\section{Spice Incident Field Excitation Model for shielded cables} \label{spice_incident_field_excitation_shielded_cables}

The development of Spice models for incident field excitation and also for transfer impedance coupling have been described in sections \ref{spice_incident_field_excitation} and \ref{transfer_impedance_model}. It has been shown in reference \cite{Xie} that a rigorous model of incident field excitation of shielded cables can be derived. This section provides an outline of the analysis presented in reference \cite{Xie} and gives the main results as implemented in the SACAMOS software.

The solution for the conductor voltage and current for a single mode, sm, in the external (source) domain may be expressed using the method of characteristics

\begin{equation} \label{eq:Einc_MOC_1a}
\begin{array}{rcl}
V_{sm}\left(z\right) & = & \frac{1}{2}\left( V_{sm}^+\left(0\right) e^{-j \beta_{sm} z} 
                                          + V_{sm}^-\left(L\right)  e^{+j \beta_{sm} z}
                                          + V_{Eim}^+\left(z\right)
                                          + V_{Eim}^-\left(z\right)
                                           \right) \\
\end{array}
\end{equation}
\begin{equation} \label{eq:Einc_MOC_1b}
\begin{array}{rcl}
I_{sm}\left(z\right) & = & \frac{1}{2 Z_{sm}}\left( V_{sm}^+\left(0\right) e^{-j \beta_{sm} z} 
                                              - V_{sm}^-\left(L\right)  e^{+j \beta_{sm} z}
                                              + V_{Eim}^+\left(z\right)
                                              - V_{Eim}^-\left(z\right)
                                           \right)
\end{array}
\end{equation}

where:

$V_{sm}^+$ is the forward mode in the source domain launched from the $z=0$ termination

$V_{sm}^-$ is the backward mode in the source domain launched from the $z=L$ termination

$V_{Eim}^+\left(z\right)$ is the contribution to the $+z$ travelling wave due to the incident field
excitation 

$V_{Eim}^-\left(z\right)$ is the contribution to the $-z$ travelling wave due to the incident field
excitation 

and $Z_{sm}$ and $\beta_{sm}$ are the characteristic impedance and propagation constant of the mode on the lossless transmission line.

The forward mode launched from the termaination at z=0, with the propagation correction for the mode applied is given by
\begin{equation} \label{eq:Einc_MOC_1a_PC}
V_{sm}^+\left(0\right) = H'_{sm} \left( V_{sm}\left( 0 \right) + Z_{sm} I_{sm}\left( 0 \right) \right)
\end{equation}
the backward (-z) propagating mode launched from the termination at z=L with the propagation correction for the mode applied is
\begin{equation} \label{eq:Einc_MOC_1b_PC}
V_{sm}^-\left(L\right) = H'_{sm} \left( V_{sm}\left( L \right) - Z_{sm} I_{sm}\left( L \right) \right)
\end{equation}

The +z propagating voltage wave due to the distributed incident field sources ($V_{F}$ and $I_{F}$ in equation \ref{eq:Spice_Einc_MTL}) is
found by integrating the contributions to this wave from 0 to z i.e.
%
\begin{equation} \label{eq:Einc_MOC_2a}
\begin{array}{rcl}
V_{Eim}^+\left(z\right)=\int_0^z \left( V_{F}\left( \tau \right) +Z_{sm} I_{F}\left( \tau \right)  \right)  e^{-j \beta_{sm} \left( z - \tau \right) } d\tau
\end{array}
\end{equation}
%
Similarly the -z propagating voltage wave due to the distributed incident field sources is found as
%
\begin{equation} \label{eq:Einc_MOC_2b}
\begin{array}{rcl}
V_{Eim}^-\left(z\right)=\int_L^z \left( V_{F}\left( \tau \right) -Z_{sm} I_{F}\left( \tau \right)  \right)  e^{+j \beta_{sm} \left( z - \tau \right) } d\tau
\end{array}
\end{equation}
%
The response on the inner (victim) domain is again written in terms of the characteristic variables launched from the terminations plus an additional term related to the transfer impedance coupling from the external domain current thus the characteristic variables incident at the terminations (and with the propagation correction for the victim domain applied) are expressed as

