\section{Spice Incident Field Excitation Model} \label{spice_incident_field_excitation}

The model for the incident field excitation is based on that described in reference \cite{PaulMTL}.
For the purposes of incident field excitaton analysis the axis of the cable bundle is assumed to be in the z direction. The direction of the incident field plane wave is specified by its wave vector in polar coordinates (figure \ref{fig:Incident_field_1})
%
\begin{figure}[h]
\centering
\includegraphics[scale=0.85]{./Imgs/Incident_field_specificaton.eps}
\caption{Specification of the incident field }
\label{fig:Incident_field_1}
\end{figure}
%

The wave vector in cartesian coordinates is
%
\begin{equation}
\begin{array}{rcl}
k_{x}=k_{0}\sin(\theta)\cos(\phi) \\[10pt]
k_{y}=k_{0}\sin(\theta)\sin(\phi) \\[10pt]
k_{z}=k_{0}\cos(\phi)
\end{array}
\end{equation}
%
where $k_{0}=\frac{\omega}{c}$
%
The wave velocity vector in cartesian coordinates is
%
\begin{equation}
\begin{array}{rcl}
v_{x}=\frac{ \omega }{k_{x}} \\[10pt]
v_{y}=\frac{ \omega }{k_{y}} \\[10pt]
v_{z}=\frac{ \omega }{k_{z}}  
\end{array}
\end{equation}
%
Given this k vector unit vectors in the $\theta$ and $\phi$ directions cen be defined. 
The Electric field polarisation direction is specified in terms of these vectors as shown 
in figure \ref{fig:Incident_field_2}
%
\begin{figure}[h]
\centering
\includegraphics[scale=1.0]{./Imgs/k_incident.eps}
\caption{Specification of the incident field polarisation}
\label{fig:Incident_field_2}
\end{figure}
%
Thus the $\theta$ and $\phi$ unit vectors are given by
%
\begin{equation}
\begin{array}{rcl}
\theta_{x}=\cos(\theta)\cos(\phi) \\[10pt]
\theta_{y}=\cos(\theta)\sin(\phi) \\[10pt]
\theta_{z}=\sin(\phi) 
\end{array}
\end{equation}
%
\begin{equation}
\begin{array}{rcl}
\phi_{x}=\sin(\phi) \\[10pt]
\phi_{y}=\cos(\phi) \\[10pt]
\phi_{z}=0 
\end{array}
\end{equation}
%
and thus the polarisation vector is
\begin{equation}
\begin{array}{rcl}
e_{x}=E_{\theta}\theta_{x}+E_{\phi}\phi_{x} \\[10pt]
e_{y}=E_{\theta}\theta_{y}+E_{\phi}\phi_{y} \\[10pt]
e_{z}=E_{\theta}\theta_{z}+E_{\phi}\phi_{z} 
\end{array}
\end{equation}
%
The frequency domain electric field is given by
\begin{equation} \label{eq:Einc_freq}
\begin{array}{rcl}
E_{x}=E_{0}(\omega) e^{-j\left( k_x x +k_y y +k_z z\right)} e_{x} \\[10pt]
E_{y}=E_{0}(\omega) e^{-j\left( k_x x +k_y y +k_z z\right)} e_{y} \\[10pt]
E_{z}=E_{0}(\omega) e^{-j\left( k_x x +k_y y +k_z z\right)} e_{z}
\end{array}
\end{equation}
%
The time domain electric field is given by
\begin{equation} \label{eq:Einc}
\begin{array}{rcl}
E_{x}=E_{0} \left( t-\frac{x}{v_x} - \frac{y}{v_y} - \frac{z}{v_z}\right) e_{x} \\[10pt]
E_{y}=E_{0} \left( t-\frac{x}{v_x} - \frac{y}{v_y} - \frac{z}{v_z}\right) e_{y} \\[10pt]
E_{z}=E_{0} \left( t-\frac{x}{v_x} - \frac{y}{v_y} - \frac{z}{v_z}\right) e_{z}
\end{array}
\end{equation}

The magnetic field components are given by
\begin{equation}
H=\frac{ \left( k \times E \right)}{Z_{0}}
\end{equation}
where $Z_{0}$ is the impedance of free space.

