\chapter{Creating a Cable Model} \label{creating_a_cable_model}

\section{Introduction}

This chapter describes the creation of a cable model from specifications. The methods by which frequency dependent cable parameters (relative permittivity, finite conductivity loss models and transfer impedance models) are specified are described before detailing the specifications for each of the available cable types in turn. An example for each cable type is provided.

\section{Cable Types} \label{Cable_types}

Models of the following cable types have been developed:

\begin{enumerate}

\item Cylindrical conductor with dielectric
\item Coaxial cable with transfer impedance and shield surface impedance loss
\item Twinax cable with transfer impedance and shield surface impedance loss
\item Twisted pair
\item Shielded twisted pair with transfer impedance and shield surface impedance loss
\item Spacewire cable with transfer impedance and shield surface impedance loss
\item Overshield with transfer impedance and shield surface impedance loss
\item flex cable

\item D connector
\end{enumerate}

All of the cable types with the exception of the D connector allow the inclusion of frequency dependent dielectrics and finite conductivity models of conductors.

The typical values required to specify each of these cable types is described in the following sub-sections. Each cable type is illustrated in a figure. The figure shows the conductor numbering used in the software e.g. conductor number 1 of a coaxial cable is the inner conductor and conductor number 2 is the shield.

In addition to these cable types a perfectly conducting ground plane is available when building a cable bundle.

\subsection{Frequency dependent models} \label{FD_cable_models}

Many of the cable models available have frequency dependent properties. The frequency dependent properties arise from frequency dependent permittivity of dielectrics, from the finite conductivity of conductors and from the frequency dependence of transfer impedance. The frequency dependent models are described in detail in the Theory Manual \cite{Theory_manual} sections 3.6 and chapter 5.

\subsubsection{Frequency dependent rational function dielectric models} \label{FD_dielectric_models}

Frequency dependent cable properties (dielectric relative permittivity or transfer impedance) are defined as rational functions of frequency. The rational function includes a normalisation constant $\omega_{0}$ which is used to prevent the coefficients
of the function becoming too large or too small i.e.
%
\begin{equation} \label{eq:rational_function}
\epsilon_{r}\left(j\omega\right)=\frac{a_{0}+a_{1}\left(\frac{j\omega}{\omega_{0}}\right)+a_{2}\left(\frac{j\omega}{\omega_{0}}\right)^{2}+\dots}{b_{0}+b_{1}\left(\frac{j\omega}{\omega_{0}}\right)+b_{2}\left(\frac{j\omega}{\omega_{0}}\right)^{2}+\dots}
\end{equation}
%
For example, a Debye dielectric model has a relative permittivity described by
%
\begin{equation} \label{eq:debye_1}
\epsilon_{r}=\epsilon_{\infty}+\frac{\epsilon_{s}-\epsilon_{\infty}}{1+j\omega\tau}
\end{equation}
%
where $\epsilon_{\infty}$ is the relative permittivity at the high frequency limit, $\epsilon_{s}$ is the relative permittivity at the low frequency limit and $\tau$ is the relaxation time of the material.

this may be written in the form of equation\ref{eq:rational_function} as
%
\begin{equation} \label{eq:debye_rational_function_1}

\epsilon_{r}\left(j\omega\right)=\frac{\epsilon_{s}+\epsilon_{\infty}\left(\frac{j\omega}{\omega_{0}}\right)}{1+\left(\frac{j\omega}{\omega_{0}}\right)}

\end{equation}
%
where
%
\begin{equation} \label{eq:debye_rational_function_2}
\omega_{0}=\frac{1}{\tau}
\end{equation}
%

The frequency dependence of relative permittivity cannot be aritrarily specified as the real and imaginary parts of the realtive permittivity are related through the Kramers-Kronig relations \cite{KramersKronig}. This relationship is due to the causality of the dielectric time response and is fundamental to linear systems. The implications for modelling dielectrics is that the permittivity model used must satisfy the Kramers-Kronig relations. The rational function representation of the relative permittivity in equation \ref{eq:rational_function} has the advantage of naturally satisfying these relations. If the real part of the permittivity is a function of frequency then the Kramer Kronig relations imply that this will be associated with some loss and vice-versa. We note that the common approximation 
in which the imaginary part of permittivity is assumed to be constant (constant $tan\left( \delta \right)$ model) is unphysical.

It is difficult to obtain the good quality complex relative permittivity data over a wide frequency band required to generate frequency dependent dielectric models, even for the dielectric materials used in the construction of cables. Unless very good data is available we recommend that a simple constant relative permittivity model be used.

If tabulated frequency domain relative permittivity data is available then a model fitting process may be applied to generate the rational function coefficients for a frequency dependent dielectric model. This process is described in section \ref{FD_model_fit}.

%
\subsubsection{Frequency dependent finite conductivity loss models} \label{FD_conductivity_model}

Cable losses arising from the finite conductivity of a conductor are incorporated into some of the cable models available \cite{Theory_manual}.

In these models the conductivity of the conductors are specified as parameters of the cable model. The contribution to
the impedance terms due to the finite conductivity is the surface impedance of the conductor \cite{Schelkunoff}.

The surface impedance is frequency dependent and incorporates the `skin effect' into the model.

For cylindrical conductors analytic expressions are available for the surface impedance of solid cylindrical conductors
and cylindrical shells (cable shields).

