\section{Transfer Impedance Model} \label{transfer_impedance_model}

The spice model for trasfer impedance coupling through cable shields uses a weak form of transfer impedance coupling i.e. the coupling is assumed
to be in one direction only from source to victim domains. This requires identification of the source and victim domains 
by the user in each scenario. Transfer admittance is not included in the shield coupling model.

In this work the transfer impedance is assumed to take the form of a rational function in $j\omega$ 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}
%
where $\omega_0$ is a normalisation constant to and the expansion can be taken up to any order required. Note that this model includes the
common $Z_T=R_T+j\omega L_T$ model as a particular case.
%
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.

The model will be derived initially for a single mode in the source and victim domain before generalising to the multi-mode 
situation. 
In the derivation losses are assumed to be zero as we are using the 'propagation correction' model
in which the effect of losses are lumped into the terminations as a lumped d.c. resistance and a propagation correction applied
to the forward and backward modes on the transmission line. 

The transfer impedance model derivation starts from the frequency domain equations for 
propagation in the victim domain. 
% 1. MTL equations with transfer impedance source functions
\begin{equation} \label{eq:MTLequa_Zt}
\begin{array}{rcl}
\frac{\partial}{\partial z}\left(\tilde{V}_v\left(z\right) \right)+ j \omega \left[L\right]\left(\tilde{I}_v\left(z\right) \right) & = &  \tilde{V}_{ZT,v}\left(z\right) \\[10pt]
\frac{\partial}{\partial z}\left(\tilde{I}_v\left(z\right) \right) + j \omega \left[C\right]\left(\tilde{V}_v\left(z\right)\right) & = & 0  
\end{array}
\end{equation}
%
in which the voltage source on the right hand side is due to transfer impedance coupling from source to victim domain and
is given by the product of the frequency dependent transfer impedance and the shield current in the source domain. 
\begin{equation} \label{eq:ZT_source_definition}
\tilde{V}_{ZT,v}\left(z\right) = \tilde{Z}'_T \left( z \right) \tilde{I}_s \left( z \right)
\end{equation}
We note here that for the transfer impedance in equation \ref{eq:ZT_source_definition}, $\tilde{Z}'_T$ the d.c. part of the transfer impedance
has been subtracted. This is a consequence of the extraction of the d.c. resistance of conducors being lumped at the
transmission line terminations where we note that the transfer impedance of a shield is equal to the d.c. resistance of the shield and
lumping this at the terminations of the transmission line means that the d.c. transfer impedance coupling is already included
in the model (we also observe that the d.c. transfer impedance coupling will always be present in the model and is included in a 
self consistent manner in the model) thus
%
\begin{equation} \label{eq:ZT_source_definition_2}
\tilde{Z}'_T \left( \omega \right) = \tilde{Z}_T \left( \omega \right)-\tilde{Z}_T\left( \omega=0 \right)
\end{equation}
%
It has been found that this approximation can lead to errors at high frequency for shielded cables whose shields are terminated with very low (comparable with the shield resistance at d.c. or less). In this situation, improved results are obtained by excluding the $R_{dc}$ value from the process of lumping the d.c. resistances at the terminations and keeping it in the transfer impedance model. A flag is available when running the software to use this alternative model. The price to be paid for this is accuracy of the model at low frequency. It is recommended that a suitable validation test be run to test which model is most appropriate in any given circumstance.

