\section{Approximate model for frequency dependent transmission lines} \label{Propagation_Correction}

The derivation of the model for frequency dependent transmission lines will be described in two stages. Firstly the model is described for single mode propagation. This model is subsequently generalised to multi-mode propagation. 
The frequency dependent properties in the models derived here arise from two mechanisms:
\begin{enumerate}
\item frequency dependent permittivity of dielectrics
\item finite conductivity of conductors. 
\end{enumerate}

\subsection{Frequency dependent dielectric models} \label{FD_dielectric_models}

Frequency dependent cable properties (dielectric relative permittivity or transfer impedance) may be included in the model using the rational function form for the relative permittivity i.e.
%
\begin{equation} \label{eq:epsr_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 the frequency dependent cable dielectric constant could be approximated by a Debye model \cite{Scaife} which has a relative permittivity (although in practice debye dielectrics would be too lossy to be used in transmission lines) 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.

A frequency dependent dielectric model will then give rise to a frequency dependent admittance in the transmission line equations. 
We note however that the frequency dependent relative permittivity function in equation \ref{eq:epsr_rational_function} will
give rise to no admittance at d.c. The only mechanism which would give rise to a d.c. admittance would be a non-zero electrical conductivity
in the dielectric. 

\subsection{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.
In these models the conductivity of the conductors are specified as parameters of the cable model. The cylindrical conductor
loss models are based on reference \cite{Schelkunoff} and the model for rectangular conductors is derived in \cite{PaulMTL}
The models of finite conductivity of conductors include both a resistance and internal inductance (due to the magnetic field penetrating the conductor). These are frequency dependent and thus a complex internal impedance contributes to the impedance matrix of the idealised cable.

It is seen that the internal resistance due to finite conductivity increases as $\sqrt(f)$ and the internal inductance decreases as $\frac{1}{\sqrt{f}}$. This is important to note for the derivation of the propagation correction. 

\subsubsection{Cylindrical Conductor}

For a cylindrical conductor of radius, $r$, and conductivity, $\sigma$, the internal impedance due to the 
magnetic field penetrating the conductor at frequency f is given by \cite{Schelkunoff},  \cite{PaulMTL}

\begin{equation} \label{eq:skin_effect_1}
Z_{int}=\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}

\subsubsection{Cylindrical Shell}

For a cylindrical shell i.e. a cable shield of radius, $r$, thickness, $t$, and conductivity, $\sigma$, the 
surface impedance (neglecting small terms related to the curvature of the conductor) may be evaluated as follows \cite{Schelkunoff}

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:rdc_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 also takes this same value.

\subsubsection{Rectangular Conductor}

The loss model for rectangular conductors assumes that the internal impedance of the conductor takes the form \cite{PaulMTL}

\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}

\subsection{Single mode propagation correction}

Assume that the frequency dependence of the transmission line parameters takes the form
%
\begin{equation} \label{eq:fd_parameters}
\begin{array}{c} 
L\left(j\omega\right)=L_{0}+L_{int}\left(j\omega\right) \\[10pt]
R\left(j\omega\right)=R_{dc}+R_{int}\left(j\omega\right) \\[10pt]
C\left(j\omega\right)=C_{0}+C_{freq}\left(j\omega\right)
 \end{array}
\end{equation}
% 
In the light of the models of the frequency dependence of the cable parameters discussed above, we emphasise the following points:

