\chapter{Cable models available} \label{cable_types}

In this section each of the cable types available will be described and any assumptions regarding the specific cable models will be described.

The cable models available are

\begin{enumerate}
\item Frequency Dependent Cylindrical conductor with dielectric
\item Frequency Dependent Coaxial cable with transfer impedance and shield surface impedance loss
\item Frequency Dependent Twinax cable with transfer impedance and shield surface impedance loss
\item Frequency Dependent Twisted pair
\item Frequency Dependent Shielded twisted pair with transfer impedance and shield surface impedance loss
\item Frequency Dependent Spacewire with transfer impedance and shield surface impedance loss
\item Frequency Dependent Overshield with transfer impedance and surface impedance loss
\item Frequency Dependent flex cable
\item D connector
\end{enumerate}

For each cable type the method of calculation for the inductance and capacitance matrices for any internal domains will be outlined.
The domain decomposition matrices which relate local domain based voltages and currents to global condcutor based voltages and currents will be 
shown. These domain decomposition matrices for cables will be augmented to include an external (reference) conductor which allows the cable to carry a common mode current.

\clearpage
\section{Frequency Dependent Cylindrical conductor with dielectric} \label{FD_Cylindrical_conductor_with_dielectric}

\begin{figure}[h]
\centering
\includegraphics[scale=1]{../USER_GUIDE/Imgs/cylindrical_with_dielectric.eps}
\caption{Frequency dependent cylindrical conductor with dielectric}
\label{fig:FD_cylindrical_with dielectric}
\end{figure}

\clearpage

The cylindrical conductor has no shielded conductors so no inductance/ capacitance matrix calculation has to be performed for internal domains.

The domain decomposition voltage matrix, $M_V$ for the single conductor is given by
%
\begin{equation} \label{eq:cylindrical_MV2}
\left( \begin{array}{c}
V_{d1\_c1} \\
V_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cc}
1 & -1    \\
0 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
V_1 \\
V_0 \\
\end{array} \right)
\end{equation}

The domain decomposition current matrix, $M_I$ for the single conductor is given by
%
\begin{equation} \label{eq:cylindrical_MI}
\left( \begin{array}{c}
I_{d1\_c1} \\
I_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cc}
1 &  0    \\
1 &  1    \\
\end{array} \right]
\left( \begin{array}{c}
I_1 \\
I_0 \\
\end{array} \right)
\end{equation}

\section{Frequency Dependent Coaxial cable with transfer impedance and shield surface impedance loss} \label{FD_Coax2}

\begin{figure}[h]
\centering
\includegraphics[scale=1]{../USER_GUIDE/Imgs/Coax2.eps}
\caption{Frequency Dependent Coaxial cable - high frequency model including shield surface impedance loss}
\label{fig:FD_coax2}    
\end{figure}

The coaxial cable has a single internal domain in which a single mode propagates. The inductance and capacitance of the coaxial mode are
calculated as
%
\begin{equation} \label{eq:coax_L}
L=\frac{\mu_0}{2\pi}ln\left(\frac{r_s}{r_w}\right)  H/m
\end{equation}
%
\begin{equation} \label{eq:coax_C}
C\left(j\omega)\right) = \frac{2 \pi \epsilon \left( j \omega \right)}{ ln \left( \frac{r_s}{r_w} \right) }  F/m
\end{equation}
%
where $r_s$ is the shield radius, $r_w$ is the inner wire radius and $eps \left(j\omega)\right)$ is the permittivity of the dielectric filling the space between the shield and the inner wire. This dielectric may be frequency dependent and in this case the capacitance will also be frequency dependent.