The voltage and current in the source domain are expressed using the sum and difference of forward and backward voltage waves (characteristic variables) as in equation \ref{eq:MOC_eqn1} i.e.

\begin{equation} \label{eq:MTLequa_source_domain}
\begin{array}{rcl}
\tilde{V}_s\left(z\right) & = &  \frac{1}{2}   \left(\tilde{V}_s^+\left(0\right) e^{-j \beta_s z} + \tilde{V}_s^-\left(L\right) e^{j \beta_s \left(z-L\right)} \right)  \\[10pt]
\tilde{I}_s\left(z\right) & = & \frac{1}{2Z_s} \left(\tilde{V}_s^+\left(0\right)  e^{-j \beta_s z} - \tilde{V}_s^-\left(L\right) e^{j \beta_s \left(z-L\right)} \right) 
\end{array}
\end{equation}
%
where the forward and backward voltage waves are 
%
\begin{equation} \label{eq:characteristics_source_domain}
\begin{array}{rcl}
\tilde{V}_s^+\left(0\right) & = & H'_s \left( \tilde{V}_s\left(0\right) + Z_s \tilde{I}_s\left(0\right) \right) \\[10pt]
\tilde{V}_s^-\left(L\right) & = & H'_s \left( \tilde{V}_s\left(L\right) - Z_s \tilde{I}_s\left(L\right) \right)
\end{array}
\end{equation}
%
where the propagation correction, $H'_s$ has been applied in the source domain to determine the characteristics in the source domain. 

