## 2 Theoretical Background and Mathematical Models
### Rusck Closed Formulation and Rusck Coupling Model
#### 2.5.1 Rusck Coupling Model  
Transmission line conditions related to the Rusck model are inferred by relating an electric field at the conductive surface with its conceivable outcomes. The corresponding power line coordination requirements given by [PERSON] are depicted through a combination of numerical conditions in Equations (2a,b) and (3) [23; 26]. \[\frac{\mathrm{d}\varphi(\mathrm{x},\mathrm{t})}{\mathrm{d}\mathrm{x}}+\mathrm{ L}^{\prime}\frac{\mathrm{d}\mathrm{i}(\mathrm{x},\mathrm{t})}{\mathrm{d} \mathrm{t}}=0 \tag{2a}\] \[\frac{\mathrm{d}\mathrm{i}(\mathrm{x},\mathrm{t})}{\mathrm{d}\mathrm{t}}+ \mathrm{C}\frac{\mathrm{d}\varphi(\mathrm{x},\mathrm{t})}{\mathrm{d}\mathrm{t} }=\mathrm{C}^{\prime}\frac{\mathrm{d}\varphi^{\mathrm{i}}(\mathrm{x},\mathrm{t })}{\mathrm{d}\mathrm{t}} \tag{2b}\]  
The variables are as follows:  
\(\varphi\): the scalar potential in the transmission line near the region where the LEMP falls. \(\varphi^{\prime}\): the total scalar potential generated by the LEMP current. t: the time constant. The entire energized voltage \(\mathrm{u}\) (\(\mathrm{x},\mathrm{t}\)) on the line, where \(\mathrm{i}\) (\(\mathrm{x},\mathrm{t}\)) is the exact line current and \(\mathrm{L}^{\prime}\) and \(\mathrm{C}^{\prime}\) are the suitable inductance and capacitance on the line, is given as  
\[\mathrm{u}(\mathrm{x},\mathrm{t})=\varphi(\mathrm{x},\mathrm{t})+\int_{0}^{ \mathrm{h}}\frac{\partial\mathrm{A}\mathrm{z}^{\mathrm{i}}(\mathrm{x},\mathrm{ z},\mathrm{t})}{\partial\mathrm{t}}\mathrm{d}\mathrm{z} \tag{3}\]  
In Equation (4a,b), \(\mathrm{h}\) represents the height of the conductors, and \(\mathrm{A}\mathrm{z}^{\mathrm{i}}\) is the orthogonal component of the vector potential. The boundary conditions for conduction are given by Equation (4a,b), where \(\mathrm{t}\) represents the time constant. (\(\mathrm{R}_{0}\): space resistance, \(\mathrm{R}_{\mathrm{L}}\): line resistance.) \[\varphi(0,\mathrm{t})=-\mathrm{R}_{0}\cdot\mathrm{i}(0,\mathrm{t})-\int_{0}^{ \mathrm{h}}\frac{\partial\mathrm{A}\mathrm{z}^{\mathrm{i}}(0,z,\mathrm{t})}{ \partial\mathrm{t}}\mathrm{d}\mathrm{z} \tag{4a}\]  
Figure 5: (**a**). A geometric representation of the coupling effect of LEMPs on ETLs resulting from a specific angle (\(\theta\)) of descent (top view, each part of the line is a spatial representation of our FDTD model) (**b**). Geometric representation of the coupling effect of LEMPs on ETLs (fall at right angles). \[\varphi(L,t)=-R_{L}\cdot i(L,t)-\int_{0}^{h}\frac{\partial Az^{i}(l,z,t)}{\partial t }dz \tag{4b}\]  
The closed formulation by [PERSON] was combined with the Heidler wave formulation, along with the boundary conditions, as shown in Equation (5). This combination resulted in more realistic results regarding the flow form by employing the Heidler current characteristic. \[\varphi(0,t)=-R_{0}\cdot\frac{I_{0}\cdot u(t)\cdot e^{\frac{-t}{2}}}{e^{-\frac {t}{2}}\cdot(\frac{n_{2}}{\
u_{1}})^{(\frac{n_{2}}{\
u_{1}})}}\cdot\frac{\frac{ t}{1}}{1+\left(\frac{t}{2}\right)}-\int_{0}^{h}\frac{\partial Az^{i}(0,z,t)}{ \partial t}dz \tag{5}\]  
A simplified equation for LEMP-initiated overvoltage in an energy transmission line located in the region nearest the LEMP impact was proposed by [PERSON]. The related mathematical equation, denoted as Equation (6), is also presented [27]:  
\[V_{max}=\frac{Z_{0}J_{0}h}{d}\left(1+\frac{1}{\sqrt{2}}\cdot\frac{v}{c}.\frac {1}{\sqrt{1-\frac{1}{2}\cdot\left(\frac{v}{c}\right)^{2}}}\right) \tag{6}\]  
\[Z=\frac{1}{4\pi}\cdot\sqrt{\frac{u_{0}}{\varepsilon_{0}}}\cong 377\;\Omega \tag{7}\]  
The variables are as follows:  
\(V_{max}\): maximum induced voltage (V). \(\varepsilon_{0}\): dielectric constant of space (F/m). \(\mu_{0}\): magnetic permeability of the space (H/m). Z: characteristic impedance (\(\Omega\)). \(I_{0}\): peak value of the LEMP current (A). d: distance of the LEMP drop location from the transmission line (m). \(v\): speed of the LEMP (m/s). c: speed of light (m/s). h: height of the line from the ground (m).