The domain decomposition voltage matrix, $M_V$ for the coaxial cable is given by
%
\begin{equation} \label{eq:coax_MV1}
\left( \begin{array}{c}
V_{d1\_c1} \\
V_{d2\_c1} \\
V_{ref}\\
\end{array} \right)
=\left[ \begin{array}{ccc}
1 & -1 &  0   \\
0 &  1 & -1  \\
0 &  0 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
V_1 \\
V_2 \\
V_0 \\
\end{array} \right)
\end{equation}

The domain decomposition current matrix, $M_I$ for the coaxial cable is given by
%
\begin{equation} \label{eq:coax_MI1}
\left( \begin{array}{c}
I_{d1\_c1} \\
I_{d2\_c1} \\
I_{ref}\\
\end{array} \right)
=\left[ \begin{array}{ccc}
1 &  0 &  0   \\
1 &  1 &  0  \\
1 &  1 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
I_1 \\
I_2 \\
I_0 \\
\end{array} \right)
\end{equation}

\clearpage

\section{Frequency Dependent Twinax cable with transfer impedance and shield surface impedance loss} \label{ZT_FD_Twinax2}

\begin{figure}[h]
\centering
\includegraphics[scale=1]{../USER_GUIDE/Imgs/ZT_FD_Twinax.eps}
\caption{Frequency dependent twinaxial cable with transfer impedance - high frequency model including shield surface impedance loss}
\label{fig:ZT_FD_Twinax}
\end{figure}

The twinax cable has a single internal domain in which two modes propagates. The inductance and capacitance matrices for the internal domain are
calculated using the Laplace solver.

The domain decomposition voltage matrix, $M_V$ for the twinax cable is given by
%
\begin{equation} \label{eq:coax_MV8}
\left( \begin{array}{c}
V_{d1\_c1} \\
V_{d1\_c2} \\
V_{d2\_c1} \\
V_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cccc}
1 & 0 & -1 &  0   \\
0 & 1 & -1 &  0  \\
0 & 0 &  1 & -1   \\
0 & 0 &  0 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
V_1 \\
V_2 \\
V_3 \\
V_0 \\
\end{array} \right)
\end{equation}

The domain decomposition current matrix, $M_I$ for the twinax cable is given by
%
\begin{equation} \label{eq:coax_MI6}
\left( \begin{array}{c}
I_{d1\_c1} \\
I_{d1\_c2} \\
I_{d2\_c1} \\
I_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cccc}
1 & 0 & 0 &  0   \\
0 & 1 & 0 &  0  \\
1 & 1 & 1 &  0   \\
1 & 1 & 1 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
I_1 \\
I_2 \\
I_3 \\
I_0 \\
\end{array} \right)
\end{equation}

\clearpage

\section{Frequency Dependent Twisted pair cable} \label{FD_Twisted_pair}

\begin{figure}[h]
\centering
\includegraphics[scale=1]{../USER_GUIDE/Imgs/FD_Twisted_pair.eps}
\caption{Frequency Dependent Twisted pair cable}
\label{fig:FD_Twisted_pair}
\end{figure}

The twisted pair cable model has a single internal domain in which the differential mode propagates. 
The inductance and capacitance of the differential mode is calculated for the two conductors in isolation as described in section \ref{twisted_pair_model}.

The domain decomposition voltage matrix, $M_V$ for the twisted pair cable is given by
%
\begin{equation} \label{eq:coax_MV7}
\left( \begin{array}{c}
V_{d1\_c1} \\
V_{d2\_c1} \\
V_{ref}\\
\end{array} \right)
=\left[ \begin{array}{ccc}
1 & -1 &  0   \\
\frac{1}{2} &  \frac{1}{2} & -1  \\
0 &  0 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
V_1 \\
V_2 \\
V_0 \\
\end{array} \right)
\end{equation}

The domain decomposition current matrix, $M_I$ for the twisted pair cable is given by
%
\begin{equation} \label{eq:coax_MI5}
\left( \begin{array}{c}
I_{d1\_c1} \\
I_{d2\_c1} \\
I_{ref}\\
\end{array} \right)
=\left[ \begin{array}{ccc}
\frac{1}{2} &  -\frac{1}{2}  &  0   \\
1 &  1 &  0  \\
1 &  1 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
I_1 \\
I_2 \\
I_0 \\
\end{array} \right)
\end{equation}

\clearpage


\section{Frequency Dependent Shielded twisted pair with transfer impedance and shield surface impedance loss} \label{Shielded_twisted_pair}

\begin{figure}[h]
\centering
\includegraphics[scale=1]{../USER_GUIDE/Imgs/ZT_FD_Shielded_twisted_pair.eps}
\caption{Frequency dependent shielded twisted pair cable with transfer impedance - high frequency model including shield surface impedance loss}
\label{fig:ZT_FD_Shielded_twisted_pair}
\end{figure}

The shielded twisted pair cable model has two internal domains, one for the differential mode of the twisted pair and one for the common mode of the twisted pair with the shield as return conductor. The inductance and capacitance matrices for the internal domains are
calculated using the Laplace solver as described in section \ref{twisted_pair_model}.