\begin{enumerate}
\item The internal inductance, $L_{int}\left(j\omega\right)$, decreases with frequency as $\frac{1}{\sqrt{f}}$.
\item The internal resistance, $R_{int}\left(j\omega\right)$, is equal to $0\Omega$ at d.c.
\item The internal resistance, $R_{int}\left(j\omega\right)$, increases with frequency as $\sqrt{f}$
\item The capacitance, $C\left(j\omega\right) \rightarrow C_{0}$ as $\omega \rightarrow \inf$ 
\end{enumerate}
%
thus as $\omega \rightarrow \inf$, we can assume that the inductance of the transmission line is $L_0$ and the capacitance of the transmission line is $C_0$.
%
The characteristic impedance of the transmission line is
%
\begin{equation} \label{eq:fd_Z0}
Z_{c}\left(j\omega\right)=\\[10pt]\sqrt{ \frac{R\left(j\omega\right)+j\omega L\left(j\omega\right) }{j\omega C\left(j\omega\right)} }
\end{equation}
%
The propagation factor of the transmission line is $e^{j\gamma L}$ where
%
\begin{equation} \label{eq:fd_gamma}
\gamma\left(j\omega\right)=\\[10pt]\sqrt{ \left(R\left(j\omega\right)+j\omega L\left(j\omega\right) \right) \left(j\omega C\left(j\omega\right)\right) }
\end{equation}
%
The method of characteristics outlined in section\ref{Method_of_characteristics} above may be generalised for frequency dependent
properties, including losses. Initially frequency domain expressions are obtained for the termination voltages as
%
\begin{equation} \label{eq:fd_ch_2}
V \left( 0,j\omega \right) =Z_{c} \left( j\omega \right) I \left(0,j\omega \right)+
e^{-\gamma L} \left( V\left(L,j\omega \right) -Z_{c}\left(j\omega \right) I\left(L,j\omega \right) \right)
\end{equation}
%
A similar process leads to an equation for the z=L end of the transmission line
%
\begin{equation} \label{eq:fd_ch_3}
V\left(L,j\omega\right)=Z_{c}\left(j\omega\right)I\left(L,j\omega \right)+
e^{-\gamma L} \left( V\left(0,j\omega\right)+Z_{c}\left(j\omega\right)I\left(0,j\omega \right) \right)
\end{equation}
%
Multiplication in the frequency domain implies a convolution in the time domain therefore the
frequency dependent propagation and the frequency dependent impedance functions require
convolution processes for time domain calculations.
It has been found that a sufficiently accurate model across a wide frequency band down to d.c.
and up into the GHz frequency range may be derived by a suitable extraction of the loss and
dispersion effects to the terminations of the transmission line model.
The first complication in equations \ref{eq:fd_ch_2} and \ref{eq:fd_ch_3}  is the frequency dependent impedance. This is
simplified firstly by assuming that the d.c. resistance of the transmission line, $R_{dc}$ , can be lumped into
two resistances $\frac{R_{dc}}{2}$ at either end of the line.
The impedance of the transmission line with the d.c. resistance removed in this way is now
%
\begin{equation} \label{eq:fd_Z}
Z_{c}\left(j\omega\right)=\sqrt{ \frac{R_{int}\left(j\omega\right)+j\omega L_{0} + j\omega L_{int}\left(j\omega\right)}{j\omega C_{0}+j\omega C_{freq}\left(j\omega\right)} }
\end{equation}
%
$R_{int}$ is 0 at d.c. and significantly less than the inductive impedance $j \omega L_{0}+j \omega L_{int}(j \omega ) $
at high frequency, we also observe that the internal inductance due to the skin effect decreases with
frequency as $\sqrt{f}$ therefore it appears to be an acceptable approximation that as far as the
transmission line characteristic impedance is concerned we may calculate it using the high frequency
inductance
%
\begin{equation} \label{eq:fd_Z0}
Z_{0}=\sqrt{ \frac{L_{0}}{C_{0}} }
\end{equation}
%
Therefore we can write
%
\begin{equation} \label{eq:fd_ch_4}
V'\left(0,j\omega\right)=Z_{0}I\left(0,j\omega\right)+
e^{\gamma'(j\omega) L} \left( V'\left(L,j\omega\right)-Z_{0}I\left(L,j\omega \right) \right)
\end{equation}
\begin{equation} \label{eq:fd_ch_5}
V'\left(L,j\omega\right)=-Z_{0}I\left(L,j\omega\right)+
e^{\gamma'(j\omega) L} \left( V'\left(0,j\omega\right)+Z_{0}I\left(0,j\omega \right) \right)
\end{equation}
%
where V' is the voltage across the modified (d.c. resistance removed) transmission line termination
and the revised propagation factor
$\gamma '$ also takes account of the extraction of the d.c. resistance and is
%
\begin{equation} \label{eq:fd_gamma_prime}
\gamma'\left(j\omega\right)=\sqrt{ \left(R_{int}\left(j\omega\right)+j\omega L\left(j\omega\right) \right) \left(j\omega C\left(j\omega\right)\right) }
\end{equation}
%
The second complication is the frequency dependence of the propagation factor. This is treated as
follows:
We define $\gamma_{0}$ using the high frequency inductance value as