In the victim domain the characteristics incident at the terminations, and subsequently corrected for the victim domain propagation by $H'_v$, are given by the sum of two contributions. The first comes from waves launched from the terminations in the victim domain and the second from the transfer impedance distributed sources i.e.
%
\begin{equation} \label{eq:characteristics_victim_domain}
\begin{array}{rcccl}
\tilde{V}_v^+\left(L\right)   & = & \tilde{V}_v\left(L\right) + Z_v \tilde{I}_v\left(L\right)  
                              & = & H'_v \tilde{V}_v^+\left(0\right) e^{-j \beta_v L} 
                                   +H'_v \tilde{V}^+_{L,ZT,v} \\[10pt]
\tilde{V}_v^-\left(0\right)   & = & \tilde{V}_v\left(0\right) - Z_v \tilde{I}_v\left(0\right) 
                              & = & H'_v \tilde{V}_v^-\left(L\right) e^{-j \beta_v L} 
                                   +H'_v \tilde{V}^-_{0,ZT,v} 
\end{array}
\end{equation}
%
where the waves launched from the terminations are found as
%
\begin{equation} \label{eq:characteristics_victim_domain2}
\begin{array}{rcl}
\tilde{V}_v^+\left(0\right)   & = & H'_v \left( \tilde{V}_v\left(0\right) + Z_v \tilde{I}_v\left(0\right) \right) \\[10pt]
\tilde{V}_v^-\left(L\right)   & = & H'_v \left( \tilde{V}_v\left(L\right) - Z_v \tilde{I}_v\left(L\right) \right) 
\end{array}
\end{equation}
%
and where $\tilde{V}^+_{L,ZT,v}$ is the voltage wave in the $+z$ direction resulting from the transfer impedance coupling, received at the $z=L$ termination. Similarly, $\tilde{V}^-_{0,ZT,v}$ is the voltage wave in the $-z$ direction resulting from the transfer impedance coupling, received at the $z=0$ termination.
%
The current in the source domain transmission line is given by equation \ref{eq:MTLequa_source_domain}. The distributed
voltage source on the victim transmission line is therefore given by
%
\begin{equation} \label{eq:ZT_source_definition_2b}
\tilde{V}_{ZT,v}\left(z\right) = \tilde{Z}'_T \left( z \right) \tilde{I}_s \left( z \right) = 
\frac{Z'_T\left( \omega \right)}{2Z_s} \left( \tilde{V}_s^+\left(0\right) e^{-j \beta_s z} - \tilde{V}_s^-\left(L\right) e^{j \beta_s \left(z-L\right)} \right) 
\end{equation}
%
At a point z on the victim transmission line, a voltage source $\tilde{V}_{ZT,v}~dz$ launches waves in both the -z and +z directions. The waves in the -z direction has amplitude
\begin{equation}
V^-dz=-\frac{\tilde{V}_{ZT,v}dz}{2}
\end{equation}
and the wave in the +z direction has amplitude
\begin{equation}
V^+dz=+\frac{\tilde{V}_{ZT,v}dz}{2}
\end{equation}
The currents associated with the waves are 
\begin{equation}
I^-dz=\frac{\tilde{V}_{ZT,v}dz}{2Z_v}
\end{equation}
and 
\begin{equation}
I^+dz=\frac{\tilde{V}_{ZT,v}dz}{2Z_v}
\end{equation}
Therefore the characteristic variables (section \ref{Method_of_characteristics}) for the -z travelling wave is
\begin{equation}
2V^-dz=\tilde{V} - Z_v \tilde{I} = -\tilde{V}_{ZT,v}dz
\end{equation}
and for the +z travelling wave it is
\begin{equation}
2V^+dz=\tilde{V} + Z_v \tilde{I} = \tilde{V}_{ZT,v}dz
\end{equation}
%
Thus the contribution to the -z travelling wave arriving at the z=0 termination of the victim transmission line is found by integrating the 
distributed contributions to the -z travelling wave from 0 to L i.e.
%
\begin{equation} \label{eq:characteristics_victim_domain_0}
\begin{array}{lcl}
\tilde{V}^-_{0,ZT,v} & = & \int_0^L 2V^-\left(z\right) dz \\[10pt]
  & = & \int_0^L -\tilde{V}_{ZT,v}\left(z\right) dz \\[10pt]
  & = & - \int_0^L \frac{Z'_T\left( \omega \right)}{2Z_s} \left( \tilde{V}_s^+\left(0\right) e^{-j \beta_s z} - \tilde{V}_s^-\left(L\right) e^{j \beta_s \left(z-L\right)} \right) e^{-j \beta_v z} dz \\[10pt]
  & = & -\frac{Z'_T\left( \omega \right)}{2Z_s}  \int_0^L \left( \tilde{V}_s^+\left(0\right) e^{-j \left( \beta_s + \beta_v \right) z} - \tilde{V}_s^-\left(L\right) e^{-j \beta_s L }e^{j \left( \beta_s - \beta_v \right) z}  \right)  dz \\[10pt]
\end{array}
\end{equation}
%
The contribution to the +z tranvelling wave at the z=L termination is found by integrating the distributed contributions from z=0 to z=L
%
\begin{equation} \label{eq:characteristics_victim_domain_L}
\begin{array}{lcl}
\tilde{V}^+_{L,ZT,v} & = & \int_0^L 2V^+\left(z\right) dz \\[10pt]
  & = & \int_0^L \tilde{V}_{ZT,v}\left(z\right) dz \\[10pt]
  & = & \int_0^L \frac{Z'_T\left( \omega \right)}{2Z_s} \left(\tilde{V}_s^+\left(0\right) e^{-j \beta_s z} -\tilde{V}_s^-\left(L\right) e^{j \beta_s \left(z-L\right)} \right) e^{-j \beta_v \left( L-z \right)} dz \\[10pt]
  & = & \frac{Z'_T\left( \omega \right)}{2Z_s}  \int_0^L \left(\tilde{V}_s^+\left(0\right) e^{-j \beta_v L }e^{-j \left( \beta_s - \beta_v \right) z} - \tilde{V}_s^-\left(L\right) e^{-j \left( \beta_s + \beta_v \right) \left( L-z \right)}  \right)  dz \\[10pt]
\end{array}
\end{equation}
%
In the time domain these equations become
%
\begin{equation} \label{eq:characteristics_victim_domain_0_time}
\begin{array}{rcl}
\tilde{V}_v^-\left(0\right)   & = & V_v\left(0,t\right)-Z_v I_v\left(0,t\right) \\[10pt]
 & = & H'_v\left(t\right)*\tilde{V}_v^-\left(L,t-T_{V}\right) +  H'_v\left(t\right)*V^-_{0,ZT,v}\left(t\right)
\end{array}
\end{equation}

%
where $T_v$  is the time delay of the victim transmission line. 

