## 2 System Description
### Formulation of Signal Propagation over the Multiconductor Cable  
The model used for the simulation is based on the multiconductor transmission line (MTL) theory, since in the frequency range of interest the radiation phenomenon is still negligible if compared to conduction. Figure 7 shows the equivalent circuit of a three-conductor transmission line relating to an elementary cell of length \(\Delta z\) (the voltages are not shown for the sake of clarity in the picture). The line model has been constructed in the frequency domain because the measured per-unit length parameters are characterized by frequency dependence and because at this stage, nonlinear behavior of the terminations is not taken into account. The equations introduced from now on are relating to a three-conductor transmission line (as shown in Figure 7) in order to simplify the readability of the paper; however, the model relates to the full cable. Figure 5: Measured conductance. Figure 6: Outline of the simulation. In the frequency domain, the MTL equations are  
\[\left\{ \begin{array}{l}\frac{\dot{U}(z)}{dz}=-\dot{\textbf{r}}\dot{\textbf{i}}(z )-j\omega\dot{\textbf{I}}\dot{\textbf{i}}(z)=-\dot{\textbf{Z}}\dot{\textbf{i}}(z )\\ \frac{\dot{d}\dot{\textbf{i}}(z)}{dz}=-\dot{\textbf{g}}\dot{V}(z)-j\omega\dot{ \textbf{e}}\dot{V}(z)=-\dot{\textbf{V}}\dot{V}(z)\end{array} \right., \tag{1}\]  
which can be rewritten as  
\[\frac{d}{dz}\left[ \begin{array}{c}\dot{V}(z)\\ \dot{\textbf{i}}(z)\end{array} \right]=-\left[ \begin{array}{cc}0&\textbf{Z} \\ \textbf{Y}&0\end{array} \right]\left[ \begin{array}{c}\dot{V}(z)\\ \dot{\textbf{i}}(z)\end{array} \right]=\textbf{A}\left[ \begin{array}{c}\dot{ V}(z)\\ \dot{\textbf{i}}(z)\end{array} \right], \tag{2}\]  
where \(\dot{V}(z)=\left[ \begin{array}{c}\dot{V}_{1}(z)\\ \dot{V}_{2}(z)\end{array} \right]\) and \(\dot{\textbf{i}}(z)=\left[\begin{array}{  
\[\left[ \begin{array}{c}\dot{V}_{1b}\\ \dot{V}_{2b}\\ \dot{I}_{1b}\\ \dot{I}_{2b}\end{array} \right]=\left[ \begin{array}{ccccc}1&0&0&0\\ 0&1&0&0\\ -1/R_{m}&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{c}\dot{V}_{1a}\\ \dot{V}_{2a}\\ \dot{I}_{1a}\\ \dot{I}_{2a}\end{array} \right]=\textbf{H}\left[ \begin{array}{c}\dot{V}_{1a} \\ \dot{V}_{2a}\\ \dot{I}_{1a}\\ \dot{I}_{2a}\end{array} \right]. \tag{6}\]  
The previous equations can be used in two different ways: the first way is to use the cascade connection between the cells (stocks and modem), which allows the calculation of the quantities at the termination in an easy way by multiplying the matrices \(\textbf{K}\) and \(\textbf{H}\); the second way is to create a higher-dimension system leaving the voltages and currents at each modem as explicit unknowns. This second strategy leads to a higher-dimension system but allows a direct calculation of the transfer function between each modem (located in each stock) and the input voltage. For our purposes, the latter way is the suitable one: the following equation shows the final algebraic matrix relating to two stocks and a modem. The \(4\times 4\) blocks whose entries are named \(K\) relate to Equation (4), that is, the behaviour of each stock; the block whose entries are named \(H\) relates to Equation (6), that is, the modem connection; and the other two rectangular blocks relate to the boundary conditions (terminations) as explained in Equation (5). \[\left|