approximation used to represent the current distribution along *k*-th segment with axis *sk*' is given with:

$$I\_k(\mathbf{s}\_{k'}) = \sum\_{m=0}^{n\_k} B\_{mk}(\mathbf{s}\_{k'} \;/\; l\_k)^m \,. \tag{12}$$

where *nk* is the polynomial degree and *Bmk* are unknown complex coefficients to be determined from the system of equations SIE-TP. By satisfying boundary condition for the tangential component of electric field, SIE-TP is obtained as:

[ ] <sup>2</sup> 0 0 0 0 0 0 0 ( )ch ( ) d ( )sh ( ) d *n n n s s n sn s s s ss s s sss* Π = = γ ϕ γ − +γ ⎡ ⎤ γ − ∫ ∫ ⎣ ⎦ [ ] 0 2 <sup>0</sup> 0 '( ) ( )sh ( ) d sh( ), *ns nn n n n s Z sI s s s s C s* = + γ −= γ ∫ (13) [] [] <sup>2</sup> 0 00 0 0 0 0 0 ( ) ( )sh ( ) d ( )ch ( ) d *n n n s s n n sn s s s s sss s sss* = = ϕ +γ ϕ γ − +γ Π γ − ∫ ∫ [ ] 0 2 <sup>0</sup> '( ) ( )ch ( ) d ch( ) *ns Z sI s s s s C s* + γ −= γ ∫ . (14)

Polynomial approximation is used to represent the current distribution along *k*-th segment with axis *sk*', so that in (13) and (14), φ(s)=-div Π JJG 0=-∂П/∂*z*0 represents the scalar potential at

*nn n n n*

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 77

The results obtained with TIA approximation are shown in Fig. 16 for real and imaginary part of the current along the vertical antenna having height *h*=300m, radius *a*=0.05m, and resistivity per unit length *R*'=0.1Ω/m, at the lossy ground of relative dielectric constant ε*r*1=2 and ε*r*1=10, specific conductivity σ1=10-1S/m or 10-5S/m, for frequency *f*=3MHz, the number of antenna segments *N*=30, and the degree of polynomial approximation *nk* =3, for

Fig. 16. Real and imaginary part of the current along the antenna for *h*=300m and radius

**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

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

*a*=0.05m, for different lossy ground parameters and *f*=3MHz

perfect dielectric of relative dielectric constant ε*r*1.

**4.4 Current distribution along the vertical antenna at the lossy ground** 

*k*=1,...,30.

the antenna surface, П*sn*(s)= П*z*0(s) Hertz vector tangential component, whereas *Zn*'(*s*) is the impedance per unit length, *sn* the matching point, and *s* the local coordinate along the *n*-th conductor, so that 0≤s≤ *ln*. П*z*0(s) and φ0(s) are potentials in the upper half-space to be calculated from:

$$\Pi\_{z0}(\mathbf{s}) = \frac{1}{4\pi\varrho\_0} \sum\_{k=1}^{N} \int\_{s\_k'=0}^{l\_k} I\_k(s\_k') [K\_0(r\_{1k}) + S\_{00}^{\upsilon}(r\_{2k})] \mathrm{d}s\_k' \,\tag{15}$$

$$\log\_{0}(s) = \frac{1}{4\pi\underline{\sigma}\_{0}} \sum\_{k=1}^{N} \int\_{-l\_{k}^{\prime}=0}^{l\_{k}^{\prime}} I\_{k}(s\_{k}^{\prime}) \frac{\partial}{\partial s\_{k}^{\prime}} [K\_{0}(r\_{1k}) - S\_{00}^{v}(r\_{2k})] \mathrm{d}s\_{k}^{\prime} \,. \tag{16}$$

for *K*0(*r*1*<sup>k</sup>*)=exp(-γ0*r*1*<sup>k</sup>*)/*r*1*<sup>k</sup>*the standard potential kernel, and S00ν (*r*2*<sup>k</sup>*) the Sommerfeld's integral kernel defined as

$$S\_{00}^v(r\_{2k}) = \int\_{a=0}^v \tilde{R}\_{z10}(a)\tilde{K}\_0(a, r\_{2k}) \, da \,. \tag{17}$$

10( ) *<sup>R</sup> <sup>z</sup>* α is the reflection coefficient in the spectral domain, for the variable 0≤α<∞, so that

$$\tilde{R}\_{z10}(\mathbf{a}) = \tilde{R}\_{z10}(\boldsymbol{\mu}\_0) = \frac{n\_{10}^2 \boldsymbol{\mu}\_0 - \boldsymbol{\mu}\_1}{n\_{10}^2 \boldsymbol{\mu}\_0 + \boldsymbol{\mu}\_1} \,\tag{18}$$

for 22 222 1 1 010 *u u* = α +γ = +γ −γ , and 2 2 0 0 *<sup>u</sup>* <sup>=</sup> α +γ , whereas 0 2 (, ) *K r* <sup>α</sup> *<sup>k</sup>* is the spectral form of the standard potential kernel

$$\tilde{K}\_0(\mathbf{a}, r\_{2k}) = \frac{\mathbf{e}^{-\mu\_0(z+s\_k)} }{\mu\_0} \mathbf{a} J\_0(\mathbf{a} \,\mathrm{p}) \tag{19}$$

and

$$K\_0(r\_{2k}) = \frac{e^{-\gamma\_0 \gamma\_{2k}}}{r\_{2k}} = \int\_{a=0}^{a} \tilde{K}\_0(a, r\_{2k}) \,\mathrm{d} \, a \,\tag{20}$$

for 2 2 <sup>2</sup> ρ ( ') *k k r zs* = ++ the distance from the field point M (ρ, ψ, *z*) to the first image of the *k*-th antenna segment for 2 2 ρ = + *x y* , and 0*<sup>J</sup>* (αρ) the Bessel function of the first kind and order zero.