The first term in these equations are the method of characteristics propagation model already described in section \ref{Method_of_characteristics}.

The time domain source terms due to the transfer impedance coupling is described term by term as follows. Firstly the two terms contributing to the -z travelling wave arriving at the z=0 termination of the victim transmission line, $V^-_{0,ZT,v}$ in equation \ref{eq:characteristics_victim_domain_0}

-z propagation, term1: contribution to $V^-_{0,ZT,v}$ from the +z wave in the source transmission line:
%
\begin{multline} \label{eq:characteristics_victim_domain_0_term1}
-\frac{Z'_T\left( \omega \right)}{2Z_s}  \int_0^L \tilde{V}_s^+\left(0\right) e^{-j \left( \beta_s + \beta_v \right) z} dz = \\
  -\frac{1}{2Z_s} \left(-\frac{Z'_T\left( \omega \right)}{j\omega} \left( \frac{L}{T_z+T_v}\right) \left( 1-e^{-j \left( \beta_s + \beta_v \right) L} \right) \tilde{V}_s^+\left(0\right) \right)
\end{multline}
%
Transforming to the time domain gives
%
\begin{multline}\label{eq:characteristics_victim_domain_0_time_term1}
 \frac{1}{2Z_s} \frac{Z'_T\left( \omega \right)}{j\omega} \left( \frac{L}{T_z+T_v}\right) \left( 1-e^{-j \left( \beta_s + \beta_v \right) L} \right) \tilde{V}_s^+\left(0\right) 
 \leftrightarrow \\
  \frac{1}{2Z_s}\left( \frac{L}{T_s+T_v}\right) H'_{ZT}*\left( V_s^+\left(0,t\right) -  V_s^+\left(0,t - T_s - T_v\right) \right) 
\end{multline}
%
where the z-domain transfer function $H'_{ZT}\left( j\omega \right)$ = $\frac{Z'_T\left( j\omega \right)}{j\omega}$ and $H'_{ZT} \left( t \right)$ is the impulse response
of this transfer function. 

-z propagation, Term2: contribution to $V^-_{0,ZT,v}$ from the -z wave in the source transmission line:
%
\begin{multline} \label{eq:characteristics_victim_domain_0_time_term2}
\frac{Z'_T\left( \omega \right)}{2Z_s}  \int_0^L \tilde{V}_s^-\left(L\right)e^{-j \left( \beta_s L \right)} e^{j \left( \beta_s - \beta_v \right) z} dz
 \leftrightarrow \\
\frac{1}{2Z_s}\left( \frac{L}{T_s-T_v}\right) H'_{ZT}*\left( V_s^-\left(L,t-T_v\right) -  V_s^-\left(L,t - T_s \right) \right) 
\end{multline}
%
In a similar manner, the characteristic variables incident at the +z termination are expressed in the time domain as
%
\begin{equation} \label{eq:characteristics_victim_domain_L_time}
\begin{array}{rcl}
\tilde{V}_v^+\left(L\right)   & = & V_v\left(L,t\right)+Z_v I_v\left(L,t\right)   \\[10pt]  
   & = & H'_v\left(t\right)*\tilde{V}_v^+\left(0,t-T_{V}\right) + H'_v\left(t\right)*V^+_{L,ZT,v}\left(t\right)
\end{array}
\end{equation}

Secondly the two terms contributing to the +z travelling wave arriving at the z=L termination of the victim transmission line, $V^+_{L,ZT,v}$ in equation \ref{eq:characteristics_victim_domain_L}.

