\section{Method of characteristics} \label{Method_of_characteristics}

The propagation of modes is implemented using the method of characteristics \cite{PaulMTL} i.e. we identify modes
propagating in the +z and -z directions at the terminations and propagate these separately. This has
advantages for incorporating incident field excitation and transfer impedance coupling processes.
In the derivation of the method of characteristics model we consider a lossless, dispersionless single
mode transmission line of length L. This analysis generalizes to multi-mode transmission lines where
each mode is treated separately and so is sufficient for the derivation of the proposed model.
If we consider the propagation of a mode with modal current, I, modal voltage, V, and modal
characteristic impedance Zc then the voltage and current may be written as the sum of waves
propagating in the +z and -z directions as
%
\begin{equation}
\begin{array}{c} 
V=V^{+}+V^{-} \\[10pt]
I=I^{+}-V^{-} 
 \end{array}
 \label{eq:MOC_eqn1}
\end{equation}
%
The voltage and current for the +z and -z waves are related by
%
\begin{equation}
\begin{array}{c} 
I^{+}=\frac{V^{+}}{Z_{c}} \\[10pt]
I^{-}=-\frac{V^{-}}{Z_{c}}
 \end{array}
\end{equation}
%
We can work out the decomposition into +z and -z travelling modes from the voltage and current at any point, z, along the transmission line
%
\begin{equation} \label{eq:ch_1}
\begin{array}{c} 
2V^{+}=V+Z_{c}I\\[10pt]
2V^{-}=V-Z_{c}I
 \end{array}
\end{equation}
%
Evaluating equation \ref{eq:ch_1} at z=0 gives
%
\begin{equation} \label{eq:ch_2}
V\left(0,t\right)=Z_{c}I\left(0,t\right)+2V^{-}\left(0,t\right)
\end{equation}
%
Similarly, at z=L we have
%
\begin{equation} \label{eq:ch_2b}
V\left(L,t\right)=-Z_{c}I\left(L,t\right)+2V^{+}\left(0,t\right)
\end{equation}
%
The -z travelling mode quantities at z=0 are the -z travelling mode quantities launched from the z=L
termination at time t-T where T is the transmission line delay
thus substituting equation \ref{eq:ch_1}  evaluated at time t-T, and z=L gives
%
\begin{equation} \label{eq:ch_3}
V\left(0,t\right)=Z_{c}I\left(0,t\right)+V\left(L,t-T\right)-Z_{c}I\left(L,t-T\right)
\end{equation}
%
A similar process leads to an equation for the z=L end of the transmission line
%
\begin{equation} \label{eq:ch_3b}
V\left(L,t\right)=-Z_{c}I\left(0,t\right)+V\left(0,t-T\right)+Z_{c}I\left(0,t-T\right)
\end{equation}
%
Equations \ref{eq:ch_2}, \ref{eq:ch_2b}  and \ref{eq:ch_3}, \ref{eq:ch_3b}  are implemented in the spice compatible circuit shown in Figure \ref{fig:spice_characteristics}
where the time delay is incorporated into the model by a perfectly matched ideal transmission line.

\begin{figure}[h]
\centering
\includegraphics[scale=0.7]{./Imgs/Method_of_characteristics_new.eps}
\caption{Spice compatible circuit implementing the method of characteristics}
\label{fig:spice_characteristics}
\end{figure}

In practice the Spice circuit which implements the calculation of the characteristic variables in equations \ref{eq:ch_1}
uses voltage controlled voltage sources. The following expression allows for the calculation of the characteristic variables in the +z direction from nodal voltage values at the $z=0$ termination.
%
\begin{equation} \label{eq:ch_4a}
2V^{+}\left(0,t\right)=V\left(0,t\right)+Z_{c}I\left(0,t\right)=2V\left(0,t\right)-\left( V\left(0,t\right) - Z_{c}I\left(0,t\right) \right)
\end{equation}
%
Similarly, the following expression allows for the calculation of the characteristic variables in the -z direction from nodal voltage values at the $z=L$ termination.
%
\begin{equation} \label{eq:ch_4b}
2V^{-}\left(L,t\right)=V\left(L,t\right)-Z_{c}I\left(L,t\right)=2V\left(L,t\right)-\left( V\left(L,t\right) + Z_{c}I\left(0,t\right) \right)
\end{equation}

\clearpage