A simplified approach to modeling the interaction between grounding grid and lightning stroke

Abstract

In this work, a new approach for the modeling of the interaction between grounding grid and lightning stroke is described. We treat the case of direct and indirect effects of lightning strike. In the case of direct impact, we inject in point of grounding system a current with bi-exponential wave shape and we calculate the distribution of potentials and currents on the grid and the electromagnetic field it will emit. For the second case, we treat a problem of electromagnetic coupling, which is to calculate the induced currents that developed on the grounding grid when this later is illuminated by a lightning channel located in its vicinity. The presented model is validated by comparing the obtained results to the results arising from the full wave (antenna) model available in literature and to the results obtained by using NEC4 software. The principal advantage of the presented approach is the simplicity of the implementation providing a direct determination of the both current and potential distribution along the grounding grid and the related electromagnetic field in an arbitrary point in the air and/or soil, as well.

Appendix A: Electric field generated by lightning channel

In general, the lightning channel is represented by a straight vertical antenna along which the lightning return stroke propagates at the return stroke speed; the ground is assumed to be flat, homogeneous and characterized by its conductivity σ _g and its relative permittivity ε _g .

1.1 Formulation of the electromagnetic field for a dipole in nonconductive medium

In the case of perfectly conducting ground, we compute electromagnetic field emitted by the lightning channel using the dipole formalism [14]; the total electromagnetic field is obtained by summing the contributions of each dipole element of lightning channel. In that case, the components of the electric and magnetic fields at the location P(r, φ, z) produced by a short vertical section of infinitesimal channel dz at height z′ carrying harmonic varying current i(z′, ω) (Fig. 26) can be computed in the frequency domain using the following relation proposed by Uman [14].

$$ d{E_r}\left( {r, z, t} \right) = \frac{{dz}}{{4\pi {\varepsilon_0}}}i\left( {z\prime, \omega } \right)\left[ {\frac{{3r\left( {z - z\prime } \right)}}{{{R^5}}}\frac{1}{{j\omega }} + \frac{{3r\left( {z - z\prime } \right)}}{{c{R^4}}} - \frac{{r\left( {z - z\prime } \right)}}{{{c^2}{R^3}}}j\omega } \right]\exp \left( { - {{{j\omega R}} \left/ {c} \right.}} \right) $$

(A.1)

$$ d{E_z}\left( {r, z, t} \right) = \frac{{dz}}{{4\pi {\varepsilon_0}}}i\left( {z\prime, \omega } \right)\left[ {\frac{{2{{\left( {z - z\prime } \right)}^2} - {r^2}}}{{{R^5}}}\frac{1}{{j\omega }} + \frac{{2{{\left( {z - z\prime } \right)}^2} - {r^2}}}{{c{R^4}}} - \frac{{{r^2}}}{{{c^2}{R^3}}}j\omega } \right]\exp \left( { - {{{j\omega R}} \left/ {c} \right.}} \right) $$

(A.2)

$$ d{H_\varphi }\left( {r,z,t} \right) = \frac{{dz}}{{4\pi }}i\left( {z\prime, \omega } \right)\left[ {\frac{r}{{{R^3}}} + \frac{r}{{c{R^2}}}j\omega } \right]\exp \left( { - {{{j\omega R}} \left/ {c} \right.}} \right) $$

(A.3)

With: $R = \sqrt {{{r^2} + {{\left( {z - z\prime } \right)}^2}}}$ is the distance from the dipole to the observation point and r is the horizontal distance between the channel and the observation point.

c :

is the velocity of light in air

ε ₀ :

is the permittivity of air.

For a perfectly conducting ground, the total field produced by the lightning return stroke is obtained by superposition of all the dipolar contributions (real and images).

1.2 Electromagnetic field radiated in the presence of a finite conductivity ground

In air, in the case of finite ground conductivity, only the horizontal component of the electric field is affected [17]. Rubinstein [18] shows (both at near areas and in remote areas) that the horizontal component of the electric field at height h taking into account the ground conductivity, can be expressed as follows:

$$ {E_r}\left( {z = h,r,j\omega } \right) = {E_{rp}}\left( {z = h,r,j\omega } \right) - {H_{\phi p}}\left( {z = 0,r,j\omega } \right) \cdot \frac{{\sqrt {{{\mu_0}}} }}{{\sqrt {{{{{{\varepsilon_g} + {\sigma_g}}} \left/ {{j\omega }} \right.}}} }} $$

(A.4)

Where:

μ ₀ and σ _g :

are the permeability of free space and the conductivity of the ground, respectively.

E_rp(r, z = h, jω):

the horizontal component of electric field at a height h for perfectly conducting ground

H _ϕp (r, z = 0, jω) :

is the azimuthal component of magnetic field for perfectly conducting ground.

1.3 Electromagnetic fields below the ground surface

The simplified expressions for the calculation of electric radiation field produced by the lightning channel below the surface of the earth at depend d, at a horizontal distance r (Fig. 27) are recently proposed by Cooray [15].

$$ {E_z}\left( {j\omega, r,d} \right) = {E_z}\left( {j\omega, r,0} \right)\frac{{{\varepsilon_0}\exp \left( { - {k_g}d} \right)}}{{{\sigma_g} + j\omega {\varepsilon_g}}} $$

(A.5)

$$ {E_r}\left( {j\omega, r,d} \right) = {E_r}\left( {j\omega, r,0} \right)\exp \left( { - {k_g}d} \right) $$

(A.6)

Where, ${k_g} = \sqrt {{{\omega^2}{\mu_g}{\varepsilon_g} + j\omega {\mu_0}{\sigma_g}}}$: propagation constant in soil.

In these expressions, the vertical and horizontal component of electric field at the ground surface E _z (jω, r, 0) and E _r (jω, r, 0) can be calculated assuming the perfectly conducting ground for vertical electric field and Rubinshtein [18] approximation for horizontal electric field.