## 2 Method for Calculating the Lightning Electromagnetic Field over Perfectly Conducting Ground
### Fundamental Formulation of Lightning Electromagnetic Field over Perfectly Conducting Ground
\right\},\left\{k=\frac{\omega}{c}\right\} \tag{5}\]  
By means of \(\int\frac{(z-z^{\prime})}{R_{z^{\prime}}^{5}}dz^{\prime}=\frac{1}{3R_{z^{ \prime}}^{3}}\) and \(\int u^{\prime}vdz=uv-\int uv^{\prime}dz\), \(E_{A}\) can be rewritten as:  
\[E_{A}=\left\{\underbrace{\frac{1}{j\omega}\cdot\frac{1}{R_{z^{\prime}}^{3}}e^ {-z^{\prime}(\frac{\omega}{v}+\frac{1}{2})}e^{-jR\kappa_{z^{\prime}}}\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
Considering \(E_{B2}+E_{A24}=0\) and \(E_{B3}+E_{C}=0\), Equation (5) can be rewritten as:  
\[E_{above}=\frac{l(0,i\omega)r}{4\pi\iota_{0}} \tag{11}\] \[+\underbrace{\frac{1}{j\omega}\cdot\left(\frac{j\omega}{v}+\frac{1 }{\alpha}\right)\frac{1}{R_{\omega}}\frac{1}{z-z^{\prime}}e^{-z^{\prime}(\frac {\omega}{v}+\frac{1}{\alpha})}e^{-jR_{\omega}r}\Bigg{|}_{0}^{h}}_{E_{A21}}\] \[E_{above}=\frac{l(0,i\omega)r}{4\pi\iota_{0}}\] (12) \[+(-\frac{1}{j\omega}\cdot\left(\frac{j\omega}{v}+\frac{1}{\alpha }\right)\frac{R_{\omega}}{r^{2}(z-z^{\prime})}e^{-z^{\prime}(\frac{\omega}{v} +\frac{1}{\alpha})}e^{-jR_{\omega}r}\Bigg{|}_{0}^{h})\] \[+(-\frac{1}{j\omega r^{2}}\cdot\left(\frac{i\omega}{v}+\frac{1}{ \alpha}\right)\!\!\left(\frac{i\omega}{v}+\frac{1}{\alpha}\right)\!\!\left( \frac{i\omega}{v}+\frac{1}{\alpha}\right)\!\!\left(\frac{e^{-j\omega}}{j}+ \frac{1}{\alpha}\right)\!\!\left(\frac{i\omega}{v}+\frac{1}{\alpha}\right)\! \!\right|}_{0}^{h})\] \[+\left[\underbrace{i\omega\frac{1}{v}\!\!\left(\!\frac{1}{c}\!- \frac{c}{v^{2}}\!\right)\!+\frac{1}{\alpha}\!\!\left(\!\frac{1}{c}\!-\!\frac{3 c}{v^{2}}\!\right)\!-\!\frac{1}{j\omega}\frac{3c}{v^{2}}\!-\!\frac{1}{(j\omega)^{2}} \frac{c}{v^{3}}\!\right]\!\cdot\frac{1}{r^{2}}\!\int_{0}^{h}e^{-z^{\prime}( \frac{\omega}{v}+\frac{1}{\alpha})}e^{-jR_{\omega}r}dz^{\prime}}\]  
and it can be transformed into the time domain, i.e. :  
\[E_{above}=E_{arithmetic}+E_{intergal}, \tag{13}\]  
where:  
\[E_{arithmetic}=\frac{1}{4\pi\iota_{0}} \tag{14}\]  
where: \(i(0,t)\)--the channel base current with delay, \(i_{integral}(t)=\frac{1}{4\pi\iota_{0}}\!\int_{0}^{h}i(0,t-\frac{\iota^{ \prime}}{v}-\frac{R_{\omega}r}{c})e^{-z^{\prime}(\frac{1}{\alpha})}dz^{\prime}\)--the integral component associated with the channel base current, \(R_{h}\)--the distance between the highest point of the lightning channel and point \(P(r,z)\), and \(R_{0}\)--the distance between the lightning strike point and point \(P(r,z)\). By means of the similar derivation, the formula for \(E_{under}\) can also be achieved, only by replacing \(z\)-\(z^{\prime}\) and \(R_{z^{\prime}}\) by \(z+z^{\prime}\) and \(R_{-z^{\prime}}\), respectively. Finally, the horizontal electric field \(E_{idail}\) can be obtained by adding them together. As can be seen from Equations (12)-(14), almost all of the integral can be evaluated in an analytical way, except for \(E_{intergal}\). Moreover, the numerical calculation of \(E_{intergal}\) is very simple, because it only contains the integral of the current along the channel and its derivatives. Generally, comparing Equations (12)-(14) with Equation (3), it can be seen that the proposed method equates the integral operation with a simple arithmetic operation and a simple integral. The arithmetic operation involves fewer steps, requires less data, and produces more accurate results. The integral is much simpler than that of the conventional method. Thus, doing this makes the proposed method more efficient and easily programmed.