# Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters
## 2 Electromagnetic Interference on Low Voltage Lines--Network Model
### Robustness of a Simple Communication Network System  
For this work, we consider a simplified model of a communication system composed by (1) a signal source, (2) a line and (3) a receiver (this communication system, as a whole, is the EMI recipient represented in Figure 1. Let us start from this one-dimensional model to grow for a general view of dynamical networks, n-dimensional, subject to any kind of external perturbations and/or internal structural features and fails, if any. Figure 2 (top) illustrates this simple communication system, where a squared signal is sent through a one-dimensional line from a source S to a receiver R. This signal could be, say, a +5 volts square traveling on a metallic transmission line towards the receiver R to communicate the meaning of \"1\", whereas a 0 volt would communicate \"0\". We then consider that same hypothetical line as a set of \"n\" nodes N interconnected by line-segments L (Figure 2, bottom), where the electrical signal travels from node to node in discrete time-steps. Figure 1: Electromagnetic interference coupling model. If we have an ideal line, the signal will travel at light speed from node to node and we can easily know the duration of each time-step according to the length of each segment. The signal will arrive at receiver R with the same shape as it was generated by S. This approach of time-steps and line-steps will afford us towards computational models to deal with a broad range of issues, as will be detailed below. However, there are no perfect lines in real world. As pointed out before when discussing interferences, external perturbations and internal characteristics could affect the signal and its readability. Figure 3 shows illustrative examples of unpredictable interferences originated by an external unknown source. The surge voltage strikes the line at an undefined node and propagates through the line. If it happens at the time when the squared signal is passing through that very node (Figure 3, top), the signal will be distorted (compared to the original) and then reaches the receiver R with a very different shape. If the surge strikes the line before the squared signal (Figure 3, middle), it changes the original signal sequence at R, leading to a different message reading. Instead of a surge, a transient noise could hit the node, causing the electrical squared signal to become completely distorted and unrecognizable (Figure 3, bottom). These three somewhat unrealistic cases are just illustrative issues of many other possible ones, and R will read very different messages than expected by the sender S.  
Despite the line's physical structure being robust (and, in this case, ideal), the line's functional process is not, considering that the message was distorted from the original by an external factor. Measuring the difference between the original and resultant signals shows us the damaging caused by some functional processes. Figure 2: A simple one-dimensional model where a squared signal is sent through a line from a source S to a receiver R (**top**). The same line (**bottom**) considered as a set of “n” nodes N interconnected by line-segments L, where the squared signal travels from node to node in time-steps. Figure 4 illustrates other cases of distortions caused by interference. For instance, a wave caused by lightning, adding a signal that changes a large sequence of bits. If the wave is too long, a whole sequence will be damaged and a very different message will be read at R. Also, other kinds of interference can multiply the original voltage by undefined factors. External interferences, their magnitude, and shapes can be unexpected and out of control. To tackle these problems, we could apply elements that modify the line _structural_ features, aiming to give better responses on the line _functional_ features. These two levels, structural features and functional features, should be helpful to generalize a measure of robustness for networks.