For a solid cylindrical conductor of radius, $r$, permeability, $\mu$ and conductivity, $\sigma$ the internal impedance due to the magnetic field penetrating the conductor at frequency f is given by

\begin{equation} \label{eq:skin_effect_1}
Z_{int\_cylinder}=\frac{1}{\sqrt{2 \pi r \sigma \delta }} \left( \frac{ber(q)+jbei(q)}{bei'(q)-jber'(q)}\right)
\end{equation}

where ber and bei are Kelvin functions, $\delta$ is the skin depth given by

\begin{equation} \label{eq:skin_depth}
\delta=\frac{1}{\sqrt{ \pi f \mu \sigma }}
\end{equation}

and q is

\begin{equation} \label{eq:skin_depth_q}

q=\sqrt{2}\frac{r}{\delta}
\end{equation}

For a cylindrical shell i.e. a cable shield, the surface impedance (neglecting small terms related to the curvature of
the conductor) may be evaluated as follows:

The d.c. resistance of a shell of radius, r, and thickness, t, is 

\begin{equation} \label{eq:rdc_shell}
R_{dc}=\frac{1}{\sqrt{ 2 \pi \sigma r t }}
\end{equation}

The complex propagation constant in the conductor is

\begin{equation} \label{eq:gamma}
\gamma=\frac{\left( 1+j \right)}{\delta}
\end{equation}

Then the surface impedance of the cylindrical shell is

\begin{equation} \label{eq:Zint_shell}

Z_{int\_shell}=R_{dc} \gamma t \: cosech \left( \gamma t  \right)
\end{equation}

It is important to note that at low frequency the surface impedance is equal to the d.c. resistance of the shield 

and that the transfer impedance should also take this value.

Note that if a transfer impedance model is included then the shield thickness parameter can be set to zero in which

case the software will calculate an `equivalent thickness' which gives the correct d.c. resistance for the shield, based on 
the d.c. transfer resistance and the specified conductivity of the shield. 

The loss model for rectangular conductors assumes that the internal impedance of the conductor takes the form

\begin{equation} \label{eq:Zint_rectangular}
Z_{int\_rectangular}=R_{dc} + B \sqrt{j \omega }
\end{equation}
where $R_{dc}$ is the d.c. resistance of a rectangular wire of width w, and height, t, is given by
\begin{equation} \label{eq:Rdc_rectangular}
R_{dc}=\frac{1}{\sigma w t}
\end{equation}
and $B$ is given by
\begin{equation} \label{eq:B_rectangular}
B=\frac{1}{2 \left( w+t \right) }\sqrt{\frac{\mu}{\sigma}}
\end{equation}

These internal impedances then contribute to the impedance matrix of the cable as appropriate for each configuration.



\subsubsection{Frequency dependent transfer impedance model} \label{FD_transfer_impedance_model}

Some shielded cable models allow the impedance of the shield conductor (transfer impedance) to be specified as

a frequency dependent function \cite{Theory_manual}. Cable models with frequency dependent transfer impedance models are:

\begin{enumerate}

\item Coaxial cable with transfer impedance and shield surface impedance loss
\item Twinax cable with transfer impedance and shield surface impedance loss
\item Shielded twisted pair with transfer impedance and shield surface impedance loss
\item Spacewire cable with transfer impedance and shield surface impedance loss
\item Overshield with transfer impedance and shield surface impedance loss

\end{enumerate}

The frequency dependent transfer impedance is represented using a rational function form i.e.
%
\begin{equation} \label{eq:ZT_rational_function}
Z_{T}\left(j\omega\right)=\frac{a_{0}+a_{1}\left(\frac{j\omega}{\omega_{0}}\right)+a_{2}\left(\frac{j\omega}{\omega_{0}}\right)^{2}+\dots}{b_{0}+b_{1}\left(\frac{j\omega}{\omega_{0}}\right)+b_{2}\left(\frac{j\omega}{\omega_{0}}\right)^{2}+\dots}
\end{equation}
%

Hence the cable specification requires the coefficients of the rational function, along with the frequency scaling, $\omega_{0}$ to be specified. This model has the often used $Z_T=R_T+j\omega L_T$ model as a special case where $a_0=R_T$, $a_1=L_T$, $\omega_{0}=1.0$ and $b_0=1.0$. The coefficients for more complex transfer impedance models may be generated from the specifications of a braided shield as in reference \cite{Kley} followed by a rational function fitting to the resulting frequency domain transfer impedance function. This process is described in section \ref{FD_transfer_impedance_models}. Alternatively the rational function coefficiants may be calculated by rational function fitting to measured (or otherwise obtained) transfer imepdance data as described in section \ref{FD_model_fit}.

%
The transfer impedance of a shield is reciprocal thus the coupling through the shield is determined by the same transfer impedance for both coupling directions.

It is important to note that for a model to be self consistent at low frequency the transfer impedance of a shield should be equal to the  d.c. resistance of the shield and hence also

the low frequency surface impedance i.e. 
%
\begin{equation} \label{eq:ZT_low_frequency}
Z_{T}\left(\omega=0\right)=\frac{a_{0}}{b_{0}}=Z_{int\_shell}=R_{dc}
\end{equation}
%

\clearpage

\subsection{Cable Specification File Format} \label{Cable_spec_file_formats}

This section describes the cable specification file format used as the input to the cable model building process.

Cable specification files have the extension \textbf{name.cable\_spec}. 

The first lines of the \textbf{.cable\_spec} file specifies the path to the cable model to be produced from the specification (i.e. the path to the cable models within MOD).
Following this come the parameters which define the cable i.e. the geometrical description of the cable cross section, dielectric properties (if required) and transfer impedance specification (if required).

In addition to the data required to specify a cable bundle, additional information and flags may be specified to influence the operation of the software. There is a choice whether to use (approximate) analytic formulae to calculate the per-unit-length parameters of shielded domains within a cable or to use the numerical Laplace solver.