The domain decomposition voltage matrix, $M_V$ for the shielded twisted pair cable is given by
%
\begin{equation} \label{eq:coax_MV6}
\left( \begin{array}{c}
V_{d1\_c1} \\
V_{d2\_c1} \\
V_{d3\_c1} \\
V_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cccc}
1 & -1 & 0 &  0  \\
\frac{1}{2} & \frac{1}{2}  & -1 &  0   \\
 & 0 &  1 & -1   \\
0 & 0 &  0 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
V_1 \\
V_2 \\
V_3 \\
V_0 \\
\end{array} \right)
\end{equation}

The domain decomposition current matrix, $M_I$ for the shielded twisted pair cable is given by
%
\begin{equation} \label{eq:coax_MI4}
\left( \begin{array}{c}
I_{d1\_c1} \\
I_{d2\_c1} \\
I_{d3\_c1} \\
I_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cccc}
\frac{1}{2} & -\frac{1}{2}  & 0 &  0   \\
1 & 1 & 0 &  0  \\
1 & 1 & 1 &  0   \\
1 & 1 & 1 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
I_1 \\
I_2 \\
I_3 \\
I_0 \\
\end{array} \right)
\end{equation}

\clearpage

\section{Frequency dependent spacewire with transfer impedance - high frequency model including shield surface impedance loss} \label{ZT_FD_Spacewire2}

\begin{figure}[h]
\centering
\includegraphics[scale=0.75]{../USER_GUIDE/Imgs/ZT_FD_spacewire.eps}
\caption{Spacewire}
\label{fig:Spacewire}
\end{figure}

Spacewire consists of 13 conductors configures as four shielded twisted pairs within an overshield. There are 5 shields, four surrounding the twisted pairs and the domain consisting of the shields surrouding each of the twisted pairs and the overshield. As each of the shielded twisted pairs is modelled using two domains (differential and common mode for the twisted pair) then there is a total of 9 internal domains in the spacewire model.

The inductance and capacitance for the shielded twisted pair differential and common mode are calculated as for the shielded twisted pair model described in section \ref{Shielded_twisted_pair}. The $4 \times 4$ inductance and capacitance matrices for the domain consisting of the twisted pair shields and the overshield are calculated using the Laplace solver.

The domain decomposition voltage matrix, $M_V$ for the spacewire cable is given by
%
\begin{equation} \label{eq:coax_MV5}
\left( \begin{array}{c}
V_{d1\_c1} \\
V_{d2\_c1} \\
V_{d3\_c1} \\
V_{d4\_c1} \\
V_{d5\_c1} \\
V_{d6\_c1} \\
V_{d7\_c1} \\
V_{d8\_c1} \\
V_{d9\_c1} \\
V_{d9\_c2} \\
V_{d9\_c3} \\
V_{d9\_c4} \\
V_{d10\_c1} \\
V_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cccccccccccccc}
       1     &     -1      &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &   0 &  0      \\
 \frac{1}{2} & \frac{1}{2} &  0 &  0 &  0 &  0 &  0 &  0 & -1 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 &       1     &     -1      &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 & \frac{1}{2} & \frac{1}{2} &  0 &  0 &  0 &  0 &  0 & -1 &  0 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 &       1     &     -1      &  0 &  0 &  0 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 & \frac{1}{2} & \frac{1}{2} &  0 &  0 &  0 &  0 & -1 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 &       1     &     -1      &  0 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 & \frac{1}{2} & \frac{1}{2} &  0 &  0 &  0 & -1 &   0 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &  0 &  0 &  -1 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &  0 &  -1 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &  -1 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  -1 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &   1 & -1      \\
  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &   0 &  1      \\
\end{array} \right]
\left( \begin{array}{c}
V_1 \\
V_2 \\
V_3 \\
V_4 \\
V_5 \\
V_6 \\
V_7 \\
V_8 \\
V_9 \\
V_10 \\
V_11 \\
V_12 \\
V_13 \\
V_0 \\
\end{array} \right)
\end{equation}

