## 2 The Model Backgrounds
The Heidler model [12] is defined by the following equation,  
\[i(0,t)=\frac{I_{01}}{\eta}\cdot\frac{(t/\tau_{1})^{2}}{(t/\tau_{1})^{2}+1}\cdot \exp(-t/\tau_{2})+I_{02}\cdot(\exp(-t/\tau_{3})-\exp(-t/\tau_{4})) \tag{13}\]  
where the values of the parameters chosen are: \(I_{01}\) = 9.9 kA, \(I_{02}\) = 7.5 kA, \(\eta\) = 0.845, \(\tau_{1}\) = 0.072 uS, \(\tau_{2}\) = 0.072 uS, \(\tau_{3}\) = 0.072 uS, \(\tau_{4}\) = 0.072 uS. The speed of the intensity wave along the channel will be taken as \(v=\frac{3}{5}c_{0}\)[13]. Thus, the time interval to consider for induction phenomena will be determined by the ratio of the channel length to the speed at which the current signal travels through it. In our case, \(t_{\max}=2\cdot 10^{-5}\)s, although the time elapsed due to the distance between the source element and the field point that traveled at the speed of light must also be taken into account. The shape of the current models and their temporal derivative at the beginning of the channel are shown in Figure 2.  
In practice, field calculations are performed by segmenting the lightning channel into portions ranging from 10 cm to 1 m in length, depending on the proximity of the field points to the channel. For each segment, the fields are evaluated using the theoretical Equations (10) and (11), and through superposition, the final result is obtained. A critical aspect of the calculation is the accurate definition of the chosen intensity model and its first derivative. In our case, the double-exponential and Heitler models have been tested, as illustrated in Figure 2, demonstrating suitable behavior for simulating inter-cloud discharges. The calculation algorithm has been implemented using MatLab, where some of the built-in routines of the software are used, although it can be adapted to any other programming language as well. Two types of application examples will be considered. On the one hand, the fields will be calculated at points close to the ground, where power transmission lines and lightning  
Figure 2: The double-exponential current model and the current first derivative at the beginning of the channel (upper panels). The Heidler current model and its first derivative in the lower panels. rods are located. On the other hand, the induced potentials in isolated metallic structures closer to the lightning channel, such as an aircraft, will be calculated. Induced Potentials at Points near Ground Level: Induced Potential Difference in a Vertical Rod with One End on the Ground  
Figure 3 represents a lightning horizontal channel under which a vertical metallic rod rests on the ground with the length \(\mathrm{Zc}\). The task is to calculate the induced potential difference that appears between the ends of the rod due to the lightning channel. The rod is \(10\) m tall and is located approximately in the middle of the channel. The induced potential difference that appears between the ends of the rod can be obtained from  
\[\Delta V_{\mathrm{c}}=\int\limits_{0}^{Z_{c}}\overset{\rightarrow}{E}\cdot d \overset{\rightarrow}{\tau} \tag{14}\]  
although other authors prefer to estimate this induced potential based on the scalar potential difference \(\Delta V=\phi(0)-\phi(Z_{c})\), both ways of defining \(\Delta V\) and \(\Delta V_{\mathrm{c}}\) only coincide for a DC current. Actually, \(\Delta V_{\mathrm{c}}\) can be interpreted as an electromotive force between the ends of the conductor, which will cause a movement of free carriers, that is, an induced current. The magnitude of this current will depend on the impedance of the conductor and can be significant if the impedance is small. The line integral (14) is numerically computed by dividing the integration path into segments, with the segment size being unimportant due to the minimal variation of vector fields along this path.