The approximate analytic formula for shielded domains uses a `wide separation' approximation i.e. it is assumed that the conductor radii are small compared to their separation.

If frequency dependent dielectric models are used in the cable specification then the filter fitting process will be required to set the elements of the cable admittance matrices for shielded domains. This is discussed in detail in the Frequency Dependent Transfer Functions chapter of the theory manual \cite{Theory_manual}. The parameters of the filter fitting process may be set following the cable specification. The filter fitting process provides a best fit model of

specified order over a specified frequency range. As a default the model order is 0 i.e. no frequency dependence is included in the admittance matrix (the permittivity used is the high frequency value of the dielectric constant).
The model order can be specified in two ways:

\begin{enumerate}
\item The order is specified as a positive integer and this is the order used
\item A negative integer is specified. In this case the order is chosen using an automatic algorithm which

attempts to choose the best order from 0 up to $|$specified order$|$

\end{enumerate}

Following the model order the frequency range is specified. The frequency scale is set to be either linear (`lin') or logarithmic (`log'), following this the minimum frequency, maximum frequency and the number of frequencies for the filter fitting process are specified. If the Laplace solution is used then the number of frequencies should not be too large as this will lead to excessive runtimes for the cable model building process. 

An example of a filter fitting specification for the admittance matrix element fit is shown below in which
the best model up to order 10 is chosen based on a fit to logarithmic frequency data over a range of 10kHz to 1GHz with 16 sample points: 
{\small

\begin{verbatim}

-10          # order for admittance matrix element fit model
log          # frequency scale  (log or lin)
1e5 1e9 16   # fmin fmax number_of_frequencies 

\end{verbatim}

}

The flags which may be applied in a \textbf{cable\_spec} file are as follows:

\begin{enumerate}

\item `verbose'    output detailed summary of the software operation and calculation results.\\
\item `use\_Laplace'    use the numerical Laplace solver to calculate inductance and capacitance matrices for the internal domains. By default, approximate analytic formulae are used. \\
\item `no\_Laplace'    use the (approximate) analytic formulae to calculate inductance and capacitance matrices for the internal domains. \\
\item `plot\_mesh'    output a vtk file which shows the mesh used in Finite Element Laplace calculations.\\

\end{enumerate}

If the Laplace solver is used then the mesh generation is controlled by the parameter
`Laplace\_surface\_mesh\_constant'  This parameter determines the number of finite element edges on a conductor surface.

                                       The edge length of elements on a cylindrical conductor of radius r is

                                       $\frac{r}{Laplace\_surface\_mesh\_constant}$. The default value is 3. \\

                                   
The default value is a compromise between accuracy and computation time for the Laplace solution. 

The default value may be overridden by the user by appending the following to the end of the \textbf{.cable\_spec} file:

\begin{verbatim}
Laplace_surface_mesh_constant
5
\end{verbatim}

\subsection{Cable models available} \label{available_cable_models}

The available cable models are described below. For each cable type a figure is provided which shows the cable cross section and the conductor numbering used for the cable. The parameters required to specify a cable are outlined and an example \textbf{.cable\_spec} file is provided.

The cable models are very general in that all shields can have a transfer impedance specified, all dielectrics can be frequency dependent and all conductors can have a finite conductivity specified. It may often be the case that not all the information required to specifiy a cable is available or only a simple model is required. If that is the case then the general model may be simplified in the following ways:

\begin{enumerate}
\item No dielectric on the outside of a cable: This may be included by setting the relative permittivity of the dielectric to 1 or by setting the dielectric radius equal to the conductor radius.
\item Lossless i.e. perfect conductors. This can be incuded in the model by setting the conductivity parameter to zero. This indicates to the software that loss is not to be included in the model.
\item No transfer impedance to be included. A transfer impedance must be specified however it can be set to zero.
\end{enumerate}

As was stated in section \ref{FD_transfer_impedance_model}, the transfer impedance of a shield at d.c. should be equal to the  d.c. resistance of the shield. In order to avoid the need for the user to perform any calculations to ensure that this is the case the thickness parameter for a shield can be set to zero in the \textbf{.cable\_spec} file. When the shield thickness parameter is set to zero the software calculates an `equivalent shield thickness' based on the d.c. value of the transfer impedance ($R_{dc}$) and the specified conductivity of the shield i.e.

%
\begin{equation} \label{eq:equivalent_thickness}
t=\frac{1}{2 \pi r_s \sigma R_{dc}}
\end{equation}
%

\subsubsection{Frequency dependent cylindrical conductor with dielectric} \label{FD_Cylindrical_conductor_with_dielectric}

Figure \ref{fig:FD_cylindrical_with dielectric} shows the cross section of the cylindrical cable model with dielectric. 

A description of the cable parameters is given in table \ref{table_FD_Cylindrical_conductor_with_dielectric} followed by an example.

\begin{figure}[h]
\centering

\includegraphics[scale=1.0]{./Imgs/cylindrical_with_dielectric.eps}

\caption{Cylindrical conductor with dielectric}
\label{fig:FD_cylindrical_with dielectric}
\end{figure}

\begin{table}[h]
\begin{center}
\begin{tabular}{| p{3cm} | p{3cm} | p{6cm} |}
\hline
example value & unit          & Comment \\ \hline
1             & integer       & Number of conductors \\ \hline
3             & integer       & Number of parameters\\ \hline
0.25e-3       & metre         & parameter 1: conductor radius \\ \hline
0.5e-3        & metre         & parameter 2: dielectric radius \\ \hline
5e7           & Siemens/metre & parameter 3: electric conductivity \\ \hline
1	      &  integer      & number of frequency dependent parameters \\ \hline
$\epsilon\left(j\omega \right)$  & rational function coefficients & Frequency dependent dielectric relative permittivity model \\ \hline
\end{tabular}
\end{center}
\caption{Cylindrical cable parameters}
\label{table_FD_Cylindrical_conductor_with_dielectric}
\end{table}

