\section{Solution to the Multi-Conductor Transmission line equations} \label{solMTL}

The SACAMOS software includes the ability to set up a validation test for the Spice transmission line models. In the validation test
a resistive termination circuit with series voltage sources is specified and the solution for conductor voltages calculated with the
Spice transmission line model can be compared with the solution obtained using the frequency domain analytic solution. The frequency domain analytic
solution is calculated on the basis of the full dimension multi-conductor transmission line equations (i.e. no domain decomposition) and therefore includes the self consistent model of transfer impedance coupling. All the parameters in the analytic solution implemented are frequency dependent. The comparison between the Spice solution and the analytic solution therfore allows the effect of the approximations made in the derivation of the Spice model to be tested.

This section outlines the solution of the transmission line equations defined in section \ref{MTLequa}. A more complete discussion of the multi-conductor transmission line equations may be found in \cite{PaulTML}.

\subsection{Solution of the transmission line equations by modal decomposition} \label{sec:TL}
The impedance and admittance matrices are given by:
%
\begin{equation} \label{eq:ZandY}
\begin{array}{rcl}
\left[\tilde{Z}\right] & = & \left[R\right]+j\omega\left[L\right] \\[10pt]
\left[\tilde{Y}\right] & = & \left[G\right]+j\omega\left[C\right].
\end{array}
\end{equation}
%
We can derive a frequency domain solution to 
the transmission line equations using a modal analysis. Equation 
\ref{eq:MTLequa_fin} can be re-written as an uncoupled second 
order equation in the following ways:
%
\begin{equation} \label{eq:MTLsecondorder}
\begin{array}{rcl}
\frac{\dif^2}{\dif z^2} \left(\tilde{V}\left(z\right)\right) & = & \left[\tilde{Z}\right]\left[\tilde{Y}\right]\left(\tilde{V}\left(z\right)\right) \\[10pt]
\frac{\dif^2}{\dif z^2} \left(\tilde{I}\left(z\right)\right) & = & \left[\tilde{Y}\right]\left[\tilde{Z}\right]\left(\tilde{I}\left(z\right)\right).
\end{array}
\end{equation}
%
A change of variables to modal quantities via a similarity 
transformation allows the decoupling (diagonalisation) of the matrix system. 
The conductor voltages and currents are related to their 
corresponding modal quantities by:
%
\begin{equation} \label{eq:ModalV}
\begin{array}{rcl}
\tilde{V}\left(z\right) & = & \left[\tilde{T}_V\right]\left(\tilde{V}_m\left(z\right)\right) \\[10pt]
\tilde{I}\left(z\right) & = & \left[\tilde{T}_I\right]\left(\tilde{I}_m\left(z\right)\right).
\end{array}
\end{equation}
%
The voltage modal transformation matrix, $\left[T_V \right]$ 
diagonalises the first equation given in (\ref{eq:MTLsecondorder}):
%
\begin{equation} \label{eq:VoltageDiagonal}
\begin{array}{rcl}
\frac{\dif^2}{\dif z^2} \left(\tilde{V}_m\left(z\right)\right) & = & \left[\tilde{T}_V\right]^{-1}\left[\tilde{Z}\right]\left[\tilde{Y}\right]\left[\tilde{T}_V\right]\left(\tilde{V}_m\left(z\right)\right) \\[10pt]
 & = & \left[\gamma^2\right]\left(\tilde{V}_m\left(z\right)\right),
\end{array}
\end{equation}
%
where $\left[\gamma^2\right]$ is a diagonal matrix with 
elements whose values are the squares of the mode propagation 
constants. The elements of the diagonal matrix $\left[\gamma^2\right]$ are found as the eigenvalues of the $[Z][Y]$ product and the columns of the matrix $\left[\tilde{T}_V\right]$ are the corresponding eigenvectors. (We note that when there are repeated eigenvalues which may originate from the symmetry of the cable cross section, then the diagonalisation is not unique. The eigenvectors corresponding to these eiganvalues are only required to span a certain subspace of the original matrix. In this situation the eigenvector solver for general complex matrices in Eispack (\cite{Eispack}) may on rare occasions return the same eigenvector for more than one eigenvalue. In order to avoid this in practice we perturb the matrix by a very small amount in a manner which prevents this degeneracy).

