# Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters
## 2 Electromagnetic Interference on Low Voltage Lines--Network Model
### Low Voltage Lines--Network Model
The wave speed propagation \(v\), for this line, which is also the speed propagation in each line segment, is  
\[v=\frac{\Delta x}{\Delta t}=\frac{1}{\sqrt{L_{d}C_{d}}} \tag{2}\]  
The wave travels from one node to the next with a time interval (time-step propagation) defined by  
\[\Delta t=\sqrt{LC} \tag{3}\]  
Each node is an independent line, which is interconnected to adjacent lines (see Figure 6). The connection between these lines is made according to the incidence and the reflection of electrical waves. Calculating iteratively, for all nodes, the incident and reflected voltages and their resultants, we can know at each point of a line, at its source and its receiver (load), at any time, the voltage and current levels. The line segment, or node, pictured in Figure 6 (bottom) is modelled as shown in Figure 8. Each line segment is characterized as a transmission line with a source \(S_{n}\) and a receiver \(R_{n}\) (we call it node \(N_{n}\)). The Thevenin equivalent for this node \(N_{n}\) shows voltage sources \(VE_{k,n}\), and \(VD_{k,n}\) and line parameters \(G\), \(R\) and \(Z_{0}\), where \(k\) is the iteration and \(n\) is the node (Figure 8, right). Losses caused by line resistance and conductance are considered and calculated within the node. This is an important feature of this method when we consider the node as an element of a dynamical network. It means that  
Figure 7: Segments of a transmission line (\(x\), \(x+\Delta x\)) with distributed parameters \(L\), \(C\), \(G\) and \(R\) (top). The same segments represented as transmission line method models, where the node “\(n\)” is a transmission line itself, with incident \(V^{i}\) and reflected \(V^{r}\) voltages for each time-step (bottom). the network behaviour, as signals travel between nodes through existent connections, will be modified by the specific characteristics of each node, and not only by the general structure of the network (in other words, it is not a view of a graph in its static mode). The voltage and current equations for this generic node \(N_{n}\), detailed in [8], are:  
\[V_{k,n}=VE_{k,n} \tag{4}\]  
\[VD_{k,n}=2 VD_{k,n}^{i}+l_{k,n}Z_{0} \tag{5}\]  
\[I_{k,n}=\frac{V_{k,n}-2 VD_{k,n}^{i}}{(R+Z_{0})} \tag{6}\]  
Figure 6:0  
From Figure 7 (bottom), we know that the relationship between incident (\(V^{i}\)) and reflected (\(V^{r}\)) voltages at iteration \(k\) for the node \(n\) can be obtained by adding their contributions, which determines its total voltage on the left (\(VE_{k,n}\)) and right (\(VD_{k,n}\)), as seen in the following equations:  
Figure 6:1  
Figure 6:2  
Reflected voltages from within the node \(N_{n}\) will be the incident voltages to the adjacent nodes \(N_{n-1}\) and \(N_{n+1}\) on the next time step \(k+1\), as follows:  
Figure 6:3  
Figure 6:4  
This allows the computation of the wave propagation (from time-step \(k\) to \(k+1\)), and any losses during the propagation will be considered internally at each node when calculating \(V_{k,n}\) (Equation (7)). The source \(S\) is considered in the first node, whilst the receiver R in the last one. The detailed explanation and development of equations can be found in [9] and [12]. Inductance and capacitance are natural filters that affect low and high frequencies, respectively, for the line described. Figure 5 (top) shows, for instance, how a high capacitance could distort the original signal. If we apply external filters to this line, like additional capacitances, we can easily see attenuation for high frequencies wherever we need them along the line. However, if we have  
Figure 6:5 Each line segment (the node \(N_{n}\)) is a transmission line with a source \(\mathsf{S_{n}}\) and a receiver \(\mathsf{R_{n}}\) (**left**). Its electrical model - Thévenin equivalent (**right**) consists of voltage sources \(VE_{k,n}\) and \(VD_{k,n}\) and line parameters \(G\), \(R\) and \(Z_{0}\), where \(k\) is the iteration.