\clearpage

\vspace{5mm}
\textbf{\underline{Example}}

\begin{verbatim}
#MOD_cable_lib_dir
.
Cylindrical
1		# number of conductors
3		# number of parameters
1.905e-4  	# parameter 1: conductor radius
0.5e-3  	# parameter 2: dielectric radius
5E7  	        # parameter 3: conductivity
1		# number of frequency dependent parameters
# dielectric relative permittivity model follows
   1E8          # w normalisation constant
           1    # a order, a coefficients follow below:
  2.60        2.25           
           1    # b order, b coefficients follow below:
  1.0         1.0    
\end{verbatim}

\clearpage

\subsubsection{Frequency Dependent Coaxial cable with Transfer Impedance and surface impedance loss } \label{ZT_FD_Coax2}

Figure \ref{fig:ZT_FD_coax2} shows the cross section of the frequency dependent coaxial cable with transfer impedance and surface impedance loss.

A description of the cable parameters is given in table \ref{table_ZT_FD_Coax2} followed by an example.

The inductance and capacitance of the coaxial mode is always calculated using the analytic formulae \cite{PaulMTL}

\begin{equation}
L=\frac{\mu_0}{2 \pi} \ln\left(\frac{r_s}{r_w}\right)
\end{equation}
\begin{equation}
C=\frac{2 \pi \epsilon_0 \epsilon_r \left(j\omega)\right)}{ \ln\left(\frac{r_s}{r_w}\right)}
\end{equation}

If the dielectric is frequency dependent then the frequency dependent capacitance is simply a scaling of the dielectric frequency dependent function specified.

\begin{figure}[h]
\centering
\includegraphics[scale=1]{./Imgs/Coax2.eps}
\caption{Coaxial cable}
\label{fig:ZT_FD_coax2}    % note that this is the coax2 cable figure which includes the shield thickness...
\end{figure}

\clearpage

\begin{table}[h]
\begin{center}
    \begin{tabular}{ |  p{3cm} | p{3cm} | p{6cm} |}
    \hline
example value     & unit          & Comment \\ \hline
2             & integer       & Number of conductors \\ \hline
6             & integer       & Number of parameters\\ \hline
0.25e-3       & metre         & parameter 1: inner conductor radius \\ \hline
1.25e-3       & metre         & parameter 2: shield radius \\ \hline
2.5e-3        & metre         & parameter 3: outer dielectric radius \\ \hline
5e7           & Siemens/metre & parameter 4: inner conductor electric conductivity \\ \hline
0.05E-3       & metre         & parameter 5: shield conductor thickness  \\ \hline
5e7           & Siemens/metre & parameter 6: shield conductor electric conductivity \\ \hline
2	      & integer      & number of frequency dependent parameters \\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent inner dielectric relative permittivity model\\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent outer dielectric relative permittivity model\\ \hline
1	      &  integer        & number of frequency dependent transfer impedance models \\ \hline
$Z_{T}\left(j\omega \right)$ & ohms (rational function coefficients) & Frequency dependent transfer impedance model\\ \hline
    \end{tabular}
\end{center}
\caption{Coaxial cable parameters}

\label{table_ZT_FD_Coax2}

\end{table}

\vspace{5mm}

\textbf{\underline{Example}}

\begin{verbatim}
#MOD_cable_lib_dir
LIBRARY_OF_CABLE_MODELS
Coax     
2		# number of conductors
6		# NUMBER OF PARAMETERS
0.00042  	# parameter 1: inner conductor radius (m)
0.00147 	# parameter 2: shield radius (m)
0.0025 	        # parameter 3: outer insulation radius (m)
5e7             # parameter 4: inner conductor electric conductivity
0.0002          # parameter 5: shield conductor thickness
5e7             # parameter 6: shield electric conductivity
2               # number of frequency dependent parameters
# Inner dielectric relative permittivity model follows
   1E7          # w normalisation constant
           1    # a order, a coefficients follow below:
  2.6        2.0           
           1    # b order, b coefficients follow below:
  1.0         1.0    
# Outer dielectric relative permittivity model follows
   1E7          # w normalisation constant
           1    # a order, a coefficients follow below:
  2.40        2.20           
           1    # b order, b coefficients follow below:
  1.0         1.0    
1               # number of frequency dependent transfer ipedance models 
# Transfer impedance model 
1.0             # angular frequency normalisation
1               # order of numerator model 
0.05  1.6E-9    # list of numerator coefficients a0 a1 a2... 
0               # order of denominator model
1.0             # list of denominator coefficients b0 b1 b2...
\end{verbatim}

\clearpage

\subsubsection{Frequency Dependent Twinaxial cable with transfer impedance with shield surface impedance loss} \label{ZT_FD_Twinax2}

Figure \ref{fig:ZT_FD_Twinax2} shows the cross section of the frequency dependent twinax cable with transfer impedance and surface impedance loss.

A description of the cable parameters is given in table \ref{table_ZT_FD_twinax2} followed by an example in which the filter fitting model order is specified and the Laplace solver used for the frequency dependent per-unit length parameter calculation.