Section \ref{incident_field_excitation} describes the solution for incident field excitation of a transmission
line in the frequency domain. The starting point for the time domain spice model derivation for the
external domain in which conductors are directly illuminated by the incident field are the frequency domain
transmission line equations with distributed sources resulting from the incident field excitation

% 1. MTL equations with forcing functions
\begin{equation} \label{eq:Spice_Einc_MTL}
\begin{array}{rcl}
\frac{\partial}{\partial z}\left(V\left(z,j\omega \right)\right) + j \omega \left[L\right] \left(I\left(z,j\omega \right)\right) & = & 
\left(V_{F}\left(z,j\omega \right)\right) \\[10pt]
\frac{\partial}{\partial z}\left(I\left(z,j\omega \right)\right) + j \omega \left[C\right] \left(V\left(z,j\omega \right)\right) & = & 
\left(I_{F}\left(z,j\omega \right)\right) .
\end{array}
\end{equation}
%
We have assumed that the transmission line is lossless here however we will introduce the propagation correction to the
incident field excitation terms in the method of characteristics solution so losses will be taken into account in the
same manner as for the propagation model.

The voltages and currents at the ends of the transmission line can be related by the chain parameters

% Chain parameter solution
\begin{equation} \label{eq:Spice_Einc_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) + 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 $z=L$. The chain parameters, $\Phi_{11}$, $\Phi_{12}$, $\Phi_{21}$ and $\Phi_{22}$ are given by equation \ref{eq:chain_parameter_2}.
The spice model will be constructed in this form with the effect of the incident field excitaiton being implemented
using source terms at one end of the transmission line.
These lumped source terms are given by the convolutions
%
\begin{equation} \label{eq:Spice_Einc_lumped_sources}
\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 spice multi-conductor propagation model is based on modal decomposition so we proceed by transforming equation \ref{eq:Spice_Einc_chain_parameter_1} to modal quantities
% Chain parameter solution
\begin{equation} \label{eq:Spice_Einc_modal_chain_parameter_1}
\begin{array}{rcl}
V_m\left( L \right) & = & \Phi_{m11}\left( L \right)  V_m\left( 0 \right) + \Phi_{m12}\left( L \right)  I_m\left( 0 \right) + V_{FTm}\left( L \right) \\[10pt]
I_m\left( L \right) & = & \Phi_{m21}\left( L \right)  V_m\left( 0 \right) + \Phi_{m22}\left( L \right)  I_m\left( 0 \right) + I_{FTm}\left( L \right) 
\end{array}
\end{equation}
%
where the (diagonal) modal chain parameter matrices are given by
\begin{equation} \label{eq:Spice_Einc_modal_chain_parameter_definitions}
\begin{array}{rcl}
\Phi_{m11}\left( L \right) & = & \left[T_V\right]^{-1}  \Phi_{11}\left( L \right) \left[T_V\right]  \\[10pt]
                           & = &  \cosh\left( \gamma_m L \right) \\
\Phi_{m12}\left( L \right) & = &  \left[T_V\right]^{-1}  \Phi_{12}\left( L \right) \left[T_I\right] \\[10pt]
                           & = & -\left[Z_{m}\right] \sinh\left( \gamma_m L \right)   \\
\Phi_{m21}\left( L \right) & = &  \left[T_I\right]^{-1}  \Phi_{21}\left( L \right) \left[T_V\right] \\ [10pt]
                           & = & -\sinh\left( \gamma_m L \right) \left[Z_{m}\right]^{-1} \\
\Phi_{m22}\left( L \right) & = &  \left[T_I\right]^{-1}  \Phi_{22}\left( L \right) \left[T_I\right] \\[10pt]
                           & = &  \cosh\left( \gamma_m L \right) 
\end{array}
\end{equation}
%
and the modal forcing functions (using the properties of the modal decomposition matrices in equations \ref{eq:Lm_Cm}) may be expressed as
%
\begin{equation} \label{eq:Spice_Einc_modal_lumped_voltage_source}
\begin{array}{rcl}
 V_{FTm}\left( L \right) & = &  \left[T_V\right]^{-1} V_{FT}\left( L \right) \\[10pt]
                         & = &  \int_0^L \cosh\left( \gamma_m L-z \right)  V_{Fm}\left( z \right) \\[10pt]
                         &   &  -\sinh\left( \gamma_m L-z \right) \left[Z_{cm}\right]I_{Fm}\left( z \right)  dz
\end{array}
\end{equation}
%
\begin{equation} \label{eq:Spice_Einc_modal_lumped_current_source}
\begin{array}{rcl}
 I_{FTm}\left( L \right) & = &  \left[T_I\right]^{-1} I_{FT}\left( L \right) \\[10pt]
                         & = &  \int_0^L -\sinh\left( \gamma_m L-z \right) \left[Z_{cm}\right]^{-1} V_{Fm}\left( z \right) \\[10pt]
                         &   &  +\cosh\left( \gamma_m L-z \right) I_{Fm}\left( z \right)  dz
\end{array}
\end{equation}
%
where $\gamma_m$ is the diagonal matrix of mode propagation constants and $Z_cm$ is the diagonal matrix of mode impedances,  $V_{Fm}=[T_V]^{-1}V_F$ and $I_{Fm}=[T_I]^{-1}I_F$.

