\chapter{The Spice Cable Model} \label{spice_cable_model}

In this chapter we describe how the transmission line theory discussed in the previous chapter may be implemented using circuit elements and implemented as a Spice model.

\begin{figure}[h]
\centering
\includegraphics[scale=0.60]{./Imgs/scenario.eps}
\caption{Multi-conductor cable modelling scenario with shielded conductors and incident field excitation}
\label{fig:scenario}
\end{figure}

The scenario which we are aiming to model is illustrated in Figure \ref{fig:scenario} The termination circuits are
arbitrary and to be specified by the user. The proposed Spice model being considered here concerns
the transmission line only, though we note that connectors, ground plane bonding straps, pigtails etc could
be modelled as short lengths of transmission line cascaded with the main transmission line.

The model has been developed so as to include frequency dependent cable parameters arising from the finite conductivity of conductors, frequency dependent dielectrics and frequencuy dependent transfer impedance effects. This is a challenging 
requirement as in general the modal decomposition used in the analysis of the previous chapter is a function of frequency.
It is shown in section \ref{Propagation_Correction} that if the d.c. resistance of the conductors is lumped at the cable terminations then the modal decomposition becomes a weak function of frequency. Simulations show that the frequency dependence of the modal decomposition can be neglected and the decomposition based on lossless high frequency inductance and capacitance matrices can be used provided that the propagation of the individual modes is corrected for the propagation loss. 

The spice model of the scenario depicted is based on a number of different parts these are:

\begin{itemize}
\item Domain decomposition: we model shielded conductor systems as separate domains, in
which the electrical quantities are referenced the shielding conductor, plus the external
domain consisting of the ground plane, unshielded conductors and the exterior surface of
shields.

\item Modal decomposition: Signal propagation on the multi-conductor system within each domain
is analysed by a modal decomposition method i.e. we diagonalise the transmission line
equations within each domain and propagate the modes on 'modal transmission lines'

\item Method of characteristics. The propagation of each mode is treated using the method of
characteristics \cite{PaulMTL} i.e. modes are propagated in the +z and -z directions and the state of the
transmission line is the combination of these +z and -z propagating modes.

\item Frequency dependent propagation characteristics are dealt with using an approximate
model in which the d.c. resistance is extracted and lumped at
the ends of each transmission line and the frequency dependence of the propagation is
dealt with by applying a frequency dependent propagation correction to the +z and -z propagating modes at the
ends of the transmission lines.

\item Incident field excitation is assumed to be a plane wave, in this model appropriate controlled
source terms lumped at the cable terminations take account of the incident field excitation.

\item Coupling through imperfect shields is modelled with transfer impedance terms and
implemented in a weak form i.e. source and victim conductors are identified and one way
coupling is implemented using appropriate controlled source terms

\item The twisted pair model is derived on the assumption that there is no coupling between the
differential mode and the other conductors within the bundle (or the shield for a shielded twisted pair) 
although coupling to the common mode is included. This model does however allow coupling from the common mode
to the differential mode and vice versa, through unbalanced terminations.
\end{itemize}

In the subsequent sections each of these stages is treated in turn and theoretical model
derivations and associated circuit diagrams will be presented.

\clearpage