Thus, for the Sommerfeld's integral kernel (7) is obtained:

$$S\_{00}^{v}(r\_{2k}) \equiv BK\_0(r\_{2k}) + A \exp(\gamma\_0 d\_{im}) K\_0(r\_{3k}) \,\tag{21}$$

for 2 2 <sup>3</sup> ρ ( ') *r zs d k k* = +++ *im* the distance from the second image to the field point, and 0 3 03 3 *Kr r r* ( ) exp( ) / *k k* = −γ *<sup>k</sup>* .

LEMF calculations were also done for the antenna models of cage structures (Javor, 2003) and in the case of lightning protection rods at a lossy ground (Javor & Rancic, 2009).

76 Fourier Transform Applications

the antenna surface, П*sn*(s)= П*z*0(s) Hertz vector tangential component, whereas *Zn*'(*s*) is the impedance per unit length, *sn* the matching point, and *s* the local coordinate along the *n*-th conductor, so that 0≤s≤ *ln*. П*z*0(s) and φ0(s) are potentials in the upper half-space to be

0 0 1 00 2

0 0 1 00 2

00 2 10 0 2 0

10 10 0 2

0 2 0 0 <sup>e</sup> (, ) ( <sup>ρ</sup>) *u zsk*

*K r <sup>k</sup> J u* − +

0 2 0 2 0 2 2 α 0 ( ) ( , )d , *<sup>k</sup> <sup>r</sup> k k k <sup>e</sup> K r K r r*

−γ ∞

α

=

∞

( ) ( ) ( , )d . *<sup>v</sup> Sr R K r kz k*

*<sup>z</sup>* α is the reflection coefficient in the spectral domain, for the variable 0≤α<∞, so that

() ( ) , *z z nu u R Ru*

<sup>0</sup> ( ')

=

<sup>2</sup> ρ ( ') *k k r zs* = ++ the distance from the field point M (ρ, ψ, *z*) to the first image of the *k*-th antenna segment for 2 2 ρ = + *x y* , and 0*<sup>J</sup>* (αρ) the Bessel function of the first kind and

<sup>3</sup> ρ ( ') *r zs d k k* = +++ *im* the distance from the second image to the field point, and

LEMF calculations were also done for the antenna models of cage structures (Javor, 2003)

and in the case of lightning protection rods at a lossy ground (Javor & Rancic, 2009).

<sup>−</sup> α= =

*k s k*

*s* <sup>=</sup> <sup>=</sup>

*k*

*k*

<sup>1</sup> ( ) ( )[ ( ) ( )]d <sup>4</sup> <sup>σ</sup>

*s I s Kr S r s*

*z k k k k k*

<sup>1</sup> ( ) ( ) [ ( ) ( )]d <sup>4</sup> <sup>σ</sup>

for *K*0(*r*1*<sup>k</sup>*)=exp(-γ0*r*1*<sup>k</sup>*)/*r*1*<sup>k</sup>*the standard potential kernel, and S00ν (*r*2*<sup>k</sup>*) the Sommerfeld's

*s I s K r S r s*

∂

*v*

*v*

<sup>=</sup> αα α ∫ (17)

0 0 *<sup>u</sup>* <sup>=</sup> α +γ , whereas 0 2 (, ) *K r* <sup>α</sup> *<sup>k</sup>* is the spectral

α = αα (19)

<sup>=</sup> = αα ∫ (20)

00 2 0 2 0 03 ( ) ( ) exp( ) ( ), *<sup>v</sup> S r BK r A d K r kk i* ≅ +γ *m k* (21)

Π = ∑ ′ ′ <sup>+</sup> ∫ , (15)

*k k k kk*

2 10 0 1

10 0 1

+ (18)

*nu u*

ϕ = ′ <sup>−</sup> ′ π ∂ ∑ ′ ∫ , (16)

0 1 ' 0

*k*

*k*

*k s*

0 1 ' 0

*N l*

π= =

1 1 010 *u u* = α +γ = +γ −γ , and 2 2

Thus, for the Sommerfeld's integral kernel (7) is obtained:

*N l*

calculated from:

integral kernel defined as

for 22 222

form of the standard potential kernel

10( ) *<sup>R</sup>*

and

for 2 2

for 2 2

0 3 03 3 *Kr r r* ( ) exp( ) / *k k* = −γ *<sup>k</sup>* .

order zero.