The domain decomposition voltage matrix, $M_I$ for the spacewire cable is given by
%
\begin{equation} \label{eq:coax_MI3}
\left( \begin{array}{c}
I_{d1\_c1} \\
I_{d2\_c1} \\
I_{d3\_c1} \\
I_{d4\_c1} \\
I_{d5\_c1} \\
I_{d6\_c1} \\
I_{d7\_c1} \\
I_{d8\_c1} \\
I_{d9\_c1} \\
I_{d9\_c2} \\
I_{d9\_c3} \\
I_{d9\_c4} \\
I_{d10\_c1} \\
I_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cccccccccccccc}
 \frac{1}{2} & -\frac{1}{2}&  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &   0 &  0      \\
       1     &      1      &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 & \frac{1}{2} & -\frac{1}{2}&  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 &       1     &      1      &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 & \frac{1}{2} & -\frac{1}{2}&  0 &  0 &  0 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 &       1     &      1      &  0 &  0 &  0 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 & \frac{1}{2} & -\frac{1}{2}&  0 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 &       1     &      1      &  0 &  0 &  0 &  0 &   0 &  0      \\
  1 &  1 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &  0 &  0 &   0 &  0      \\
  0 &  0 &  1 &  1 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &   0 &  0      \\
  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &   0 &  0      \\
  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &   1 &  0      \\
  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &  1 &   1 &  1      \\
\end{array} \right]
\left( \begin{array}{c}
I_1 \\
I_2 \\
I_3 \\
I_4 \\
I_5 \\
I_6 \\
I_7 \\
I_8 \\
I_9 \\
I_10 \\
I_11 \\
I_12 \\
I_13 \\
I_0 \\
\end{array} \right)
\end{equation}

\clearpage

\section{Frequency Dependent Overshield with transfer impedance and surface impedance loss} \label{over_shield}

\begin{figure}[h]
\centering
\includegraphics[scale=1]{../USER_GUIDE/Imgs/overshield.eps}
\caption{Over shield}
\label{fig:Overshield}
\end{figure}

The overshield is a single conductor which may enclose other conductors within a cable bundle. As the internal conductor structure is not known at the cable model building stage there is no internal domain information at this stage. For the same reason the domain decomposition matrices only describe the external domain. Information relating to any enclosed conductors will be built at the cable bundle building stage.

The domain decomposition voltage matrix, $M_V$ for the overshield is given by
%
\begin{equation} \label{eq:cylindrical_MV}
\left( \begin{array}{c}
V_{d1\_c1} \\
V_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cc}
1 & -1    \\
0 &  1   \\
\end{array} \right]
\left( \begin{array}{c}
V_1 \\
V_0 \\
\end{array} \right)
\end{equation}

The domain decomposition current matrix, $M_I$ for the overshield is given by
%
\begin{equation} \label{eq:cylindrical_MI2}
\left( \begin{array}{c}
I_{d1\_c1} \\
I_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cc}
1 &  0    \\
1 &  1    \\
\end{array} \right]
\left( \begin{array}{c}
I_1 \\
I_0 \\
\end{array} \right)
\end{equation}


\clearpage

\section{Flex cable} \label{flex_cable}

\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{../USER_GUIDE/Imgs/ML_flex_cable.eps}
\caption{flex cable}
\label{fig:flex_cable}
\end{figure}

