# Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters
## 4 Results, Discussion and Perspectives  
### Results of Practical Experiments and Simulations  
The experiments prepared to test our model considered a communication metallic line such as the one pictured in Figure 9, with the nonlinear varistor filter applied on the last node, close to the receiver R.  
Table 1 shows the distributed line parameters (resistance, capacitance, and inductance per metre), and Table 2 shows the characteristics for the zinc-oxide varistors S05K11, S10K11 and S10K14 [24] used in the tests. Their constants \(\alpha\) and \(\beta\) were determined by voltages and currents at points \(a\) and \(b\) according to technical plots from [24], values are shown in Table 1 (see also Figure 11). Simulations were done with the computer Python code available at [26]. Using the same line parameters of Table 2, with a varistor S05K11 protecting the receiver load, a practical experiment was done in the laboratory. A 50 V squared wave, 6 \(\upmu\)s, was applied to the line as an alien attack and the result is shown in Figure 13, where clamping values after filtering vary around 31 volts (compare to clamping simulations shown in Appendix A, Figure A1, top-left). \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{4}{c}{**Line Parameters (Distributed)**} \\ \hline \multicolumn{2}{c}{**R (\(\Omega\)/m)**} & \multicolumn{1}{c}{**C (F/m)**} & \multicolumn{1}{c}{**L (H/m)**} \\ \hline \multicolumn{2}{c}{\(1.0\times 10^{-5}\)} & \multicolumn{1}{c}{\(1.0\times 10^{-10}\)} & \multicolumn{1}{c}{\(2.5\times 10^{-7}\)} \\ \hline \end{tabular}
\end{table} 
Table 1: Distributed line parameters (resistance, capacitance and inductance per metre). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \multicolumn{1}{c}{**Variator Model [Idk]**} & \begin{tabular}{c} **Metal** \\ **Oxide** \\ **(mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Disk** \\ **Diameter** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **C (nF)** \\ **Voltage** \\ **(V)** \\ \end{tabular} & \multicolumn{1}{c}{\(\boldsymbol{L_{a}}\) **(A)**} & \multicolumn{1}{c}{\(\boldsymbol{V_{b}}\) **(V)**} & \multicolumn{1}{c}{\(\boldsymbol{l_{b}}\) **(A)**} & \multicolumn{1}{c}{\(\boldsymbol{\alpha}\)} & \multicolumn{1}{c}{\(\boldsymbol{\beta}\)} \\ \hline S05K11 & ZnO & 5 & 1.6 & 11 & 23 & 0.01 & 34 & 1 & 11.78 & 9.04 \\ \hline S10K11 & ZnO & 10 & 6.8 & 11 & 23 & 0.01 & 30 & 0.5 & 14.72 & \(8.93\times 10^{-23}\) \\ \hline S10K14 & ZnO & 10 & 5.2 & 14 & 28 & 0.01 & 36 & 0.5 & 15.56 & \(2.97\times 10^{-25}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Characteristics for the zinc-oxide varistors used in the tests. Several simulations of our dynamical nonlinear model using zinc-oxide varistors as examples of application are shown at Annexes A, B and C [26]. Annex A shows the results for the ZnO varistor model S05K11 [24]. This commercial product is composed of a metallic disk of 5 mm diameter comprising the zinc-oxide ceramic, nominal voltage 11 V and capacitance 1.6 nF. The left side pictures at Annex A show squared pulses of 6 \(\upmu\)s (50, 75, 100 and 150 V high) achieving the load and the resultant voltages after the nonlinear filter clamping. The right side shows atmospheric surges and correspondent clampings. The atmospheric surge is a standard definition to represent typical lightning surges, characterized as an impulse with the format \(1.2\times 50\)\(\upmu\)s, which means that the surge rises from 10% to 90% of the maximum voltage in 1.2 \(\upmu\)s, decreasing to 50% in 50 \(\upmu\)s [9]. This curve format is coded to simulate the lightning electromagnetic interference at [26]. The top-left plot is the simulated computational correspondent to the practical result shown in Figure 13. Comparing pictures from the left to the right, one can realize the consistency of the clamping values for both surge types and how, for the atmospheric surge example, it smoothly decreases as the curve goes down. Annex B shows the results for the ZnO varistor model S10K11 [24].