\chapter{Multi-Conductor Transmission line theory} \label{MTLtheory}

In this chapter we present the multi-conductor transmission line 
equations which underly the models which are used in the SACAMOS project.
The fundamental approximations applied throughout this document 
are that the models produced will obtain quasi TEM propagation on cables and  
that the cables are uniform along their length \cite{PaulMTL}. The only
non-uniformity along the length of the transmission lines that will be considered involves twisting
of wire pairs.

\section{Multi-Conductor Transmission line equations} \label{MTLequa}

The electrical state of a multi-conductor cable bundle is described by the voltage 
and current on each of the conductors. We assume that a multi-conductor 
cable bundle (including the ground plane) carries no common mode current, i.e. 
at any cross section of the cable normal to the propagation direction, $z$ and 
at any time, $t$:
%
\begin{equation}
\sum_{n=1}^{n_{conductors}} I_n\left(z,t\right) = 0.
\end{equation}
%
We note that this assumption does not preclude the use of the current 
distribution on the conductors in 3D space from being used to provide an 
estimate of radiation from a cable bundle. The currents induced by 
external fields may be common mode on the cables but differential with respect to ground. 
Finally it is assumed that there is no field interaction between the circuit subsystems 
(i.e. the interconnect can be broken down into separate elements).

\begin{figure}[h]
\centering
\includegraphics[width=0.85 \textwidth]{./Imgs/MTLvoltagescurrents.eps}
\caption{A multi-conductor cable showing the differential mode voltages and currents.}
\label{fig:MTLVandI}
\end{figure}

The differential mode voltage, $V$, and current, $I$, on the conductors of a z directed 
multi-conductor cable bundle in the absence of sources may be described in the time 
domain by the following equations:

\begin{equation} \label{eq:MTLequa}
\begin{array}{rcl}
\frac{\partial}{\partial z}\left(V\left(z,t\right)\right) & = & -\left[R\right]\left(I\left(z,t\right)\right) - \left[L\right]\frac{\partial}{\partial t}\left(I\left(z,t\right)\right) \\[10pt]
\frac{\partial}{\partial z}\left(I\left(z,t\right)\right) & = & -\left[G\right]\left(V\left(z,t\right)\right) - \left[C\right]\frac{\partial}{\partial t}\left(V\left(z,t\right)\right).
\end{array}
\end{equation}

Here the cable bundle per-unit-length parameter matrices are inductance, $[L]$, 
capacitance, $[C]$, Resistance, $[R]$ and conductance, $[G]$.
We have implicitly assumed for the purposes of this initial discussion that 
these matrices are constant. We note that this is not the 
case in practice and the cable properties are frequency dependent which 
implies a convolution process in the time domain. The discussion of the 
generalisation of this theory to more realistic situations will be described in 
subsequent sections of this document. 

For an $n+1$ conductor system the 
vectors and matrices are of dimension n where one of the conductors is 
chosen as a reference conductor. Typically the ground plane would be 
chosen as this reference.
The voltage and current directions are shown in figure \ref{fig:MTLVandI}.

Values for these per-unit-length parameters can be obtained for some configurations by
analytic expressions, such as for cylindrical conductors that are widely spaced (separation
significantly larger than wire radius). In general these matrices can be obtained
by numerical methods for the capacitance matrix, from which the other parameters can also be obtained.
This involves the solution of the Laplace equation in the 2D cable cross section and can take account of dielectric materials inclusing dielectric losses. This is described in chapter \ref{Laplace}. 

The frequency domain equations corresponding to equation (\ref{eq:MTLequa}) are expressed as:

\begin{equation} \label{eq:MTLequa_freq}
\begin{array}{rcl}
\frac{\dif}{\dif z} \left(\tilde{V}\left(z\right)\right) & = & -\left[R\right]\left(\tilde{I}\left(z\right)\right) - j\omega\left[L\right]\left(\tilde{I}\left(z\right)\right) \\[10pt]
\frac{\dif}{\dif z} \left(\tilde{I}\left(z\right)\right) & = & -\left[G\right]\left(\tilde{V}\left(z\right)\right) - j\omega\left[C\right]\left(\tilde{V}\left(z\right)\right).
\end{array}
\end{equation}

The resistance and inductance terms are conveniently combined 
into a single frequency dependent series impedance, similarly the capacitance and conductance 
terms are combined into a single frequency dependent admittance term, this yields:

\begin{equation} \label{eq:MTLequa_fin}
\begin{array}{rcl}
\frac{\dif}{\dif z} \left(\tilde{V}\left(z\right)\right) & = & -\left[\tilde{Z}\right]\left(\tilde{I}\left(z\right)\right) \\[10pt]
\frac{\dif}{\dif z} \left(\tilde{I}\left(z\right)\right) & = & -\left[\tilde{Y}\right]\left(\tilde{V}\left(z\right)\right),
\end{array}
\end{equation}
where
\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}

This matrix formulation of the multi-conductor transmission
line equations in frequency domain will be used in the next chapters
to solve for currents and voltages on each conductor. It is also used
for generating the models with frequency dependent cable parameters. 

\cleardoublepage