\section{Domain decomposition} \label{Domain_decomposition}

The basis for computation of the systems being addressed in this work is the multi-conductor cable
bundle. A bundle consists of multiple individual cables. These cables may consist of shielded and/ or
un-shielded conductors perhaps with the addition of over-shielding and a ground plane (satellite
panel).

The spice cable bundle model decomposes the transmission line system into domains. The domains consist of an
external domain whose conductors are exposed to the external EM environment and internal domains
in which conductors are shielded from the external environment. These domains are assumed to be only weakly coupled through transfer impedance terms. 

The domain decomposition method along with the weak coupling assumption leads to a smaller system of coupled equations and has the advantage of removing one of the strong frequency dependent effects, that of transfer impedance coupling, from the main propagation model.

In an internal domain the natural reference conductor is the shield enclosing the domain. The model allows for multiple levels of
shielding hence an internal domain may contain conductors which are themselves the shields of other
domains as seen in Figure \ref{fig:cable_bundle}. In the case of imperfect shielding these domains are coupled through
transfer impedance and admittance terms. In this work we assume that the coupling through transfer
admittance may be neglected.

\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{./Imgs/cable_bundle.eps}
\caption{Illustration of a cable bundle built with multiple layers of shielding}
\label{fig:cable_bundle}
\end{figure}

When applying this decomposition technique the voltages on shielded cables are naturally referenced
to their respective shields and the current return is also assumed to be on the inside of the shield.
This can aid the propagation analysis and also the analysis of coupling from an external illuminating
field. The internal and external propagation problems can be coupled by transfer impedance and
admittance terms. We note that there may be more than one level of shielding in a particular cable
installation and we must be able to take this into account in our analysis.
Figure \ref{fig:coax_domain_decomp} shows a shielded cable with voltages and currents reference to their natural (local)
references in blue on the right hand side and with voltages and currents referenced to the ground
plane (global reference) on the right hand side.

\begin{figure}[h]
\centering
\includegraphics[scale=0.9]{./Imgs/coax_domain_decomp.eps}
\caption{A shielded cable with alternative voltage and current definitions}
\label{fig:coax_domain_decomp}
\end{figure}

We can define a domain decomposition matrix which transforms conductor voltages referenced to
their local reference (shield for shielded conductors, ground for external conductors) and voltages
referenced to the global reference (ground).
%
\begin{equation} \label{eq:dd_voltage}
\left(V_{local}\right)=\left[M_{V}\right] \left(V_{global}\right)
\end{equation}
%
For the coaxial cable pictured
%
\begin{equation} \label{eq:dd_v1a}
\begin{array}{c}
V_{w}=V_{2}-V_{1} \\[10pt]
V_{s}=V_{1}
\end{array}
\end{equation}
%
therefore
%
\begin{equation}
\left(\begin{array}{c} V_{w} \\ V_{s} \end{array}\right)
=\left[\begin{array}{cc} -1 & 1 \\ 1 & 0 \end{array} \right] 
\left(\begin{array}{c} V_{1} \\ V_{2} \end{array}\right)
\end{equation}
%
Similarly we can define a domain decomposition matrix which transforms conductor currents whose
return path is their local reference (shield for shielded conductors, ground for external conductors)
and total conductor currents with reference to the global ground.
%
\begin{equation} \label{eq:dd_current}
\left(I_{local}\right)=\left[M_{I}\right] \left(I_{global}\right)
\end{equation}
%
For the coaxial cable pictured
%
\begin{equation}
\begin{array}{c}
I_{w}=I_{2} \\[10pt]
I_{s}=I_{1}+I_{2}
\end{array}
\end{equation}
%
therefore
%
\begin{equation}
\left(\begin{array}{c} I_{w} \\ I_{s} \end{array}\right)
=\left[\begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array} \right] 
\left(\begin{array}{c} I_{1} \\ I_{2} \end{array}\right)
\end{equation}

Figure \ref{fig:cable_bundle} shows a complex cable bundle with multiple levels of shielding. In this case there are
multiple local voltage references. The transformation matrices for these complex situations can be
defined in a similar manner as for the coaxial cable.


The domain decomposition process is achieved by a matrix multiplication. This may be implemented
in a Spice model using controlled voltage and current sources.
In the example of the coaxial cable above a ground plane there are three conductors. We define the
global voltage variables with respect to the ground plane as
%
\begin{equation} 
\begin{array}{c}
V_{1}=V_{shield} \\[10pt]
V_{2}=W_{wire}
\end{array}
\end{equation}
%
The model is divided into two domains, the inner domain consisting of the inner wire and the shield
(domain 1) and the external domain consisting of the outer shield and the ground (domain 2). Each of
the domains has a single conductor in addition to its reference. The domain based voltage variables
are
%
\begin{equation} \label{eq:dd_v1b}
\begin{array}{c}
V_{d1\_c1}=V_{wire}-V_{shield}= V_{2}-V_{1} \\[10pt]
V_{d2\_c1}=W_{shield}= V_{1}
\end{array}
\end{equation}
%
The equation relating the global voltage variables and domain voltage variables is
%
\begin{equation}
\left(\begin{array}{c} V_{d1\_c1} \\[10pt] V_{d2\_c1} \end{array}\right)
=\left[\begin{array}{cc} -1 & 1 \\[10pt] 1 & 0 \end{array} \right] 
\left(\begin{array}{c} V_{1} \\[10pt] V_{2} \end{array}\right)
=\left[M_{V}\right] 
\left(\begin{array}{c} V_{1} \\[10pt] V_{2} \end{array}\right)
\end{equation}
%
Inverting this equation gives
%
\begin{equation}\label{dd_v_eqn}
\left(\begin{array}{c} V_{1} \\[10pt] V_{2} \end{array}\right)
=\left[\begin{array}{cc} 1 & 1 \\[10pt] 1 & 0 \end{array} \right] 
\left(\begin{array}{c} V_{d1\_c1} \\[10pt] V_{d2\_c1} \end{array}\right)
=\left[M_{V}\right]^{-1}
\left(\begin{array}{c} V_{d1\_c1} \\[10pt] V_{d2\_c1} \end{array}\right)
\end{equation}
%
Similarly the equation relating the global current variables and domain current variables is
%
\begin{equation}\label{dd_i_eqn}
\left(\begin{array}{c} I_{d1\_c1} \\[10pt] I_{d2\_c1} \end{array}\right)
=\left[\begin{array}{cc} 0 & 1 \\[10pt] 1 & 1 \end{array} \right] 
\left(\begin{array}{c} I_{1} \\[10pt] I_{2} \end{array}\right)
=\left[M_{I}\right] 
\left(\begin{array}{c} I_{1} \\[10pt] I_{2} \end{array}\right)
\end{equation}
%
Figure \ref{fig:spice_domain_decomposition} shows a Spice circuit model which implememnts equations \ref{dd_v_eqn} and \ref{dd_i_eqn}. Note that the voltage sources on the left hand side of the figure have a value of zero volts, they are only required as Spice uses such voltage sources as current sensing elements for the current controlled voltage sources.

\begin{figure}[h]
\centering
\includegraphics[scale=0.7]{./Imgs/Domain_decomposition_example.eps}
\caption{Spice circuit model which implements domain decomposition for a shielded cable}
\label{fig:spice_domain_decomposition}
\end{figure}

\clearpage