\begin{multline} \label{eq:Einc_MOC_3a}
\tilde{V}_{vm}^+\left(L\right)    =  V_{vm}\left(L\right) + Z_{vm} I_{vm}\left( L \right) \\
 = H'_{vm}\left( V_{vm}\left( 0 \right) - Z_{vm} I_{vm}\left( 0 \right) \right)e^{-j \beta_{vm} L} + H'_{vm}V^+_{L,Einc,ZT} \\
 = H'_{vm}\left( \tilde{V}_{vm}^+\left( 0 \right)   \right)e^{-j \beta_{vm} L} + H'_{vm}V^+_{L,Einc,ZT}
\end{multline}

\begin{multline} \label{eq:Einc_MOC_3b}
\tilde{V}_{vm}^-\left(0\right)    = V_{vm}\left(0\right) - Z_{vm} I_{vm}\left( 0 \right) \\
= H'_{vm}\left( V_{vm}\left( L \right) - Z_{vm} I_{vm}\left( L \right) \right)e^{-j \beta_{vm} L} + H'_{vm}V^-_{0,Einc,ZT} \\
= H'_{vm}\left( \tilde{V}_{vm}^-\left(L\right) \right)e^{-j \beta_{vm} L} + H'_{vm}V^-_{0,Einc,ZT}
\end{multline}

In the time domain, these equations become

\begin{equation} \label{eq:Einc_MOC_3a_time}
\tilde{V}_{vm}^+\left(L,t\right) = H'_{vm}*\left( \tilde{V}_{vm}^+\left(0,t-T_{vm}\right) \right) + H'_{vm}*V^+_{L,Einc,ZT}
\end{equation}

\begin{equation} \label{eq:Einc_MOC_3b_time}
\tilde{V}_{vm}^-\left( 0,t \right) = 
H'_{vm}*\left( \tilde{V}_{vm}^-\left( 0,t - T_{vm} \right)  \right) + H'_{vm}*V^-_{0,Einc,ZT}
\end{equation}

Where the transfer impedance coupling sources may be expressed as
\begin{equation} \label{eq:Einc_MOC_4a}
V^-_{0,ZT}=\frac{H_{ZT}}{2Z_{sm}}\left(j\omega\right) \left[ -V_{0,ext}^{+} + V_{0,ext}^{-}  \right]
\end{equation}
\begin{equation} \label{eq:Einc_MOC_4b}
V^+_{L,ZT}=\frac{H_{ZT}}{2Z_{sm}}\left(j\omega\right) \left[  V_{L,ext}^{+} - V_{L,ext}^{-}  \right]
\end{equation}

Transforming equations \ref{eq:Einc_MOC_4a} and \ref{eq:Einc_MOC_4a} into the time domain gives

\begin{equation} \label{eq:Einc_MOC_5a}
V^-_{0,ZT}=\frac{1}{2Z_{sm}} H_{ZT}(t)*\left[ -V_{0,ext}^{+}(t) + V_{0,ext}^{-}(t)  \right]
\end{equation}
\begin{equation} \label{eq:Einc_MOC_5b}
V^+_{L,ZT}=\frac{1}{2Z_{sm}} H_{ZT}(t)*\left[  V_{L,ext}^{+}(t) - V_{L,ext}^{-}(t)  \right]
\end{equation}

where $ V_{0,ext}^{+}$ and $V_{0,ext}^{-}$ are the contributions to the source term at z=0 due to the +z and -z propagating characteristics on the external conductor and
      $ V_{L,ext}^{+}$ and $V_{L,ext}^{-}$ are the contributions to the source term at z=L due to the +z and -z propagating characteristics on the external conductor 

