## 2 Method for Calculating the Lightning Electromagnetic Field over Perfectly Conducting Ground
### Fundamental Formulation of Lightning Electromagnetic Field over Perfectly Conducting Ground  
The widely used model of the lightning electromagnetic field over a perfect conducting ground can be illustrated in Figure 2[25]. The lightning channel can be regarded as a combination of an above-ground lightning channel and its mirror in the free space. Regarding the source as a vertical electric dipole of current \(i(z^{\prime},t)dz^{\prime}\) located at \(z^{\prime}\), the formulas of the electromagnetic field at an observation point generated by the lightning channel can be obtained and shown as:  
\[\left\{ \begin{array}{l}E_{ideal}=\frac{1}{4\pi\epsilon_{0}}\left\{\begin{array} []{l}\int_{0}^{t}\left[\begin{array}{l}\frac{3\tau(z-z^{\prime})}{R_{z^{ \prime}}^{3}}\int_{0}^{t}i(z^{\prime},\tau-R_{z^{\prime}}/c)d\tau\\ +\frac{\delta(z-z^{\prime})}{R_{z^{\prime}}^{3}}i(z^{\prime},t-R_{z^{\prime}}/ c)\\ +\frac{\tau(z-z^{\prime})}{cR_{z^{\prime}}^{3}}\frac{\partial i(z^{\prime},t-R_{z^{ \prime}}/c)}{\partial t}\end{array}\right]dz^{\prime}+\int_{0}^{t}\left[ \begin{array}{l}\frac{3\tau(z+z^{\prime})}{R_{z^{\prime}}^{3}}\int_{0}^{t}i (z^{\prime},\tau-R_{z^{\prime}}/c)d\tau\\ +\frac{\tau(z-z^{\prime})}{cR_{z^{\prime}}^{3}}\frac{\partial i(z^{\prime},t-R _{z^{\prime}}/c)}{\partial t}\end{array}\right]dz^{\prime}\\ H_{ideal}=\frac{1}{4\pi\epsilon}\left\{\begin{array}{l}\int_{0}^{t}\frac{R_{z ^{\prime}}}{R_{z^{\prime}}^{3}}i(z^{\prime},t-R_{z^{\prime}}/c)+\frac{r}{cR_{ z^{\prime}}^{2}}\frac{\partial i(z^{\prime},t-R_{z^{\prime}}/c)}{\partial t} \end{array}\right\}dz^{\prime}\\ +\int_{0}^{t}\left[\begin{array}{l}\frac{T}{R_{z^{\prime}}^{3}}i(z^{\prime}, t-R_{z^{\prime}}/c)+\frac{r}{cR_{z^{\prime}}^{2}}\frac{\partial i(z^{\prime},t-R_{z^{ \prime}}/c)}{\partial t}\end{array}\right]dz^{\prime}\\ +\int_{0}^{t}\left[\begin{array}{l}\frac{T}{R_{z^{\prime}}^{3}}i(z^{\prime },t-R_{z^{\prime}}/c)+\frac{r}{cR_{z^{\prime}}^{2}}\frac{\partial i(z^{\prime },t-R_{z^{\prime}}/c)}{\partial t}\end{array}\right]dz^{\prime}\\ \end{array}\right\}dz^{\prime}\end{array} \right\}, \tag{3}\]  
where: \(E_{ideal}(\varphi,\,r,\,z,\,t)\)--ideal horizontal electric field at observation point \(P(r,\,z)\), \(H_{ideal}(\varphi,\,r,\,z,\,t)\)--ideal tangential magnetic field at observation point \(P(r,\,z)\), \(h\)--the height of the lightning channel, \(\epsilon_{0}\)--vacuum permittivity, \(r\)--horizontal distance from the lightning channel to point \(P(r,\,z)\), \(R_{z^{\prime}}\) and \(R_{-z^{\prime}}\)--distance from the dipole to point \(P(r,\,z)\), \(z\) and \(z^{\prime}\)--the z-coordinate of point \(P(r,\,z)\) and the dipole, respectively, and \(c\)--speed of light in vacuum. \begin{table}
\begin{tabular}{c c c c c c c} \hline \(i_{01}\) & \(i_{02}\) & \(\tau_{11}\) & \(\tau_{12}\) & \(\tau_{21}\) & \(\tau_{22}\) & \(n_{1}\) & \(n_{2}\) \\ \hline \(10.7\times 10^{3}\) A & \(6.5\times 10^{3}\) A & \(0.25\times 10^{-6}\) s & \(2.5\times 10^{-6}\) s & \(2\times 10^{-6}\) s & \(230\times 10^{-6}\) s & \(2\) & \(2\) \\ \hline \end{tabular}
\end{table} 
Table 1: The basic parameters for the channel base current function. Figure 2: Model for calculating the lightning electromagnetic field. The calculation of Equation (3) is commonly based on a numerical integration due to the lack of the analytical formulae of Equation (3), which is called the conventional method in this paper. As can be seen, the integrands in Equation (3) are functions of two variables (\(t\) and \(z^{\prime}\)), and the propagation of the current along the channel must be considered when implementing the numerical integration. Therefore, it is somewhat complicated and not easily programmed. Experience indicates that a sufficiently small computational step \(dz^{\prime}\) is essential to get an accurate result, but this leads to a large number of calculations, resulting in a lengthy calculation time.