Then equations \ref{eq:fd_ch_4} and \ref{eq:fd_ch_5} may be written
%
\begin{equation} \label{eq:fd_ch_6}
V'\left(0,j\omega\right)=Z_{0}I\left(0,j\omega\right)+
e^{\gamma_{0} L} e^{\left(\gamma'(j\omega)- \gamma_{0} \right) L}\left( V'\left(L,j\omega\right)-
Z_{0}I\left(L,j\omega \right) \right)
\end{equation}
\begin{equation} \label{eq:fd_ch_7}
V'\left(L,j\omega\right)=-Z_{0}I\left(L,j\omega\right)+
e^{\gamma_{0} L} e^{\left(\gamma'(j\omega)- \gamma_{0} \right) L}\left( V'\left(0,j\omega\right)+
Z_{0}I\left(0,j\omega \right) \right)
\end{equation}
%
The $e^{j\gamma_{0} l}$ propagation term is a simple delay of T seconds where
%
\begin{equation} \label{eq:fd_ch_8}
T=\sqrt{L_{0}C_{0}}
\end{equation}
%
The propagation factor
$e^{j\gamma_{0} L}e^{j\left(\gamma'(j\omega)-\gamma_{0} \right) L} $
can be thought of as a delay with a subsequent frequency dependent propagation correction applied.

The propagation of the characteristic variables can thus be implemented as a simple delay line in
Spice followed by a frequency dependent correction factor $H(j\omega)=e^{j\left(\gamma'(j\omega)-\gamma_{0} \right) L}$
which may be implemented in Spice using the s-domain transfer function element.
The corresponding time domain model equations are now
%
\begin{equation} \label{eq:fd_ch_9}
V'\left(0,t\right)=Z_{0}I\left(0,t\right)+
H(j\omega)^{*}\left( V'\left(L,t-T\right)-Z_{0}I\left(L,t-T \right) \right)
\end{equation}
%
\begin{equation} \label{eq:fd_ch_10}
V'\left(L,t\right)=-Z_{0}I\left(L,t\right)+
H(j\omega)^{*}\left( V'\left(0,t-T\right)+Z_{0}I\left(L,t-T \right) \right)
\end{equation}
%
where
\begin{equation} \label{eq:fd_H_defn}
H(j\omega)=e^{j\left(\gamma'(j\omega)-\gamma_{0} \right) L}
\end{equation}
%
The Spice model for the method of characteristics propagation model with the propagation correction
included is shown in Figure \ref{fig:spice_FD_characteristics}
%
\begin{figure}[h]
\centering
\includegraphics[scale=0.7]{./Imgs/FD_Method_of_characteristics.eps}
\caption{Method of characteristics Spice model with propagation correction}
\label{fig:spice_FD_characteristics}
\end{figure}
%
The transmission line model may be thought of as a lossless transmission line with a network at either
end which models the effect of the frequency dependent line losses.
It is found that the individual termination models implemented as shown in figure \ref{fig:spice_FD_characteristics} is not reciprocal i.e. at the
termination of the lossless line the propagation correction is applied to the wave in only one direction.
The model can be made reciprocal by a termination network which applies half of the
propagation correction $\left( \sqrt{H(j\omega)} \right)$ at each end of the lossless transmission line as shown in Figure \ref{fig:spice_RFD_characteristics}. Each part of the propagation correction corrects for propagation along half the length of the transmission line i.e. the propagation correction applied at each end of the method of characteristics delay line is
%
\begin{equation} \label{eq:fd_H_defn}
H'(j\omega)=e^{j\left(\gamma'(j\omega)-\gamma_{0} \right) \frac{L}{2}}
\end{equation}
%
\begin{figure}[h]
\centering
\includegraphics[scale=0.7]{./Imgs/FD_Reciprocal_method_of_characteristics.eps}
\caption{Reciprocal method of characteristics Spice model with propagation correction}
\label{fig:spice_RFD_characteristics}
\end{figure}
%
\subsection{Multi-mode propagation correction}