\begin{equation} \label{eq:Einc_MOC_6a}
V_{0,ext}^{+}=\int_0^L V_{sm}^+\left(0\right) e^{-j \left( \beta_{sm}+\beta_{vm}\right) z} dz +
          \int_0^L V_{Eim}^+\left(z\right) e^{-j \left( \beta_{vm}\right) z} dz
\end{equation}

\begin{equation} \label{eq:Einc_MOC_6b}
V_{0,ext}^{-}=\int_0^L V_{sm}^-\left(L\right)e^{-j \beta_{sm} L} e^{j \left( \beta_{sm}-\beta_{vm}\right) z} dz +
          \int_0^L V_{Eim}^-\left(z\right) e^{-j \left( \beta_{vm}\right) z} dz
\end{equation}

\begin{equation} \label{eq:Einc_MOC_6c}
V_{L,ext}^{+}=\int_0^L V_{sm}^+\left(0\right) e^{-j \beta_{vm} L}e^{j \left( \beta_{vm}-\beta_{sm}\right) z} dz +
          \int_0^L V_{Eim}^+\left(z\right)e^{-j \beta_{vm} L}e^{j \beta_{vm} z} dz
\end{equation}

\begin{equation} \label{eq:Einc_MOC_6d}
V_{L,ext}^{-}=\int_0^L V_{sm}^-\left(L\right)e^{-j \beta_{vm+sm} L} e^{j \left( \beta_{sm}+\beta_{vm}\right) z} dz +
          \int_0^L V_{Eim}^-\left(z\right) e^{j \beta_{vm}\left( L- z\right)} dz
\end{equation}
      
      
Each of these terms has a contribution from the characteristic variables scattered from the terminations ( $V_{sm}$ terms ) and from
characteristic variables directly launched by the incident field source terms ( $V_{Eim}$ terms ) i.e. the source terms (including the propagation correction for the victim domain propagation in equations \ref{eq:Einc_MOC_3a} and \ref{eq:Einc_MOC_3b} may be expressed as

\begin{equation} \label{eq:Einc_MOC_5a_v2}
H'_{vm}*V^-_{0,ZT}=\frac{1}{2Z_{sm}} H'_{vm}H_{ZT}\left[ V^-_{0,ZT,vm}  + V^-_{0,Einc,ZT,sm,vm} \right]
\end{equation}

\begin{equation} \label{eq:Einc_MOC_5b_v2}
H'_{vm}*V^+_{L,ZT}=\frac{1}{2Z_{sm}} H'_{vm}H_{ZT}\left[ V^+_{L,ZT,vm} + V^+_{L,Einc,ZT,sm,vm} \right]
\end{equation}

The contributions from the characteristic variables scattered from the terminations $V^-_{0,ZT,v} $ and $V^+_{L,ZT,v}$ have already been described in section \ref{transfer_impedance_model}. Here we will describe how the additional terms related to the ( $V_{Eim}$ terms in equations \ref{eq:Einc_MOC_6a} to \ref{eq:Einc_MOC_6d} ) i.e. the direct coupling of the incident field through the shield before any interation with the terminations, may be included in the Spice cable bundel model. It is worth noting here that the influence of the incident field will be allowed to couple through the shield via the transfer impedance once it has scattered from the terminations when it is becomes a part of the  $V_{sm}^+$ and $V_{sm}^-$ terms. The additional terms discussed here are a refinement of the incident field coupling model and caould be considered to be a 'direct transfer impedance incident field coupling' model.