If the approximate analytic solution is used to calculate the per-unit-length parameters of the internal modes then

the solution assumes that the space within the shield is completely filled with the inner dielectric. If the inner dielectric is frequency dependent then the frequency dependence is neglected and the high frequency permittivity value is used. The analytic solution uses the wide separation approximation \cite{PaulMTL}. 

\begin{figure}[h]
\centering
\includegraphics[scale=1]{./Imgs/ZT_FD_Twinax.eps}
\caption{Twinaxial cable}
\label{fig:ZT_FD_Twinax2}
\end{figure}

\clearpage

\begin{table}[h]
\begin{center}
    \begin{tabular}{ |  p{3cm} | p{3cm} | p{6cm} |}
    \hline
example value & unit          & Comment \\ \hline
3             & integer       & Number of conductors \\ \hline
8             & integer       & Number of parameters\\ \hline
0.25e-3       & metre         & parameter 1: inner conductor radius \\ \hline
0.40e-3       & metre         & parameter 2: inner dielectric radius \\ \hline
1.0e-3        & metre         & parameter 3: inner conductor separation \\ \hline
2.0e-3        & metre         & parameter 4: shield radius \\ \hline
0.1e-3        & metre         & parameter 5: shield thickness \\ \hline
2.5e-3        & metre         & parameter 6: outer dielectric radius \\ \hline
5e7           & Siemens/metre & parameter 7: inner conductor electric conductivity \\ \hline
5e7           & Siemens/metre & parameter 8: shield electric conductivity \\ \hline
2	      &  integer      & number of frequency dependent parameters \\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent inner dielectric relative permittivity model\\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent outer dielectric relative permittivity model\\ \hline
1	      &  integer        & number of frequency dependent transfer impedance models \\ \hline
$Z_{T}\left(j\omega \right)$ & ohms (rational function coefficients) & Frequency dependent transfer impedance model\\ \hline
    \end{tabular}
\end{center}
\caption{Twinaxial cable parameters}

\label{table_ZT_FD_twinax2}

\end{table}

\vspace{5mm}
\textbf{\underline{Example}}

\begin{verbatim}
#MOD_cable_lib_dir
.
Twinax
3              # Number of conductors 
8              # Number of parameters
0.25e-3        # parameter 1: inner conductor radius 
0.40e-3        # parameter 2: inner dielectric radius 
1.0e-3         # parameter 3: inner conductor separation 
2.0e-3         # parameter 4: shield radius 
0.1e-3         # parameter 5: shield thickness
2.5e-3         # parameter 6: outer dielectric radius 
5e7            # parameter 7: inner conductor electric conductivity
5e7            # parameter 8: shield electric conductivity
2	       # number of frequency dependent parameters 
# Inner dielectric relative permittivity model follows
   1E7         # w normalisation constant
           1   # a order, a coefficients follow below:
  2.6        2.0           
           1   # b order, b coefficients follow below:
  1.0         1.0    
# Outer dielectric relative permittivity model follows
   1E7         # w normalisation constant
           1   # a order, a coefficients follow below:
  2.40        2.20           
           1   # b order, b coefficients follow below:
  1.0         1.0    
1              # number of frequency dependent transfer ipedance models 
# Transfer impedance model 
1.0            # angular frequency normalisation
1              # order of numerator model 
0.05  1.6E-9   # list of numerator coefficients a0 a1 a2... 
0              # order of denominator model
1.0            # list of denominator coefficients b0 b1 b2...

-10    ! order for filter fit solution
lin   # frequency range type for filter fitting type (lin or dB)
1e3 1e9 10 # fmin fmax number_of_frequencies for filter fitting
use_Laplace

\end{verbatim}

\clearpage

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

Figure \ref{fig:FD_Twisted_pair} shows the cross section of the frequency dependent twisted pair model.

A description of the cable parameters is given in table \ref{table_FD_Twisted_pair} followed by an example. Note that there is no parameter for the twisted pair model which relates to the number of twists per unit length. This is due to the nature of the model which assumes that there is no interaction between the differential mode and the other conductors in the bundle thus the twisting period is not required for the model. 

The analytic formulae used to calculate the inductance and capacitance of the differential mode are \cite{PaulMTL}

\begin{equation}

C=\frac{ \pi \epsilon_0 }{ \ln\left( \frac{s}{2r_w}+\sqrt{ \left(\frac{s}{2r_w}\right)^2-1 }  \right)}

\end{equation}
\begin{equation}
L=\frac{\mu_0 \epsilon_0}{C}
\end{equation}

where the dielectric coating of the conductors is neglected.

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

\clearpage

\begin{table}[!htb]

\begin{center}
    \begin{tabular}{ |  p{3cm} | p{3cm} | p{6cm} |}
    \hline
example value & unit          & Comment \\ \hline
2             & integer       & Number of conductors \\ \hline
4             & integer       & Number of parameters\\ \hline
0.25e-3       & metre         & parameter 1: conductor radius, $r_w$ \\ \hline
1.0e-3        & metre         & parameter 2: conductor separation, $s$ \\ \hline
0.5e-3        & metre         & parameter 3: dielectric radius \\ \hline
5e7           & Siemens/metre & parameter 4: inner conductor electric conductivity \\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent dielectric relative permittivity model\\ \hline
    \end{tabular}
\end{center}
\caption{Twisted pair cable parameters}

\label{table_FD_Twisted_pair}

\end{table}

\vspace{5mm}
\textbf{\underline{Example}}

\begin{verbatim}
#MOD_cable_lib_dir
.
Twisted_pair
2		# number of conductors
4		# number of parameters
0.25e-3  	# parameter 1: conductor radius
1.0e-3  	# parameter 2: conductor separation
0.45e-3  	# parameter 3: dielectric radius
5e7             # parameter 4: inner conductor electric conductivity
# Dielectric relative permittivity model follows
   1E7          # w normalisation constant
           1    # a order, a coefficients follow below:
  2.40        2.20           
           1    # b order, b coefficients follow below:
  1.0         1.0    
\end{verbatim}

\clearpage

\subsubsection{Frequency Dependent shielded twisted pair cable with transfer impedance and shield surface impedance loss} \label{Shielded_twisted_pair2}

Figure \ref{fig:ZT_FD_Shielded_twisted_pair} shows the cross section of the frequency dependent shielded twisted pair cable with transfer impedance and shield surface impedance loss model.