We follow the analysis of the frequency dependent single mode transmission line in our development
of a frequency dependent multi-conductor transmission line model.
Assume that the frequency dependence of the transmission line parameters takes the form
%
\begin{equation} \label{eq:Mfd_parameters}
\begin{array}{c} 
\left[ L\left(j\omega\right) \right]=\left[ L_{0} \right]+\left[ L_{int}\left(j\omega\right) \right] \\[10pt]
\left[ R\left(j\omega\right) \right]=\left[ R_{dc} \right]+\left[ R_{int}\left(j\omega\right)  \right]\\[10pt]
\left[ C\left(j\omega\right) \right]=\left[ C_{0} \right]+\left[ C_{freq}\left(j\omega\right) \right]
 \end{array}
\end{equation}
%
Where square brackets denote matrix quantities.
We follow the same process as for the single mode model development i.e. firstly the d.c. resistance
for each conductor is lumped at either end of the MTL..
The remaining transmission line is assumed to have the properties
%
\begin{equation} \label{eq:Mfd_parameters1}
\begin{array}{c} 
\left[ L\left(j\omega\right) \right]=\left[ L_{0} \right]+\left[ L_{int}\left(j\omega\right) \right] \\[10pt]
\left[ R\left(j\omega\right) \right]=\left[ R_{int}\left(j\omega\right)  \right]\\[10pt]
\left[ C\left(j\omega\right) \right]=\left[ C_{0} \right]+\left[ C_{freq}\left(j\omega\right) \right]
 \end{array}
\end{equation}
%
In deriving the modal decomposition and the mode impedances, the transmission line is
approximated as a lossless line with the frequency independent properties
%
\begin{equation} \label{eq:Mfd_parameters2}
\begin{array}{c} 
\left[ L \right]=\left[ L_{0} \right] \\[10pt]
\left[ C\right]=\left[ C_{0}  \right]
\end{array}
\end{equation}
%
The transmission line impedance and admittance matrices are
%
\begin{equation} \label{eq:Mfd_Z}
\begin{array}{c} 
\left[ \tilde{Z}\right]=j\omega\left[ L_{0}\right] \\[10pt]
\left[ \tilde{Y}\right]=j\omega\left[ C_{0}\right]
\end{array}
\end{equation}
%
In this lossless case we can derive a frequency domain solution to the transmission line equations
using a modal analysis as outlined in section \ref{solMTL}