+z propagation, term1: contribution to $V^+_{L,ZT,v}$ from the +z wave in the source transmission line:
%
\begin{multline} \label{eq:characteristics_victim_domain_L_time_term1}
\frac{Z'_T\left( \omega \right)}{2Z_s}  \int_0^L \tilde{V}_s^+\left(0\right)e^{-j \left( \beta_v L \right) } e^{-j \left( \beta_s - \beta_v \right) z} dz
 \leftrightarrow \\
 \frac{1}{2Z_s}\left( \frac{L}{T_s-T_v}\right) H'_{ZT}*\left( V_s^+\left(0,t-T_v \right) -  V_s^+\left(0,t - T_s \right) \right)
\end{multline}
%
+z propagation, term2: contribution to $V^+_{L,ZT,v}$ from the -z wave in the source transmission line:
%
\begin{multline} \label{eq:characteristics_victim_domain_L_time_term2}
-\frac{Z'_T\left( \omega \right)}{2Z_s}  \int_0^L \tilde{V}_s^-\left(L\right) e^{-j \left( \beta_s + \beta_v \right) \left( L-z \right)}  dz
 \leftrightarrow \\
 -\frac{1}{2Z_s}\left( \frac{L}{T_s+T_v}\right) H'_{ZT}*\left( V_s^-\left(L,t\right) -  V_s^-\left(L,t - T_s - T_v\right) \right)
\end{multline}
%

\subsection{Multiple mode propagation}

The above theory is readily extended to the case of multi-mode systems, both in the source and victim domains. First we will discuss how to manage a multi-mode source domain before going on to derive the model for multi-mode victim domain.

\subsubsection{multiple modes in the source domain}\label{MM_source}

If there is more than one mode in the source domain then we must take into account that multiple modes can contribute to the shield current. There are two scenarios in which the shield is or is not the reference conductor for the source domain. These two scenarios must be treated separately. 

If the shield is \textbf{not} the reference conductor in the source domain then the conductor currents are related to the mode currents by
%
\begin{equation} 
\left(\tilde{I}\right)  =  \left[\tilde{T_{I}}\right]\left(\tilde{I_{m}}\right) 
\end{equation}

thus the shield current can be expressed in terms of the individual mode currents as
%
\begin{equation} 
\tilde{I_{shield}}  =  \sum_{i=1}^{N_m} \tilde{T_{I,shield,i}}\tilde{I_{m,i}}
\end{equation}

If the shield \textbf{is} the reference conductor then the reference conductor current is found as minus the sum of the currents on all the other conductors in the domain i.e.
%
\begin{equation} 
\begin{array}{rcl}
\tilde{I_{shield}} & = & -\sum_{j=1}^{N_m} \tilde{I_{j}} \\[10pt]
                   & = & -\sum_{j=1}^{N_m} \left( \sum_{i=1}^{N_m} \tilde{T_{I,j,i}}\tilde{I_{m,i}}  \right) \\[10pt]
                   & = &  \sum_{i=1}^{N_m} \left( -\sum_{j=1}^{N_m} \tilde{T_{I,j,i}}\right) \tilde{I_{m,i}}
\end{array}
\end{equation}

In either case the shield current is expressed as a linear combination of the individual mode currents in the source domain i.e.
\begin{equation}\label{Zt_weight_s}
\tilde{I_{shield}} = \sum_{i=1}^{N_m} P_{s,i} \tilde{I_{m,i}}
\end{equation}
where $P_{s,i}$ is the coefficient of the $i^{th}$ mode contribution. 

\subsubsection{multiple modes in the victim domain}\label{MM_victim}

The voltage source in the victim domain appears only on the shield conductor therefore
each victim domain mode $vm$ may recieve a contribution from this source term .
source and victim domains. 

If there are multiple modes in the victim domain there are again two situations which need to be treated separately dependent on whether the shield is or is not the reference conductor of the domain. In the victim domain the weak transfer impedance coupling model gives a voltage source on the shield conductor $V_{s,ZT}=Z_T I_{shield}$.