A description of the cable parameters is given in table \ref{table_ZT_FD_Shielded_twisted_pair} followed by an example in which the filter fitting model order is specified and the Laplace solver used for the frequency dependent per-unit length parameter calculation. Note that there is no parameter for the number of twists per unit length. This is due to the nature of the model which assumes that there is no interaction between the differential mode and the other conductors in the bundle thus the twisting period is not required for the model. 

If the approximate analytic solution is used to calculate the per-unit-length parameters of the internal modes then

the solution assumes that the space within the shield is completely filled with the inner dielectric. If the inner dielectric is frequency dependent then the frequency dependence is neglected and the high frequency permittivity value is used. The analytic solution uses the wide separation approximation \cite{PaulMTL}
. 

\begin{figure}[h]
\centering
\includegraphics[scale=1]{./Imgs/ZT_FD_Shielded_twisted_pair.eps}
\caption{Shielded twisted pair}
\label{fig:ZT_FD_Shielded_twisted_pair}
\end{figure}

\clearpage

\begin{table}[h]
\begin{center}
    \begin{tabular}{ |  p{3cm} | p{3cm} | p{6cm} |}
    \hline
example value & unit          & Comment \\ \hline
3             & integer       & Number of conductors \\ \hline
8             & integer       & Number of parameters\\ \hline
0.25e-3       & metre         & parameter 1: inner conductor radius \\ \hline
0.40e-3       & metre         & parameter 2: inner dielectric radius \\ \hline
1.0e-3        & metre         & parameter 3: inner conductor separation \\ \hline
2.0e-3        & metre         & parameter 4: shield radius \\ \hline
0.1e-3        & metre         & parameter 5: shield thickness \\ \hline
2.5e-3        & metre         & parameter 6: outer dielectric radius \\ \hline
5e7           & Siemens/metre & parameter 7: inner conductor electric conductivity \\ \hline
5e7           & Siemens/metre & parameter 8: shield electric conductivity \\ \hline
2	      &  integer      & number of frequency dependent parameters \\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent inner dielectric relative permittivity model\\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent outer dielectric relative permittivity model\\ \hline
1	      &  integer        & number of frequency dependent transfer impedance models \\ \hline
$Z_{T}\left(j\omega \right)$ & ohms (rational function coefficients) & Frequency dependent transfer impedance model\\ \hline
    \end{tabular}
\end{center}
\caption{Shielded twisted pair parameters}

\label{table_ZT_FD_Shielded_twisted_pair}

\end{table}

\vspace{5mm}
\textbf{\underline{Example}}

\begin{verbatim}
#MOD_cable_lib_dir
.
Shielded_twisted_pair
3             # Number of conductors 
8             # Number of parameters
0.25e-3       # parameter 1: inner conductor radius 
0.40e-3       # parameter 2: inner dielectric radius 
1.0e-3        # parameter 3: inner conductor separation 
2.0e-3        # parameter 4: shield radius 
0.1e-3        # parameter 5: shield thickness
2.5e-3        # parameter 6: outer dielectric radius 
5e7           # parameter 7: inner conductor electric conductivity
5e7           # parameter 8: shield electric conductivity
2	      # number of frequency dependent parameters 
# Inner dielectric relative permittivity model follows
   1E7        # w normalisation constant
           1  # a order, a coefficients follow below:
  2.6        2.0           
           1  # b order, b coefficients follow below:
  1.0         1.0    
# Outer dielectric relative permittivity model follows
   1E7        # w normalisation constant
           1  # a order, a coefficients follow below:
  2.40        2.20           
           1  # b order, b coefficients follow below:
  1.0         1.0    
1             # number of frequency dependent transfer ipedance models 
# Transfer impedance model 
1.0           # angular frequency normalisation
1             # order of numerator model 
0.05  1.6E-9  # list of numerator coefficients a0 a1 a2... 
0             # order of denominator model
1.0           # list of denominator coefficients b0 b1 b2...

-10    ! order for filter fit solution
lin   # frequency range type for filter fitting type (lin or dB)
1e3 1e9 10 # fmin fmax number_of_frequencies for filter fitting
use_Laplace

\end{verbatim}


\clearpage

\subsubsection{Frequency dependent Spacewire cable with transfer impedance and shield surface impedance loss} \label{ZT_FD_Spacewire}

Figure \ref{fig:ZT_FD_spacewire} shows the cross section of the frequency dependent spacewire cable with transfer impedance and shield surface impedance loss model.

A description of the cable parameters is given in table \ref{table_ZT_FD_spacewire} followed by an example in which the filter fitting model order is specified and the Laplace solver used for the frequency dependent per-unit length parameter calculation. Note that there is no parameter for the number of twists per unit length in the shielded twisted pairs. This is due to the nature of the model which assumes that there is no interaction between the differential mode and the other conductors in the bundle thus the twisting period is not required for the model. 

If the approximate analytic solution is used to calculate the per-unit-length parameters of the internal modes then

the solution assumes that the space within the twisted pair shields is completely filled with the inner dielectric. If the inner dielectric is frequency dependent then the frequency dependence is neglected and the high frequency permittivity value is used. Similarly the solution assumes that the space within the outer shield is completely filled with the inner shield dielectric. If the inner shield dielectric is frequency dependent then the frequency dependence is neglected and the high frequency permittivity value is used. The analytic solution uses the wide separation approximation \cite{PaulMTL}.