The diagonal modal inductance and capacitance matrices on the lossless line are given by \cite{PaulMTL}
%
\begin{equation} \label{eq:Mfd_parameters3}
\begin{array}{c} 
\left[ L_{m} \right]=\left[ T_{V} \right]^{-1} \left[ L_{0}\right] \left[ T_{I} \right]\\[10pt]
\left[ C_{m} \right]=\left[ T_{I} \right]^{-1} \left[ C_{0}\right] \left[ T_{V} \right]
\end{array}
\end{equation}
%
The impedance of the ith mode is then
%
\begin{equation} \label{eq:MZfd_0}
Z_{m,i}=\sqrt{ \frac{L_{m,i}}{C_{m,i} } }
\end{equation}
%
The modal transformation matrices and the modal impedances derived from the lossless line
parameters are independent of frequency.
It now remains to calculate the propagation correction which takes into account the frequency
dependence of the transmission line losses.
Taking into account the loss terms but excluding the d.c. resistance which has already been removed,
the impedance and admittance matrices are
%
\begin{equation} \label{eq:Mfd_parameters5}
\begin{array}{c} 
\left[ \tilde{Z}'\right]=\left[ R_{int} \left(j\omega\right)  \right]+
j\omega\left[ \left[ L_{0} \right]+ \left[ L_{int}\left(j\omega\right) \right] \right] \\[10pt]
\left[ \tilde{Y}'\right]=j\omega\left[ \left[ C_{0} \right]+ \left[ C_{freq}\left(j\omega\right) \right] \right]
\end{array}
\end{equation}
%
Having subtracted the d.c. resistance we know that
$R_{int}(j\omega)=0$  when $\omega=0$. Also the contribution to
the impedance due to the internal inductance (which is due to the field penetrating the
conductors) becomes negligible compared to the inductance $L_0$ as  $\omega\rightarrow \inf$. The impedance matrix is assumed to
be weakly varying with frequency which provides justification for the approximation that the modal
decomposition and mode impedances are independent of frequency. In order to determine a
propagation correction which incorporates the effects of loss we perform the modal decomposition on
the product of the frequency dependent impedance and admittance i.e. diagonalise the matrix system
%
\begin{equation} \label{eq:Mfd_parameters6}
\left[ \tilde{Z}'\right]\left[ \tilde{Y}'\right]=
\left[ \tilde{T_{V}}\right]\left[ \tilde{\gamma^{2}}\right]\left[ \tilde{T_{V}}\right]^{-1}
\end{equation}
%
As for the single mode transmission line the propagation correction for mode i is given by
%
\begin{equation} \label{eq:Mfd_parameters7}
H_{i}\left(j\omega \right)=e{-\left(\gamma_{i}'\left(j\omega\right)-\gamma_{0,i}\right) L}
\end{equation}
%
Again, the propagation correction can be applied at both ends of the transmission line to give a reciprocal and symmetrical model. The propagation correction in this case is the square root of $H'(j\omega)$ i.e.
%
\begin{equation} \label{eq:Mfd_parameters8}
H_{i}'\left(j\omega \right)=e{-\left(\gamma_{i}'\left(j\omega\right)-\gamma_{0,i}\right) \frac{L}{2}}
\end{equation}
%

The implementation of the propagation correction for each mode is identical to that described in the
previous section.
The derivation of the correction factor as described above relies on being able to uniquely identify
each mode of the frequency dependent transmission line with a corresponding mode of the frequency
independent transmission line. The voltage transformation matrix $[T_{v}]$ which
diagonalises the frequency dependent system in equation (7.57) has been found to be a weak
function of frequency and hence the mode identification has been straightforward. We note that identifying the corresponding modes
of the lossy transmission line and the lossless transmission line may require some care. In the implementation of this 
process we identify the modes on the lossy transmission line by identifying the closest mode 
of the lossless transmission line (using the vector norm as a distance measure).
%
The propagation correction for the twisted pair model must take proper account of the domain decomposition. The
propagation correction is developed as follows:

In the first stage the d.c. resistance of the conductors is lumped at the ends of the cable bundle model before
the domain decomposition.

As derived in section \ref{Propagation_Correction}
the propagation of the differential mode can be implemented as a simple delay line followed by a frequency dependent correction factor $H(j\omega)=e^{j\left(\gamma'(j\omega)-\gamma_{0} \right) L}$
where
$\gamma '$ takes account of the extraction of the d.c. resistance for the differential mode. The 
differential mode current sees the twice the impedance contribution of a single wire i.e. $\left( 2 R_{dc} \right)$, twice the resistance of a single wires is subtracted.
%
\begin{equation} \label{eq:fd_gamma_prime}
\gamma'\left(j\omega\right)=\sqrt{ \left(Z_{int}\left(j\omega\right)+j\omega L_{DM}\left(j\omega\right) \right) \left(j\omega C_{DM}\left(j\omega\right)\right) }
\end{equation}

The common mode is slightly different in that the common mode current is distributed between the two twisted pair conductors
in parallel so the contribution to the loss of the common mode is $\left( \frac{1}{2} R_{dc} \right)$, i.e half the resistance of a single wires is subtracted (note that any resistance in the common mode return must be subtracted too).


\clearpage