A flex cable may have an arbirary number of conductors, $N_c$. The dimension of the domain decomposition matrices is $N_c+1$.
The domain decomposition voltage matrix, $M_V$ for a flex cable is given by
%
\begin{equation} \label{eq:coax_MV4}
\left( \begin{array}{c}
V_{d1\_c1}  \\
V_{d1\_c2}  \\
\vdots      \\
V_{d1\_cNc} \\
V_{ref}\\
\end{array} \right)
=\left[ \begin{array}{ccccc}
  1     &     0   & \cdots &     0   &  -1    \\
  0     &     1   & \cdots &     0   &  -1    \\
 \vdots & \vdots  & \ddots & \vdots  & \vdots \\
  0     &     0   & \cdots &     1   &  -1    \\
  0     &     0   & \cdots &     0   &   1    \\
\end{array} \right]
\left( \begin{array}{c}
V_1    \\
V_2    \\
\vdots \\
V_{Nc} \\
V_0    \\
\end{array} \right)
\end{equation}
%
The domain decomposition current matrix, $M_I$ for a flex cable is given by
%
\begin{equation} \label{eq:coax_MV3}
\left( \begin{array}{c}
I_{d1\_c1}  \\
I_{d1\_c2}  \\
\vdots      \\
I_{d1\_cNc} \\
I_{ref}\\
\end{array} \right)
=\left[ \begin{array}{ccccc}
  1     &     0   & \cdots &     0   &  0     \\
  0     &     1   & \cdots &     0   &  0     \\
 \vdots & \vdots  & \ddots & \vdots  & \vdots \\
  0     &     0   & \cdots &     1   &  0     \\
  1     &     1   & \cdots &     1   &  1     \\
\end{array} \right]
\left( \begin{array}{c}
I_1    \\
I_2    \\
\vdots \\
I_{Nc}   \\
I_0    \\
\end{array} \right)
\end{equation}
%

\clearpage
\section{D connector} \label{Dconnector}

\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{../USER_GUIDE/Imgs/Dconnector.eps}
\caption{D connector}
\label{fig:Dconnector}
\end{figure}

A D connector may have an arbirary number of conductors, $N_c$ consisting of $N_c-1$ internal conductor pins and the outer shell. 

The internal conductor pins are all shielded by the connector shell therefore there is a single internal domain. The inductance and capacitance matrices for the internal domain are calculated using the Laplace solver.

The dimension of the domain decomposition matrices is $N_c+1$
The domain decomposition voltage matrix, $M_V$ for a flex cable is given by
%
\begin{equation} \label{eq:coax_MV2}
\left( \begin{array}{c}
V_{d1\_c1}  \\
V_{d1\_c2}  \\
\vdots      \\
V_{d1\_cNc-1} \\
V_{d2\_c1} \\
V_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cccccc}
  1     &     0   & \cdots &     0   &  -1    &    0    \\
  0     &     1   & \cdots &     0   &  -1    &    0    \\
 \vdots & \vdots  & \ddots & \vdots  & \vdots & \vdots   \\
  0     &     0   & \cdots &     1   &  -1    &    0    \\
  0     &     0   & \cdots &     0   &   1    &   -1    \\
  0     &     0   & \cdots &     0   &   0    &    1    \\
\end{array} \right]
\left( \begin{array}{c}
V_1    \\
V_2    \\
\vdots \\
V_{Nc-1} \\
V_{Nc} \\
V_0    \\
\end{array} \right)
\end{equation}
%
The domain decomposition voltage matrix, $M_I$ for a flex cable is given by
%
\begin{equation} \label{eq:coax_MI2}
\left( \begin{array}{c}
I_{d1\_c1}  \\
I_{d1\_c2}  \\
\vdots      \\
I_{d1\_cNc-1} \\
I_{d2\_c1} \\
I_{ref}\\
\end{array} \right)
=\left[ \begin{array}{cccccc}
  1     &     0   & \cdots &     0   &   0    &   0    \\
  0     &     1   & \cdots &     0   &   0    &   0    \\
 \vdots & \vdots  & \ddots & \vdots  & \vdots & \vdots   \\
  0     &     0   & \cdots &     1   &   0    &   0    \\
  0     &     0   & \cdots &     0   &   1    &   0    \\
  1     &     1   & \cdots &     1   &   1    &   1    \\
\end{array} \right]
\left( \begin{array}{c}
I_1    \\
I_2    \\
\vdots \\
I_{Nc-1} \\
I_{Nc} \\
I_0    \\
\end{array} \right)
\end{equation}
%



\clearpage