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

\clearpage

\begin{table}[h]
\begin{center}
    \begin{tabular}{ |  p{3cm} | p{3cm} | p{6cm} |}
    \hline
example value     & unit          & Comment \\ \hline
13             & integer      & Number of conductors \\ \hline
13             & integer       & Number of parameters\\ \hline
0.25e-3       & metre         & parameter 1: inner conductor radius \\ \hline
0.40e-3       & metre         & parameter 2: inner dielectric radius \\ \hline
1.0e-3        & metre         & parameter 3: inner conductor separation \\ \hline
2.0e-3        & metre         & parameter 4: inner shield radius \\ \hline
0.1e-3        & metre         & parameter 5: inner shield thickness \\ \hline
2.25e-3        & metre        & parameter 6: inner shield jacket radius \\ \hline
3.25e-3        & metre        & parameter 7: shielded twisted pair radius \\ \hline
5.65e-3        & metre         & parameter 8: outer shield radius \\ \hline
0.15e-3        & metre        & parameter 9: outer shield thickness \\ \hline
6.25e-3        & metre        & parameter 10: outer dielectric radius \\ \hline
5e7           & Siemens/metre & parameter 11: inner conductor electric conductivity \\ \hline
5e7           & Siemens/metre & parameter 12: inner shield electric conductivity \\ \hline
5e7           & Siemens/metre & parameter 13: outer shield electric conductivity \\ \hline
3	      &  integer      & number of frequency dependent parameters \\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent inner dielectric relative permittivity model\\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent inner shield dielectric relative permittivity model\\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent outer dielectric relative permittivity model\\ \hline
2	      &  integer        & number of frequency dependent transfer impedance models \\ \hline
$Z_{T}\left(j\omega \right)$ & ohms (rational function coefficients) & Inner shield Frequency dependent transfer impedance model\\ \hline
$Z_{T}\left(j\omega \right)$ & ohms (rational function coefficients) & Outer shield Frequency dependent transfer impedance model\\ \hline
    \end{tabular}
\end{center}
\caption{Spacewire parameters}

\label{table_ZT_FD_spacewire}

\end{table}

\clearpage

\textbf{\underline{Example}}

\begin{verbatim}
#MOD_cable_lib_dir
.
Spacewire
13		# number of conductors
13              # Number of parameters
0.25e-3         # parameter 1: inner conductor radius
0.40e-3         # parameter 2: inner dielectric radius
1.0e-3          # parameter 3: inner conductor separation
2.0e-3          # parameter 4: inner shield radius
0.1e-3          # parameter 5: inner shield thickness
2.25e-3         # parameter 6: inner shield jacket radius 
3.25e-3         # parameter 7: shielded twisted pair radius 
5.65e-3          # parameter 8: outer shield radius 
0.1e-3          # parameter 9: outer shield thickness
6.25e-3         # parameter 10: outer dielectric radius
5e7             # parameter 11: inner conductor electric conductivity
5e7             # parameter 12: inner shield electric conductivity
5e7             # parameter 13: outer shield electric conductivity
3	       number of frequency dependent parameters 
# Inner dielectric relative permittivity model follows
   1E7          # w normalisation constant
           1    # a order, a coefficients follow below:
  2.6        2.0           
           1    # b order, b coefficients follow below:
  1.0         1.0    
# Inner shield dielectric relative permittivity model follows
   1E7          # w normalisation constant
           1    # a order, a coefficients follow below:
  2.6        2.0           
           1    # b order, b coefficients follow below:
  1.0         1.0    
# Outer dielectric relative permittivity model follows
   1E7          # w normalisation constant
           1    # a order, a coefficients follow below:
  2.40        2.20           
           1    # b order, b coefficients follow below:
  1.0         1.0    
2               # number of frequency dependent transfer ipedance models 
# Inner shield Transfer impedance model
1.0             # angular frequency normalisation
1               # order of numerator model 
0.05  1.6E-9    # list of numerator coefficients a0 a1 a2... 
0               # order of denominator model
1.0             # list of denominator coefficients b0 b1 b2...
# Outer shield Transfer impedance model
1.0             # angular frequency normalisation
1               # order of numerator model 
0.002  2.8E-9   # list of numerator coefficients a0 a1 a2... 
0               # order of denominator model
1.0             # list of denominator coefficients b0 b1 b2...

-10    ! order for filter fit solution
lin   # frequency range type for filter fitting type (lin or dB)
1e3 1e9 10 # fmin fmax number_of_frequencies for filter fitting
use_Laplace

\end{verbatim}


\clearpage

\subsubsection{Over-Shield} \label{over_shield}

Figure \ref{fig:Overshield} shows the cross section of the overshield.

A description of the overshield parameters is described in table \ref{table_overshield} followed by an example.