If the shield is \textbf{not} the reference conductor then the modal voltage can be related to the individual conductor voltages by
%
\begin{equation} 
\left(\tilde{V_m}\right)  =  \left[\tilde{T_{V}^{-1}}\right]\left(\tilde{V}\right) 
\end{equation}

thus the modal voltage source for mode i is found as
%
\begin{equation} 
\left(\tilde{V_{m,i}}\right)  =  \tilde{T_{V,i,shield}^{-1}}\left(\tilde{V_{s,ZT}}\right) 
\end{equation}

If the shield \textbf{is} the reference conductor in the victim domain then all the other conductors have a source voltages -$V_{s,ZT}$ on them. The modal voltage source for mode i is found as
%
\begin{equation} 
\left(\tilde{V_{m,i}}\right)  = -\sum_{j=1}^{N_m} \tilde{T_{V,i,j}^{-1}}\left(\tilde{V_{s,ZT}}\right) 
\end{equation}

Again, in either case, the contribution to mode i has a weighting factor $P_{v,i}$

\begin{equation}\label{Zt_weight_v}
\left(\tilde{V_{m,i}}\right)  = P_{v,i} \tilde{V_{s,ZT}}
\end{equation}

\subsubsection{multiple modes in the source domain and victim domains}

In order to generalise the transfer impedance coupling model to multiple source and victim domains, equations \ref{eq:characteristics_victim_domain_0_time} and \ref{eq:characteristics_victim_domain_L_time} must be generalised. This is done by recognising that if there are $ns$ source domain modes and $nv$ victim domain modes then there are $ns \times nv$ contributions to the transfer impedance coupling i.e. there is a contribution relating to every combination 
of source and victim modes.

The contribution from source mode, $s_{i}$, to victim mode, $v_{j}$, is readiliy found using the weighted values in equations \ref{Zt_weight_s} and \ref{Zt_weight_v}.

In this case equations \ref{eq:characteristics_victim_domain_0_time} and \ref{eq:characteristics_victim_domain_L_time} become:
%
\begin{equation} \label{eq:characteristics_victim_domain_L_time_generalised}
\begin{array}{rcl}
\tilde{V}^+_{vj}\left(L,t\right)   & = & V_{vj}\left(L,t\right)+Z_{vj} I_{vj}\left(L,t\right) \\
 & = & H'_{vj}\left(t\right)*\tilde{V}_{vj}^+\left(0,t-T_{Vj}\right) +\frac{P_{s,i}P_{v,j}}{2Z_si}  H_{ZT}*H'_{vj}\left(t\right)* \\
 & & \left( \left( \frac{L}{T_{si}-T_{vj}}\right) \left( V_{si}^+\left(0,t-T_v \right) -  V_{si}^+\left(0,t - T_{si} \right) \right) \right. \\
 & & \left.-\left( \frac{L}{T_{si}+T_{vj}}\right) \left( V_{si}^-\left(L,t\right) -  V_{si}^-\left(L,t - T_{si} - T_{vj} \right) \right) \right) 
\end{array}
\end{equation}
%
and
%
\begin{equation} \label{eq:characteristics_victim_domain_0_time_generalised}
\begin{array}{rcl}
\tilde{V}^-_{vj}\left(0,t\right)   & = &  V_{vj}\left(0,t\right)-Z_{vj} I_{vj}\left(0,t\right) \\
 & = & H'_{vj}\left(t\right)*\tilde{V}_{vj}^-\left(L,t-T_{V}\right)  -\frac{P_{s,i}P_{v,j}}{2Z_{si}}  H_{ZT}*H'_{vj}\left(t\right)*\\
 & & \left( \left( \frac{L}{T_{si}+T_{vj}}\right) \left( V_{si}^+\left(0,t\right) -      V_{si}^+\left(0,t - T_{si} - T_{vj}\right) \right) \right. \\
 & & \left.-\left( \frac{L}{T_{si}-T_{vj}}\right) \left( V_{si}^-\left(L,t-T_{vj}\right) -  V_{si}^-\left(L,t - T_{si}      \right) \right) \right) 
\end{array}
\end{equation}
%