In the lossless case the impedance matrices and admittance matrices 
are imaginary and symmetric, and the $\left[Z\right]\left[Y\right]$ product is real and symmetric. 
The modal transformation matrices are real and are independent of 
frequency and the diagonal $\left[\gamma^2\right]$ matrix has negative, real values of the 
form $-\frac{\omega^2}{v^2}$, where $v$ is the mode velocity. Modal decomposition for a lossless transmission line will
be important in the derivation of the Spice multi-conductor transmission line model derived in section \ref{spice_cable_model}.

The solution for the modal voltages at any point $z$ on the 
multi-conductor transmission line may be written in terms of 
modes travelling in the $+z$ and $- z$ directions as:
%
\begin{equation}
\tilde{V}_m\left(z\right) = \left[ \rm e^{ - \gamma z} \right]\left(\tilde{V}^+_m\right) + \left[ \rm e^{+\gamma z}\right]\left(\tilde{V}^-_m\right).
\end{equation}
%
The general solution for the actual conductor voltages is then:
%
\begin{equation}
\tilde{V}\left(z\right) = \left[\tilde{T}_V\right]\left(\left[\rm e^{- \gamma z}\right]\left(\tilde{V}^+_m\right) + \left[\rm e^{+\gamma z}\right]\left(\tilde{V}^-_m\right)\right).
\end{equation}

Similarly, the current modal transformation matrix [TI] diagonalises 
the second equation in (\ref{eq:MTLsecondorder}):
%
\begin{equation} \label{eq:CurrentDiagonal}
\begin{array}{rcl}
\frac{\dif^2}{\dif z^2} \left(\tilde{I}_m\left(z\right)\right) & = & \left[\tilde{T}_I\right]^{-1}\left[\tilde{Y}\right]\left[\tilde{Z}\right]\left[\tilde{T}_I\right]\left(\tilde{I}_m\left(z\right)\right) \\[10pt]
 & = & \left[\gamma^2\right]\left(\tilde{I}_m\left(z\right)\right).
\end{array}
\end{equation}

It can be shown \cite{PaulMTL} that the voltage and current modal 
transformation matrices can be related by:
%
\begin{equation}
\left[\tilde{T}_I\right]^T = \left[\tilde{T}_V\right]^{-1}.
\end{equation}
%
The solution for the modal currents at any point $z$
on the multi-conductor transmission line may be written 
in terms of modes travelling in the $+z$ and $-z$ directions as:
%
\begin{equation}
\tilde{I}_m\left(z\right) = \left[ \rm e^{-\gamma z} \right]\left(\tilde{I}^+_m\right) - \left[\rm e^{+\gamma z} \right]\left(\tilde{I}^-_m\right).
\end{equation}
%
The general solution for the actual conductor currents is then:
\begin{equation} \label{eq:CurrentSol}
\tilde{I}\left(z\right) = \left[\tilde{T}_I \right]\left(\left[\rm e^{- \gamma z} \right]\left(\tilde{I}^+_m\right) - \left[\rm e^{+\gamma z}\right]\left(\tilde{I}^-_m\right)\right).
\end{equation}
%
The conductor voltage vector may be written in terms of the 
modal current vectors as:
%
\begin{equation} \label{eq:VoltageSol}
\tilde{V}\left(z\right) = \left[\tilde{Z}_C\right]\left[\tilde{T}_I\right] \left(\left[\rm e^{- \gamma z}\right] \left(\tilde{I}^+_m\right) + \left[\rm e^{+\gamma z}\right] \left(\tilde{I}^-_m\right) \right),
\end{equation}
%
where $\left[Z_C\right]$ is the characteristic impedance matrix defined as:
%
\begin{equation}
\left[\tilde{Z}_C\right] = \left[\tilde{Y}\right]^{-1} \left[\tilde{T}_I\right] \left[\gamma \right] \left[\tilde{T}_I\right]^{-1}
 = \left[\tilde{Z}\right] \left[\tilde{T}_I\right] \left[\gamma \right]^{-1} \left[\tilde{T}_I\right]^{-1}.
\end{equation}


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