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

\begin{table}[h]
\begin{center}
    \begin{tabular}{ |  p{3cm} | p{3cm} | p{6cm} |}
    \hline
example value     & unit          & Comment \\ \hline
1             & integer       & Number of conductors \\ \hline
3             & integer       & Number of parameters \\ \hline
5.5e-3        & metre           & parameter 1: overshield radius \\ \hline
0.05E-3       & metre           & parameter 2: overshield conductor thickness  \\ \hline
5e7           & Siemens/metre   & parameter 3: overshield conductor electric conductivity \\ \hline
0	      &  integer        & number of frequency dependent parameters \\ \hline
1	      &  integer        & number of frequency dependent transfer impedance models \\ \hline
$Z_{T}\left(j\omega \right)$ & ohms (rational function coefficients) & overshield Frequency dependent transfer impedance model\\ \hline
    \end{tabular}
\end{center}
\caption{Overshield parameters}

\label{table_overshield}

\end{table}

\clearpage

\vspace{5mm}
\textbf{\underline{Example}}

\begin{verbatim}
#MOD_cable_lib_dir
LIBRARY_OF_CABLE_MODELS
Overshield   
1	       # number of conductors
3	       # number of parameters
0.005	       # parameter 1: overshield radius
0.0001	       # parameter 2: overshield thickness
5E7	       # parameter 3: overshield conductivity
0              # number of frequency dependent parameters
1              # number of frequency dependent transfer impedance models
# Transfer impedance model 
1.0            #  angular frequency normalisation
1              #  order of numerator model
0.05  1.6E-9   #  list of numerator coefficients a0 a1 a2... 
0              #  order of denominator model 
1.0            #  list of denominator coefficients b0 b1 b2... 
\end{verbatim}


\clearpage

\subsubsection{Flex cable} \label{flex_cable}

Figure \ref{fig:flex_cable} shows the cross section of the flex cable model.

A description of the model parameters is described in table \ref{table_flex_cable} followed by an example. The flex cable differs from most other cable models in that the number of conductors is a parameter of the model. The conductors are numbered from left to right. 

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

\begin{table}[h]
\begin{center}
    \begin{tabular}{ |  p{3cm} | p{3cm} | p{6cm} |}
    \hline
example value     & unit          & Comment \\ \hline
8             & integer       & Number of conductors - can be any number of conductors in a flex cable model \\ \hline
6             & integer       & Number of parameters, always 6 for flex cables \\ \hline
1.0e-3       & metre         & parameter 1: conductor width (x) \\ \hline
0.2e-3       & metre         & parameter 2: conductor height (y) \\ \hline
0.6e-3       & metre         & parameter 3: conductor separation (x)\\ \hline
1.0e-3       & metre         & parameter 4: dielectric offset in x \\ \hline
0.2e-3       & metre         & parameter 5: dielectric offset in y \\ \hline
5E7          & Siemens/metre & parameter 6: conductivity \\ \hline
1	      &  integer      & number of frequency dependent parameters \\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent dielectric relative permittivity model\\ \hline
    \end{tabular}
\end{center}
\caption{Flex cable parameters}

\label{table_flex_cable}

\end{table}

\vspace{5mm}

\clearpage

\textbf{\underline{Example}}

\begin{verbatim}
#MOD_cable_lib_dir
.
Flex_cable
3		# number of conductors
6		# number of parameters
1.0e-3  	# parameter 1: conductor width (x dimension)
0.25e-3  	# parameter 2: conductor height (y dimension)
0.5e-3  	# parameter 3: conductor separation (x dimension)
0.2e-3  	# parameter 4: dielectric offset x
0.1e-3  	# parameter 5: dielectric offset y
5E7  	        # parameter 6: conductivity
1		# number of frequency dependent parameters
# dielectric relative permittivity model follows
   1E9          # w normalisation constant
           1    # a order, a coefficients follow below:
  2.2      2.0     
           1    # b order, b coefficients follow below:
  1.0      1.0
\end{verbatim}

\clearpage

\subsubsection{D connector} \label{Dconnector}

Figure \ref{fig:Dconnector} shows the cross section of the D connector model.
A description of the model parameters is described in table \ref{table_Dconnector} followed by an example. The D connector model differs from most other cable models in that the number of conductors is a parameter of the model. The conductors are numbered from left to right on the top row, then left to right on the bottom row and finally the D shaped shell conductor. The minimum number of conductors is 5.

The Laplace solution must always be used to calculate the per-unit-length parameters of the D-connector as there is no appropriate analytic solution available. 

\begin{figure}[h]
\centering

\includegraphics[scale=0.70]{./Imgs/Dconnector.eps}

\caption{D connector}
\label{fig:Dconnector}
\end{figure}

\clearpage

\begin{table}[h]
\begin{center}
    \begin{tabular}{ |  p{3cm} | p{3cm} | p{6cm} |}
    \hline
example value     & unit          & Comment \\ \hline
8             & integer       & Number of conductors - can be any number of conductors greater than 5 \\ \hline
4             & integer       & Number of parameters, always 4 for D connectors \\ \hline
0.5e-3        & metre         & parameter 1: conductor radius \\ \hline
1.5e-3        & metre         & parameter 2: conductor pitch (separation in x) \\ \hline
1.5e-3        & metre         & parameter 3: conductor separation in y \\ \hline
1.0e-3        & metre         & parameter 4: offset from conductors to shell \\ \hline
0	      &  integer      & number of frequency dependent parameters \\ \hline
0	      &  integer      & number of transfer impedance models \\ \hline
$\epsilon\left(j\omega \right)$ & rational function coefficients & Frequency dependent dielectric relative permittivity model\\ \hline
    \end{tabular}
\end{center}
\caption{D connector parameters}

\label{table_Dconnector}

\end{table}

\vspace{5mm}

\textbf{\underline{Example}}

\begin{verbatim}
#MOD_cable_lib_dir
.
Dconnector
10		# number of conductors
4		# number of parameters
0.5e-3  	# parameter 1: conductor radius
1.5e-3  	# parameter 2: conductor pitch (separation in x)
1.5e-3  	# parameter 3: conductor separation in y
1.0e-3  	# parameter 4: offset from conductors to shell
0		# number of frequency dependent parameters
0		# number of transfer impedance models

use_Laplace

\end{verbatim}

\cleardoublepage