The modal characteristic varibles at $z=0$ and $z=L$ are found by first substituting the modal chain parameters 
(equation \ref{eq:Spice_Einc_modal_chain_parameter_definitions}) into the expressions for
the modal voltage and current expressions (equation \ref{eq:Spice_Einc_modal_chain_parameter_1})
%                                                          
\begin{equation} \label{eq:Spice_Einc_modal_characteristics}
\begin{array}{rcl}
V_m\left( L \right) & = &  \cosh\left( \gamma_m L \right) V_m\left( 0 \right)
                          -\sinh\left( \gamma_m \left( z \right) \right) \left[Z_{m}\right] I_m \left( 0 \right) + V_{FTm} \left( L \right) \\[10pt]
\left[Z_{m}\right] I_m\left( L \right) & = & -\sinh\left( \gamma_m z \right) V_m\left( 0 \right)             
                                             +\cosh\left( \gamma_m z \right) \left[Z_{m}\right] I_m\left( 0 \right) 
                                             +\left[Z_{m}\right]I_{FTm}\left( L \right) 
\end{array}
\end{equation}
%
The characteristic variables are found by adding and subtracting these equations to give
%
\begin{equation} \label{eq:Spice_Einc_modal_characteristics_2}
\begin{array}{rcccl}
V^-_{m,0} & = & V_m\left( 0 \right)- \left[Z_{m}\right] I_m\left( 0 \right) & = &  e^{-\gamma_m L} \left[V_m\left( L \right)- \left[Z_{m}\right] I_m\left( L \right) \right] 
                                                                   + E_{inc0}\left( L \right)  \\[10pt]
V^+_{m,L} & = & V_m\left( L \right)+ \left[Z_{m}\right] I_m\left( L \right) & = &  e^{-\gamma_m L} \left[V_m\left( 0 \right)+ \left[Z_{m}\right] I_m\left( 0 \right) \right] 
                                                                   + E_{incL}\left( L \right)    
\end{array}
\end{equation}
%
The incident field source terms are 
%
\begin{equation} \label{eq:Spice_Einc_modal_characteristics_sources_1}
\begin{array}{lcl}
E_{inc0}\left( L \right) & = & -e^{-\gamma_m L} \left[ V_{FTm}\left( L \right) - \left[Z_{m}\right] I_{FTm}\left( L \right) \right] \\[10pt]
E_{incL}\left( L \right) & = & \left[ V_{FTm}\left( L \right) + \left[Z_{m}\right] I_{FTm}\left( L \right) \right]  
\end{array}
\end{equation}
%
The exponential in equations \ref{eq:Spice_Einc_modal_characteristics_2} and \ref{eq:Spice_Einc_modal_characteristics_sources_1}
may be recognised as a time delay therefore equation \ref{eq:Spice_Einc_modal_characteristics_2} may be expressed in the time domain 
for mode i as
%
\begin{equation} \label{eq:Spice_Einc_modal_characteristics_2}
\begin{array}{rcl}
V^-_{m,0}(t) & = & V_{mi}\left( 0,t \right)- \left[Z_{mi}\right] I_{mi}\left( 0,t \right) \\[10pt] 
  & = &  \left[V_{mi}\left( L,t-T_{mi} \right)- \left[Z_{mi}\right] I_{mi}\left( L,t-T_{mi}  \right) \right] 
                                                                   + E_{inc0,i}\left( L,t \right)  \\[10pt]
V^+_{m,0}(t) & = & V_{mi}\left( L,t \right)+ \left[Z_{mi}\right] I_{mi}\left( L,t \right) \\[10pt]
 & = &   \left[V_{mi}\left( 0,t -T_{mi} \right)+ \left[Z_{mi}\right] I_{mi}\left( 0,t-T_{mi}  \right) \right] 
                                                                   + E_{incL,i}\left( L,t \right)    
\end{array}
\end{equation}
%
$T_{mi}$ is the delay in mode i, propagating a distance L.
\begin{equation} \label{eq:Spice_Einc_Tmi}
T_{mi}=\frac{L}{v_{mi}}=\frac{\gamma_{mi} L}{j \omega }
\end{equation}

