-g_{ab}                     & g_{ab}+g_{bc}+g_{bo}  & -g_{bc}                           \\
-g_{ac}                     & -g_{bc}                   & g_{bc}+g_{ac}+g_{co}          \\
\end{bmatrix} \notag\\
\label{eq:L_C_R_G_matrices}\end{aligned}$$ are the PUL parameters matrices for the resistance, inductance, capacitance and conductance, respectively. We remark that $\bm{R}$ includes both the ohmic and radiation resistances. Solving as presented in [@versolatto2011an] gives the voltage and current propagation equations

$$\begin{aligned}
\bm{I}(x) =& \bm{Z_C^{-1}}\bm{T}(e^{-\bm{\Gamma }x}\bm{V^+}_m - e^{\bm{\Gamma }x}\bm{V^-}_m) \notag\\
\bm{V}(x) =& \bm{T}e^{-\bm{\Gamma }x}\bm{V^+}_m + e^{\bm{\Gamma }x}\bm{V^-}_m ,
\label{eq:V_I_forw_back}\end{aligned}$$

where $\bm{\Gamma }$ and $\bm{Z_C}$ are the propagation constant and the characteristic impedance of the cable respectively. $\bm{T}$ is a known transformation matrix and $\bm{V^+}_m = \bm{\rho}_L\bm{V^-}_m$, where $\bm{\rho}_L$ is the reflection coefficient given by the load $\bm{Z_L}$ as

$$\begin{aligned}
\bm{\rho}_L = \left(\bm{Z_L}-\bm{Z_C}\right)^{-1}\left(\bm{Z_L}+\bm{Z_C}\right).
\label{eq:rho}\end{aligned}$$

MTL scattering matrix {#sec:mtlscat}
---------------------

A power line branch can be in general represented as a multiport system where the signal at N input ports is transferred to N output ports. To derive the scattering matrix of this system, has to be solved for $x=0$ and $x=l$, where $l$ is the length of the line, respectively. The voltages and currents at the input side of the MTL are $$\begin{aligned}
\bm{I}_I =& \bm{Z_C}^{-1}(\bm{V^+}_m - \bm{V^-}_m) \\
\bm{V}_I =& (\bm{V^+}_m + \bm{V^-}_m),
\label{eq:V_I_left_zero}\end{aligned}$$ while the voltages and currents at 