The contributions due to the direct transfer impedance incident field coupling model are
\begin{equation}
\begin{array}{rcl}
H'_{vm}*V^-_{0,Einc,ZT} & = & \frac{1}{2Z_{sm}} H'_{vm}H_{ZT} V^-_{0,Einc,ZT,sm,vm} \\[10pt]
 & = &\frac{1}{2Z_{sm}} H'_{vm}H_{ZT}\left[-E_{0\_Eim}^{+}+E_{0\_Eim}^{-}\right]
\end{array}
\end{equation}

where

\begin{equation} \label{eq:Einc_MOC_7a}
\begin{array}{rlc}
V_{0\_Eim}^{+} & = & \int_0^L V_{Eim}^+\left(z\right) e^{-j \beta_{vm} z} dz \\[10pt]
V_{0\_Eim}^{-} & = & \int_0^L V_{Eim}^-\left(z\right) e^{-j \beta_{vm} z} dz
\end{array}
\end{equation}

and

\begin{equation}
\begin{array}{rlc}
H'_{vm}*V^+_{L,Einc,ZT} & = & \frac{1}{2Z_{sm}} H'_{vm}H_{ZT} V^+_{L,Einc,ZT,sm,vm}\\[10pt]
 & = & \frac{1}{2Z_{sm}} H'_{vm}H_{ZT}\left[V_{L\_Eim}^{+}-V_{L\_Eim}^{-}\right]
\end{array}
\end{equation}

where

\begin{equation} \label{eq:Einc_MOC_7c}
\begin{array}{c}
V_{L\_Eim}^{+}=\int_0^L V_{Eim}^+\left(z\right)e^{-j \beta_{vm} L}e^{j \left( \beta_{vm}\right) z} dz \\[10pt]
V_{L\_Eim}^{-}=\int_0^L V_{Eim}^-\left(z\right)e^{-j \beta_{vm} L}e^{j \left( \beta_{vm}\right) z} dz
\end{array}
\end{equation}

If we assume a plane wave excitation as defined in equation \ref{eq:Einc_freq} in the time domain and \ref{eq:Einc} for the frequency domain
and identify the distributed source terms in the external domain as in equation \ref{eq:time_domain_forcing_function_VF} and \ref{eq:time_domain_forcing_function_IF}
then we can substitute \ref{eq:time_domain_forcing_function_VF} and \ref{eq:time_domain_forcing_function_IF} into  \ref{eq:Einc_MOC_2a} and \ref{eq:Einc_MOC_2b}.

Using the resulting expressions for $V_{Eim}^+\left(z\right)$ and $V_{Eim}^-\left(z\right)$ in equations \ref{eq:Einc_MOC_7a} and \ref{eq:Einc_MOC_7c} and evaluating the resulting double integrals leads to a frequency domain solution. Recognising the $e^{j \beta z}$ terms as representing time delays then leads to a time domain solution for the contributions to the terminations at z=0 and z=l as

\begin{equation}\label{EincZt_end1_source}
H'_{vm}*V^-_{0,Einc,ZT}=\frac{1}{2Z_{sm}} H'_{ZT_vm}*\left[-V_{0\_Eim}^{+}\left(t\right)+V_{0\_Eim}^{-}\left(t\right)\right]
\end{equation}

\begin{equation}\label{EincZt_end2_source}
H'_{vm}*V^+_{L,Einc,ZT}=\frac{1}{2Z_{sm}} H'_{ZT_vm}*\left[V_{L\_Eim}^{+}\left(t\right)-V_{L\_Eim}^{-}\left(t\right)\right]
\end{equation}


where $H'_{ZT_v}$ is the impulse response of the transfer function $\frac{H'_{vm}H_{ZT}}{j \omega}$ and 

\begin{multline} \label{eq:Einc_MOC_9a}
V_{0\_Eim}^{+}\left(t\right)=2dL\frac{e_t\left(T_z-T_{sm} \right)-e_z T_x}{T_z-T_{sm}}\left\{  
 \frac{E_0\left(t\right)-E_0\left( t-\left( T_{sm}+T_{vm} \right)\right)}{T_{sm}+T_{vm}}  \right. \\[10pt]
\left. -\frac{E_0\left(t\right)-E_0\left( t-\left( T_{z}+T_{vm} \right)\right)}{T_{z}+T_{vm}}\right\}
\end{multline}

