\section{Incident Field Excitation Model} \label{incident_field_excitation}

An electromagnetic field illuminating a transmission line can couple energy onto the line and 
therefore be a cause of interferance. The effect of an incident field excitation may
be included in the transmission line equations by distributed forcing functions $I_{F}$ and $V_{F}$ in the
transmssion line equations as described in references \cite{Agrawal}, \cite{PaulMTL} thus the transmission line equations with incident field excitation are written as

% 1. MTL equations with forcing functions
\begin{equation} \label{eq:MTLequa_einc}
\begin{array}{rcl}
\frac{\partial}{\partial z}\left(V\left(z,t\right)\right) & = & -\left[R\right]\left(I\left(z,t\right)\right) - \left[L\right]\frac{\partial}{\partial t}\left(I\left(z,t\right)\right) + V_{F}\left(z,t\right) \\[10pt]
\frac{\partial}{\partial z}\left(I\left(z,t\right)\right) & = & -\left[G\right]\left(V\left(z,t\right)\right) - \left[C\right]\frac{\partial}{\partial t}\left(V\left(z,t\right)\right)+ I_{F}\left(z,t\right) .
\end{array}
\end{equation}
%
Assuming that the conductors are widely separated, 
the time domain forcing functions may be written in terms of the incident electric field as

% 2. definition of forcing functions
\begin{equation} \label{eq:time_domain_forcing_function_VF}
V_{F}\left(z,t\right) = -\frac{\partial}{\partial z} E_{T}\left(z,t\right) + 
E_{L}\left(z,t\right)
\end{equation}
where for the i=th conductor
\begin{equation} \label{eq:time_domain_forcing_function_VF2}
E_{L}\left(z,t\right)_{i} = E^{inc}_{z}\left(ith\: conductor,z,t\right)-E^{inc}_{z}\left(reference\: conductor,z,t\right)
\end{equation}
and 
\begin{equation} \label{eq:time_domain_forcing_function_VF3}
E_{T}\left(z,t\right)_{i} = \int_{ref}^{conductor i} E^{inc}\left(z,t\right) \cdot dl
\end{equation}
and 
% 2. definition of forcing functions
\begin{equation} \label{eq:time_domain_forcing_function_IF}
I_{F}\left(z,t\right) =.-\left[G\right]\left(E_{T}\left(z,t\right)\right) - \left[C\right]\frac{\partial}{\partial t}\left(E_{T}\left(z,t\right)\right)
\end{equation}

Reference \cite{Agrawal} notes the following assumptions made in the derivation of this model for incident field excitation:

\begin{enumerate}
\item The propagation is transverse magnetic
\item The sum of the induced currents in any cross section, including the reference conductor is zero
\item $\frac{\partial E^s_z }{\partial z}$ is small compared with $\nabla_t \dot E_t$ 
\item The conductivity of the medium surrounding the conductors is homogeneous.
\end{enumerate}

A solution may be obtained in the frequency domain by the use of chain parameter matrices \cite{PaulMTL} which relate the
voltages and currents at one end of the transmission line to the volages and currents at the other. In
the absence of distributed forcing functions we can write

% 3. chain parameter solution
\begin{equation} \label{eq:chain_parameter_1}
\begin{array}{rcl}
V\left( L \right) & = & \Phi_{11}\left( L \right)  V\left( 0 \right) + \Phi_{12}\left( L \right)  I\left( 0 \right) \\[10pt]
I\left( L \right) & = & \Phi_{21}\left( L \right)  V\left( 0 \right) + \Phi_{22}\left( L \right)  I\left( 0 \right)
\end{array}
\end{equation}
%
where
\begin{equation} \label{eq:chain_parameter_2}
\begin{array}{rcl}
\Phi_{11}\left( L \right) & = & \left[Y\right]^{-1}\left[T\right]  \left[\cosh\left( \gamma L \right) \right] \left[T\right]^{-1}\left[Y\right]  \\[10pt]
\Phi_{12}\left( L \right)  & = & -\left[Y\right]^{-1}\left[T\right]  \left[ \gamma \sinh\left( \gamma L \right) \right] \left[T\right]^{-1}  \\[10pt]
          & = & -\left[Z_{C}\right]\left[T\right]  \left[  \sinh\left( \gamma L \right) \right] \left[T\right]^{-1}  \\[10pt]
\Phi_{21}\left( L \right)  & = & -\left[T\right]  \left[\sinh\left( \gamma L \right)  \gamma ^{-1} \right] \left[T\right]^{-1}\left[Y\right]  \\[10pt]
          & = & -\left[T\right]  \left[sinh\left( \gamma L \right)  \right] \left[T\right]^{-1}\left[Y_{C}\right]  \\[10pt]
\Phi_{22}\left( L \right)  & = & \left[T\right]  \left[\cosh\left( \gamma L \right) \right] \left[T\right]^{-1} \\[10pt]
\end{array}
\end{equation}

