# Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters
## 3 Linear and Nonlinear Filters
### The Dynamic Model of A ZnO Variator Filter  
The question that arises about the nonlinear filter is: where and how to insert it in the finite-difference time-domain TLM model above described? We saw that all parameters are calculated within the nodes at each time-step, and the dynamic occurs when reflected signals are considered the new incident signals on the next time-step, forwarding the process on. For this reason, the model of any component must be embedded in the node where it is activated. That is to say that the nonlinear filter will be computed within the node where it is connected. This procedure agrees with the concept of bringing all \"decision-making\" about the signal processing to within the node in this model. However, its nonlinearity brings a novel problem for the modelling, which occurs just when the time-steps are going forward, and the nonlinearity results are considered and calculated within one time-step, and not between two consecutive time-steps. As we see in [12], computing a nonlinear element directly into a node can lead to unwanted oscillations. This happens because the variator response at a given time step may give rise to a large voltage variation for the next iteration (within the node where it is attached), which in turn does not allow an adequate voltage to be found at the next calculation, causing the process not to converge as expected. The solution to this problem is to solve the nonlinear equation within the node, but only during the connection between two consecutive time-steps (e.g., \(k\) and \(k+1\)) [12]. The nonlinear component model considers the component (varistor) divided into two parts: firstly, a capacitor of linear behaviour (detailed at [8]) and, secondly, a resistor with a nonlinear equation to be solved by a specific calculation process. The capacitor is modelled as a stub, to where the voltage is reflected at each iteration (\(k\)) and where a new incident voltage returns to the circuit in the next iteration (\(k+1\)). In the stub model, the capacitor is introduced as a shunt. Thus, the signal does not cross it continuously. Instead, it is forced to shift inwards and then back to the line, using a total time \(\Delta t\) to complete this process. The capacitor extremities are represented as an open circuit, which the wave reaches at time \(\Delta t/2\), reflecting and then returning to the line at another \(\Delta t/2\), with a reflection coefficient equals +1 [8]. The resistor nonlinear behaviour calculation is done in this range, which implies that the new incident voltage on the node is different from the previously reflected voltage, i.e., \(Vvar^{i}_{k+1}\) is different from \(Vvar^{i}_{k}\), differently to what happens for a linear capacitor filter [8; 12]. Then, taking into account the reflected and incident voltage differences within the nonlinear component, the current _Ivar_ circulating through the variator will be given by the equation [12]. \[Ivar=\frac{Vvar^{r}_{k}-Vvar^{i}_{k+1}}{Zvar} \tag{15}\]  
where Zvar is the variator characteristic impedance (which considers its capacitance), \(Vvar^{r}_{k}\) is the reflected voltage to the variator at time k, and \(Vvar^{i}_{k+1}\) is the incident voltage coming from the variator to the node where it is connected at moment \(k+1\)[12]. The voltage in the variator is given by  
\[Vvar=Vvar^{r}_{k}+Vvar^{i}_{k+1} \tag{16}\]  
Figure 12: Section of a variator showing the ZnO grains and the composition of “microvaristors” encapsulated between metallic discs. By substituting the expressions (15) and (16) in Equation (12), we can finally get  
\[\frac{Vvar_{k}^{r}-Vvar_{k+1}^{i}}{Zvar}=\beta\big{(}Vvar_{k}^{r}+Vvar_{k+1}^{i} \big{)}^{\alpha} \tag{17}\]  
The resolution of Equation (15) must be done at each iteration to provide the new incident voltage \(Vvar_{k+1}^{i}\), thus allowing the connection and continuity of the calculation process. Due to the nonlinear and non-quadratic condition of Equation (15), a specific method of calculation was developed and described in [12] and implemented in [26].