\begin{multline} \label{eq:Einc_MOC_9b}
V_{0\_Eim}^{-}\left(t\right)=2dL\frac{e_t\left(T_z+T_{sm} \right)-e_z T_x}{T_z+T_{sm}} \left\{ 
-\frac{E_0\left(t\right)-E_0\left( t-\left( T_{z}+T_{vm} \right)\right)}{T_{z}+T_{vm}}\right. \\[10pt]
\left. -\frac{E_0\left( t-\left( T_{z}+T_{vm} \right)\right)-E_0\left( t-\left( T_{z}+T_{sm} \right)\right)}{T_{vm}-T_{sm}} \right\}
\end{multline}

\begin{multline} \label{eq:Einc_MOC_9c}
V_{L\_Eim}^{+}\left(t\right)=2dL \frac{e_t\left( T_z-T_{sm} \right)-e_z T_x}{T_z-T_{sm}} \left\{
 \frac{E_0\left(t-T_z\right) - E_0 \left( t-T_{vm}\right)}{T_{z}-T_{vm}} \right. \\[10pt]
\left.  -\frac{E_0\left(t-T_{vm}\right)-E_0\left( t-T_{sm} \right)}{T_{vm}-T_{sm}} \right\}
\end{multline}

\begin{multline} \label{eq:Einc_MOC_9d}
V_{L\_Eim}^{-}\left(t\right)=2dL\frac{e_t\left(T_z+T_{sm} \right)-e_z T_x}{T_z+T_{sm}}\left\{  
\frac{E_0\left(t-T_z\right)-E_0\left( t-T_{vm}\right)}{T_{z}-T_{vm}} \right. \\[10pt]
\left. +\frac{E_0\left(t-T_{z}\right) - E_0 \left( t- \left( T_{z}+T_{vm}+T_{sm} \right)\right)}{T_{vm}+T_{sm}} \right\}
\end{multline}

where $e_t$ and $e_z$ are the components of the Electric field in the plane of the cable from reference to the conductor in question and the z directions respectively, $d$ is the distance from the reference conductor to the conductor in question, $T_z=\frac{L}{v_z}$ and $T_x=\frac{L}{v_x}$ and $v_z$ and $v_x$ are the velocities of propagation of the incident field in the x and z directions. 

\subsection{Incident field excitation with a ground plane}

If there is a ground plane present then the incident field may be expressed as the superposition of a direct wave and the wave reflected from the ground and the source terms are found as described in section \ref{Einc_GP}.

\subsection{Multiple mode propagation}

As for the transfer impedance coupling model, the above theory is readily extended to the case of multi-mode systems, both in the external (source) domain and the shielded (victim) domain. The calculation of the contribution from each mode to the shield current is discussed in section \ref{MM_source}, similarly the distribution of a voltage source (due to transfer impedance coupling) to the modes in the victim (sheilded) domain is discussed in section \ref{MM_victim}. We can summarise these discissions by noting that:
\vspace{5mm}

1. The shield current is expressed as a linear combination of the individual mode currents in the source domain i.e.

\begin{equation}\label{EincZt_weight_s}
\tilde{I_{shield}} \sum_{i=1}^{N_m} P_{s,i} \tilde{I_{m,i}}
\end{equation}

2. the voltage source contribution to victim mode i has a weighting factor $P_{v,i}$

\begin{equation}\label{EincZt_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 direct transfer impedance incident field coupling model to multiple source and victim domains, equations \ref{EincZt_end1_source} and \ref{EincZt_end2_source} 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, smi, to victim mode, vmj, is readily found using the weighted values in equations \ref{EincZt_weight_s} and \ref{EincZt_weight_v}.