The distributed sources in equation \ref{eq:MTLequa_einc} launch waves in the $\pm z$ directions. The cumuulative effect of these waves may be calculated using the chain parameters for the transmission line and by this means, the effect of the incident field excitation on the transmission line may be included as lumped voltage and current sources at one end of the transmission line

% 3. chain parameter solution
\begin{equation} \label{eq:chain_parameter_3}
\begin{array}{rcl}
V\left( L \right) & = & \Phi_{11}\left( L \right)  V\left( 0 \right) + \Phi_{12}\left( L \right)  I\left( 0 \right) + V_{FT}\left( L \right) \\[10pt]
I\left( L \right) & = & \Phi_{21}\left( L \right)  V\left( 0 \right) + \Phi_{22}\left( L \right)  I\left( 0 \right) + I_{FT}\left( L \right) 
\end{array}
\end{equation}

where the incident field contribution appears as lumped voltage and current sources at the line termination $z=L$. These lumped source terms are given by the convolutions

\begin{equation} \label{eq:chain_parameter_4}
\begin{array}{rcl}
 V_{FT}\left( L \right) = \int_0^L \Phi_{11}\left( L-z \right) V_{F}\left( z \right) + \Phi_{12}\left( L-z \right) I_{F}\left( z \right) dz \\[10pt]
 I_{FT}\left( L \right) = \int_0^L \Phi_{21}\left( L-z \right) V_{F}\left( z \right) + \Phi_{22}\left( L-z \right) I_{F}\left( z \right) dz
\end{array}
\end{equation}
%
The source and load may be represented using Th\'{e}venin equivalent termination conditions:
%
\begin{equation} \label{eq:Thevenin_inc}
\begin{array}{rcl}
\left(\tilde{V}\left(0\right)\right) & = & \left(\tilde{V}_S\right) - \left[\tilde{Z}_S\right] \left(\tilde{I}\left(0\right)\right) \\[10pt]
\left(\tilde{V}\left(L\right)\right) & = & \left(\tilde{V}_L\right) + \left[\tilde{Z}_L\right] \left(\tilde{I}\left(L\right)\right),
\end{array}
\end{equation}
%
An equivalent circuit model of the incident field excitation is shown in figure \ref{Einc_equivalent_circuit}
%
\begin{figure}[h]
\centering
\includegraphics[scale=1.0]{./Imgs/Incident_field_equivalent_curcuit.eps}
\caption{Equivalent circuit for the incident field excitaiton solution }
\label{Einc_equivalent_circuit}
\end{figure}
%
In the second part of the figure the includent field sources have been combined into the termination Th\'{e}venin equivalent
circuit where
\begin{equation} \label{eq:Thevenin_inc_2}
\begin{array}{rcl}
\left[\tilde{Z}_L'\right] = \left[\tilde{Z}_L\right] \\[10pt]
\left(V'_{S}\right)=\left(V_{S}\right)+\left[\tilde{Z}_L\right] \left(I_{FT}\right)-\left(V_{FT}\right)
\end{array}
\end{equation}
Thus the solution may be found as for the unexcited transmission line for the termination voltages with one variation.
The solution for the termination voltage in the lower part of figure \ref{Einc_equivalent_circuit} includes $V_{FT}$ so to determine the actual transmission line voltages we have
\begin{equation} \label{eq:Thevenin_inc_3}
\begin{array}{rcl}
\left(V \left( L \right)\right) = \left(V' \left( L \right) \right)- \left(V_{FT}\right)
\end{array}
\end{equation}



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