\subsection{Spice circuit for Transfer Impedance model}\label{Zt_circuit}

The transfer impedance model equations may be implemented in spice using controlled sources, delay lines plus s-domain transfer functions to implement the frequency dependent transfer functions/ time domain convolutions. The Spice model for the main method of characteristics propagation model for souce domain mode smi, and victim domain model, vmi are seen in figure \ref{fig:spice_transfer_impedance_model_main}

\begin{figure}[h]
\centering
\includegraphics[scale=0.70]{./Imgs/spice_transfer_impedance_model.eps}
\caption{Spice circuit for the simulation of souce and victim domain modes with the method of characteristics with additional source term for transfer impedance coupling}
\label{fig:spice_transfer_impedance_model_main}
\end{figure}

The following delay lines are also required in addition to the delay lines required to implement the method of characteristics in the source and victim domains:

\begin{enumerate}
\item Delay line for $V_{si}^+\left(0,t - T_{si} - T_{vi}\right)$
\item Delay line for $V_{si}^+\left(0,t - T_{vi}\right)$
\item Delay line for $V_{si}^-\left(0,t - T_{si} - T_{vi}\right)$
\item Delay line for $V_{si}^-\left(0,t - T_{vi}\right)$
\end{enumerate}

The Spice model for these elements is shown in figure \ref{fig:spice_transfer_impedance_model_2}

\begin{figure}[h]
\centering
\includegraphics[scale=0.75]{./Imgs/spice_transfer_impedance_model2.eps}
\caption{Spice circuit for the simulation of souce and victim domain modes with the method of characteristics: Delay lines for transfer impedance cource term}
\label{fig:spice_transfer_impedance_model_2}
\end{figure}

When $T_s=T_v$ there is a problem implementing equations \ref{eq:characteristics_victim_domain_0_time_term2} and \ref{eq:characteristics_victim_domain_L_time_term1} in Spice. in this case it can be shown that
as $T_s \rightarrow T_v$ equation \ref{eq:characteristics_victim_domain_0_time_term2} becomes
%
\begin{multline} \label{eq:characteristics_victim_domain_0_time_term2b}
\frac{1}{2Z_s}\left( \frac{L}{T_s-T_v}\right) H_{ZT}*\left( V_s^-\left(L,t-T_v\right) -  V_s^-\left(L,t - T_s \right) \right)  \rightarrow \\
\frac{L}{2Z_s} H_{ZT}*\frac{dV_s^-\left(0,t-T_v \right)}{dt}
\end{multline}
%
The time derivative may be implemented in Spice using a 1H inductor in parallel with a current source of value $V_s^+\left(0,t-T_v \right)$ whose voltage will provide the time derivative as shown in  figure \ref{fig:spice_ZT_cct_special_1}
%
\begin{figure}[h]
\centering
\includegraphics[scale=0.75]{./Imgs/spice_transfer_impedance_model2_special.eps}
\caption{Spice circuit for the special case $T_s=T_v$}
\label{fig:spice_ZT_cct_special_1}
\end{figure}
%
Similarly, equation \ref{eq:characteristics_victim_domain_L_time_term1} becomes
%
\begin{multline} \label{eq:characteristics_victim_domain_L_time_term1_special}
 \frac{1}{2Z_s}\left( \frac{L}{T_s-T_v}\right) H_{ZT}*\left( V_s^+\left(0,t-T_v \right) -  V_s^+\left(0,t - T_s \right) \right) \rightarrow \\
 \frac{L}{2Z_s}\frac{dV_s^+\left(0,t-T_v \right)}{dt}
\end{multline}
%
and the Spice circuit takes a similar form to that shown in figure \ref{fig:spice_ZT_cct_special_1}

Controlled sources with s-domain transfer functions implement the convolutions in equations \ref{eq:characteristics_victim_domain_0_time_term1},  \ref{eq:characteristics_victim_domain_0_time_term2},  \ref{eq:characteristics_victim_domain_L_time_term1} and \ref{eq:characteristics_victim_domain_L_time_term2}. 



\clearpage
