## 1 Introduction
### Determination of the Component of the Overvoltage That Is Transmitted by Electromagnetic Induction  
The standard pulse wave is depicted in Figure 1. The actual definition is precise: for the lightning pulse, the front time (T\({}_{1}\)) is 1.2 us and the back time (T\({}_{2}\)) is 50 us. In the field, the lightning pulse wave can take various shapes [1; 2; 3; 5; 6; 7; 8].  
The configuration of the circuit that was analyzed to evaluate the overvoltage that is transmitted by electromagnetic induction is described in Figure 2, where \(e_{1}(t)\) is the standard lightning impulse; its analytical expression is as follows [23]:  
\[e_{1}(t)=U_{0}\cdot\left(e^{-\alpha_{1}\cdot t}-e^{-\alpha_{2}\cdot t}\right) \tag{1}\]  
where\(\alpha_{1}\) and \(\alpha_{2}\) are specific time constants of the lightning impulse t;  
\(U_{0}\) is the maximum amplitude of the lightning impulse;  
\(R_{IT}\) is the resistance of the high-voltage (IT) winding;  
\(L_{IT}\) is the inductance of the high-voltage winding;  
\(R_{MT}\) is the resistance of the medium-voltage (MT) winding;  
\(L_{MT}\) is the inductance of the medium-voltage winding;  
\(M\) is the mutual inductance between the IT and MT windings.  
The equations describing the circuit shown in Figure 2 are as follows:  
\[e_{1}(t)=R_{IT}\cdot i+L_{IT}\cdot\frac{di}{dt} \tag{2}\]  
\[u_{MT}(t)=M\cdot\frac{di}{dt} \tag{3}\]  
In the Laplace domain, we have the following relations [1; 32; 33; 34]:  
\[\mathcal{L}(i)=\frac{\mathcal{L}(u_{MT})}{s\cdot M} \tag{4}\]  
\[\mathcal{L}(e_{1})=R_{IT}\cdot\frac{\mathcal{L}(u_{MT})}{s\cdot M}+s\,\frac{L _{IT}}{s\cdot M}\cdot\mathcal{L}(u_{MT}) \tag{5}\]  
\[\mathcal{L}(u_{MT})=\frac{s\cdot M}{R_{IT}+s\cdot L_{IT}}\cdot\mathcal{L}(e_ {1}) \tag{6}\]  
\[\mathcal{L}(e_{1})=U_{0}\cdot\left(\frac{1}{s+\alpha_{1}}-\frac{1}{s+\alpha_{ 2}}\right) \tag{7}\]  
\[\mathcal{L}(u_{MT})=\frac{s\cdot\frac{M}{R_{IT}}}{1+s\cdot\frac{L_{IT}}{R_{IT }}}\cdot U_{0}\left(\frac{1}{s+\alpha_{1}}-\frac{1}{s+\alpha_{2}}\right) \tag{8}\]  
Figure 1: Standard impulse wave.  
Figure 2: Circuit for the evaluation of the surge voltage that is transmitted by electromagnetic induction.  
We note the time constant of the high-voltage winding as follows:  
\[\tau_{IT}=\frac{L_{IT}}{R_{IT}} \tag{9}\]  
\[\mathcal{L}(U_{MT})=\frac{L_{0}\cdot M}{\tau_{IT}\cdot R_{IT}}\Bigg{[}\frac{s}{ \Big{(}s+\frac{1}{\tau_{IT}}\Big{)}(s+\alpha_{1})}-\frac{s}{\Big{(}s+\frac{1}{ \tau_{IT}}\Big{)}(s+\alpha_{2})}\Bigg{]} \tag{10}\]  
Knowing that the Laplace image \(\dfrac{s}{(s+a)(s+b)}\) has in the time domain the expression  
\[\frac{1}{b-a}\Big{(}b\cdot e^{-b\cdot t}-a\cdot e^{-a\cdot t}\Big{)} \tag{11}\]  
the voltage that is transmitted by electromagnetic induction [20; 23]:  
\[u_{MTind}(t)=\frac{U_{0}\cdot M}{\tau_{IT}\cdot R_{IT}}\Bigg{[}\frac{1}{\alpha_ {1}-\frac{1}{\tau_{IT}}}\bigg{(}\alpha_{1}\cdot e^{-\alpha_{1}\cdot t}-\frac{ 1}{\tau_{IT}}\cdot e^{-\frac{1}{\tau_{IT}}\cdot t}\bigg{)}-\frac{1}{\alpha_{2} -\frac{1}{\tau_{IT}}}\bigg{(}\alpha_{2}\cdot e^{-\alpha_{2}\cdot t}-\frac{1} {\tau_{IT}}\cdot e^{-\frac{1}{\tau_{IT}}\cdot t}\bigg{)}\Bigg{]} \tag{11}\]  
Note that the value of \(R_{IT}\) has a value up to \(t\) = 2 \(\upmu\)s, given by Equation (12), and for times exceeding this value, \(R_{IT}\) will be determined according to the following [24]:  
\[R_{IT}\cong R_{\omega}=R_{cc}\cdot\frac{S_{cu}}{S_{cu\delta}} \tag{12}\]  
where:  
* \(R_{\omega}\) is the resistance value of the conductor in the pulsating electric field \(\omega\);
* \(R_{cc}\) is the resistance value when a direct current flows through the conductor;
* \(S_{cu}\) is the cross-section of the conductor;
* \(S_{cu\delta}\) is the surface through which the pulsating electric current flows, \(\omega\).