## 2 Method for Calculating the Lightning Electromagnetic Field over Perfectly Conducting Ground
### Fundamental Formulation of Lightning Electromagnetic Field over Perfectly Conducting Ground
Therefore, we doubt whether or not some of the integral can be implemented in an analytical way; if so, the efficiency can be improved, especially for the evaluation of the lighting-induced voltages of a distributed network, in which lightning electromagnetic fields with a great amount of observation points are required to be calculated. In the following sections, an efficient method is achieved, which can calculate most of the integrals in Equation (3) analytically. Proposed Method for Calculating the Lightning Electromagnetic Field over Perfectly Conducting Ground  
In order to simplify Equation (3), firstly, we transform the system from time varying to time invariant. The process of the channel base current propagating along the lightning channel is time varying, but if regarded it as a current model with \(i(z^{\prime},\,t)=i_{base}(t\text{-}z^{\prime}/v)\) distribution at all points along the channel direction, the system can be transformed into a time-invariant system. Then, we can transform the time-invariant system into the frequency domain based on the Fourier transform. Finally, the analytical formulae for most of the integral in Equation (3) can be obtained by means of some derivations. Substituting Equations (1) and (2) into (3) and representing the equations in the frequency domain allows the formulas to be rearranged as:  
\[E_{ideal}=\frac{1}{4\pi i\alpha_{0}}\left\{\int_{0}^{h}\left[ \begin{array}{c} \frac{3\gamma(z-z^{\prime})}{R_{z^{\prime}}^{5}}\frac{I(0,i\omega)}{j\omega}e^ {-z^{\prime}(\frac{\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{\prime}}}\\ +\frac{3\gamma(z-z^{\prime})}{cR_{z^{\prime}}^{5}}I(0,i\omega)e^{-z^{\prime}( \frac{\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{\prime}}}\\ +i\omega\frac{\gamma(z-z^{\prime})}{cR_{z^{\prime}}^{5}}I(0,j\omega)e^{-z^{ \prime}(\frac{\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{\prime}}}\end{array}  \right]dz^{\prime}+\int_{0}^{h}\left[ \begin{array}{c}\frac{3\gamma(z+z^{ \prime})}{R_{z^{\prime}}^{5}}\frac{I(0,i\omega)}{j\omega}e^{-z^{\prime}(\frac {\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{\prime}}}\\ +\frac{3\gamma(z+z^{\prime})}{cR_{z^{\prime}}^{5}}I(0,j\omega)e^{-z^{\prime}( \frac{\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{\prime}}}\\ +i\omega\frac{\gamma(z+z^{\prime})}{cR_{z^{\prime}}^{5}-z^{\prime}(0,i\omega) }e^{-z^{\prime}(\frac{\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{\prime}}} \end{array} \right]dz^{\prime}\right\} \tag{4}\]  
As can be seen in Equation (4), \(E_{above}\) and \(E_{under}\) have the same form, so only the derivation for \(E_{above}\) is provided. The above-ground lightning channel-generated horizontal electric field can be divided into three parts, i.e., \(E_{A}\), \(E_{B}\), and \(E_{C}\), as shown in Equation (5):  
\[E_{above}=\frac{I(0,i\omega)r}{4\pi i\alpha_{0}}\left\{ \begin{array}{c}\frac {3}{j\omega}\int_{0}^{h}\frac{(z-z^{\prime})}{R_{z^{\prime}}^{5}}\cdot e^{-z ^{\prime}(\frac{\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{\prime}}}dz^{\prime }+\underbrace{\frac{3}{c}\int_{0}^{h}\frac{(z-z^{\prime})}{R_{z^{\prime}}^{4} }\cdot e^{-z^{\prime}(\frac{\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{\prime}}} dz^{\prime}}_{E_{B}}\\ +\underbrace{\mu\omega}_{c}\frac{1}{c^{2}}\int_{0}^{h}\frac{(z-z^{\prime})}{R_{ z^{\prime}}^{3}}\cdot e^{-z^{\prime}(\frac{\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{ \prime}}}dz^{\prime}\\ E_{C}\end{array}