In this case equations \ref{EincZt_end1_source} and \ref{EincZt_end2_source} become:
%
\begin{equation}\label{EincZt_end1_source_MM}
H'_{vmj}*V^-_{0,Einc,ZT,smi,vmj}=\frac{1}{2Z_{smi}}P_{s,i}P_{v,j} H'_{ZT_vmj}*
\left[-E_{0\_Eim,smi,vmj}^{+}\left(t\right)+E_{0\_Eim,smi,vmj}^{-}\left(t\right)\right]
\end{equation}

\begin{equation}\label{EincZt_end2_source_MM}
H'_{vmj}*V^+_{L,Einc,ZT,smi,vmj}=\frac{1}{2Z_{smi}}P_{s,i}P_{v,j} H'_{ZT_vmj}* \left[E_{L\_Eim,smi,vmj}^{+}\left(t\right)-E_{L\_Eim,smi,vmj}^{-}\left(t\right)\right]
\end{equation}

where the expressions for $E_{0\_Eim,smi,vmj}^{+}\left(t\right)$, $E_{0\_Eim,smi,vmj}^{-}\left(t\right)$, $E_{L\_Eim,smi,vmj}^{+}\left(t\right)$ and $E_{L\_Eim,smi,vmj}^{-}\left(t\right)$ are evaluated using equations \ref{eq:Einc_MOC_9a} to \ref{eq:Einc_MOC_9d} with delays $T_{vm}=T_{vmi}$ and $T_{sm}=T_{smi}$

\subsection{Spice circuit for direct transfer impedance incident field coupling model}

The Spice circuit for the direct transfer impedance incident field coupling model is a variation on the circuit for transfer impedance coupling seen in figures \ref{fig:spice_transfer_impedance_model_main} and \ref{fig:spice_transfer_impedance_model_2}. The only change to the main circuit in figure \ref{fig:spice_transfer_impedance_model_main} is that there is an addional source term in series with the ususal transfer impedance csource term at the $z=0$ and $z=L$ terminations of the victim transmission line as shown in figure \ref{fig:Einc_ZT_1}. 

\begin{figure}[h]
\centering
\includegraphics[scale=0.75]{./Imgs/spice_Einc_ZT_model_1.eps}
\caption{Spice circuit for the victim domain mode direct transfer impedance incident field coupling model}
\label{fig:Einc_ZT_1}
\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 $E_{0}\left(t - T_{z} - T_{si} - T_{vi}\right)$
\item Delay line for $E_{0}\left(t - T_{si} - T_{vi}\right)$
\item Delay line for $E_{0}\left(t - T_{z} - T_{vi}\right)$
\item Delay line for $E_{0}\left(t - T_{z} - T_{si}\right)$
\item Delay line for $E_{0}\left(t - T_{vi}\right)$
\item Delay line for $E_{0}\left(t - T_{si}\right)$
\item Delay line for $E_{0}\left(t - T_{z}\right)$
\end{enumerate}

The Spice model for these elements is shown in figure \ref{fig:Einc_ZT_2}

\begin{figure}[h]
\centering
\includegraphics[scale=0.75]{./Imgs/spice_Einc_ZT_model_2.eps}
\caption{Delay lines required to implement the direct transfer impedance incident field coupling model}
\label{fig:Einc_ZT_2}
\end{figure}

As for the incident field model and the transfer impedance model, there are circumstances in which terms in equations \ref{eq:Einc_MOC_9a} to \ref{eq:Einc_MOC_9d} need to be modified i.e. when $T_z-T_{sm}=0$, $T_{vm}-T_{sm}=0$ or $T_{z}-T_{vm}=0$. In these cases the terms required are again (as in sections \ref{Einc_circuit} and \ref{Zt_circuit}) shown to become time derivatives of the delayed incident field function which can be realised by circuits similar to that in figure \ref{fig:spice_incident_field_cct_special_1} using a delay line and a 1H inductor in parallel with a controlled current source.

\clearpage