# 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  
Going further with the simple telecommunication line above described, let us see it as a metallic transmission line as illustrated in Figure 6, where a transmission line connects source S to receiver R and is split into n nodes N (Figure 6, top). The transmission line parameters are its inductance and capacitance, and losses are determined by its resistance and conductance. Figure 6 (middle) shows a representation of the line segments and their distributed inductances and capacitances (for design clarity the resistance and conductance are not shown). These parameters are specific structural characteristics. Figure 5: Line structural variations could modify the signal sent by S in such a way that R could misunderstand it (**top**) or even do not receive it (**bottom**). These inherent characteristics affect the communication of internal factors. A metallic line has a different and increasing capacitance from node N5 and beyond, significantly distorting the signal (top example), or a failure that intermittently interrupts the line-segment between N5 and N6 (bottom example). Back to the approach of time-steps and line-steps, each node N and its boundary line segment is modelled as a transmission line itself, considering distributed sources S\({}_{n}\) and receivers \(Rn\) (Figure 6, bottom). This approach is called the transmission line method (TLM) and is well established [8; 9; 10; 11; 12], standing as a finite difference method in the time domain. The finite difference methods are based on space discretization (grids) that allows computational solutions based on matrix algebra techniques [13]. Each node \(N_{n}\) is seen as a transmission line itself connected to the previous (\(N_{n-1}\)) and the next transmission line (\(N_{n+1}\)), from the source to the receiver. The signal moves from node to node in time-steps--that is, from one transmission line segment to the following transmission line segment. The basics of the whole model, in one, two and three-dimensional networks, were described in [9; 11], whereas a different three-dimensional cell was presented by [12]. Our approach considers the example showed in Figure 7, which is derived from [8; 12]. The top picture shows segments \(x\) and \(x+\Delta x\) of a transmission line with parameters \(L\), \(C\), \(G\) and \(R\) (inductance, capacitance, conductance, and resistance respectively). The node \(N_{n}\) comprises these parameters around the point \(x\), which is modelled as an impedance \(Z_{0}\) with losses represented by \(R\) and \(G\) (respectively resistance and conductance for the segment with length \(\Delta x\) (Figure 7 bottom). This representation is called the transmission line method model, where the node \(N_{n}\) is a transmission line itself, with the incident and reflected voltages for each time-step. A complete description of the one-dimensional method and its equations were given in [8; 12]. Below, we summarize its most important concepts and equations, which will be useful to model the system's dynamical behaviour and the nonlinear filter in the sections following. Figure 6: Metallic transmission line connecting source S to receiver R, considering \(n\) nodes \(N\) (**top**). The transmission line segments have distributed inductances and capacitances, their specific structural characteristics (**middle**). Each line segment can be seen as a transmission line itself, considering also distributed sources \(S_{n}\) and receivers \(R_{n}\) (**bottom**). This line segment is called a “node” of the network. For each node, the characteristic impedance \(Z_{0}\) is given by  
\[Z_{0}=\sqrt{\frac{L_{d}}{C_{d}}}=\sqrt{\frac{L}{C}} \tag{1}\]  
where \(L_{d}\) and \(C_{d}\) are the distributed line inductance and capacitance, respectively.