Equations \ref{eq:Spice_Einc_modal_characteristics_2} express the method of characteristics based model as described in section \ref{Method_of_characteristics} with the addition of sources which model the effect of the incident
field coupling. In order to develop a viable Spice model for incident field coupling, we need to develop a circuit model for the time domain source terms $E_{inc0,i}\left( L,t \right)$ and $E_{incL,i}\left( L,t \right)$  

If the incident field takes the form of a plane wave then the source terms $E_{inc0,i}\left( L,t \right)$ and $E_{incL,i}\left( L,t \right)$ may be 
derived analytically to give a complete time domainspice model for incident field excitation of multi-conductor transmission lines which
may be implemented using delay lines and controlled sources. 

%The time domain incident field sources are
%\begin{equation} \label{eq:Spice_Einc_sources}
%\begin{array}{rcl}
%E_{inc0,i}\left( L,t \right) & = & - V_{FTmi}\left( L,t-T_{mi} \right) + Z_{mi}I_{FTmi}\left( L,t-T_{mi} \right)  \\
%E_{incL,i}\left( L,t \right) & = &   V_{FTmi}\left( L,t \right) + Z_{mi}I_{FTmi}\left( L,t \right) 
%\end{array}
%\end{equation}

Substituting the expressions for the distributed incident field sources \ref{eq:time_domain_forcing_function_VF} and \ref{eq:time_domain_forcing_function_IF}, transformed to the frequency domain, with the Electric field expressions for a plane wave (\ref{eq:Einc_freq}) allows the evaluation of the modal forcing functions in equations \ref{eq:Spice_Einc_modal_lumped_voltage_source} and \ref{eq:Spice_Einc_modal_lumped_current_source}. The modal incident field transmission line source functions may be written in terms of the transverse and longitudinal electric field contributions $E_T\left(z\right)$ and $E_L\left(z\right)$ respectively
%
\begin{equation} \label{eq:Spice_Einc_modal_characteristics_sources_1}
\begin{array}{rcl}
E_{inc0}\left( L \right) & = & -e^{-\gamma_m L} \left[ V_{FTm}\left( L \right) - \left[Z_{m}\right] I_{FTm}\left( L \right) \right] \\[10pt]
                         & = &  -\int_0^L e^{-\gamma_m z}\left[T_{I}\right]^{t}E_L\left(z\right) dz                                 \\[10pt]
                         &   &  -e^{-\gamma_m L}\left[T_{I}\right]^{t}E_T\left(L\right)-\left[T_{I}\right]^{t}E_T\left(0\right)     \\[10pt]
E_{incL}\left( L \right) & = & \left[ V_{FTm}\left( L \right) + \left[Z_{m}\right] I_{FTm}\left( L \right) \right]        \\[10pt]
                         & = &  \int_0^L e^{-\gamma_m (L-z)}\left[T_{I}\right]^{t}E_L\left(z\right) dz       \\[10pt]
                         &   &  -\left[T_{I}\right]^{t}E_T\left(L\right)+e^{-\gamma_m L}\left[T_{I}\right]^{t}E_T\left(0\right)
\end{array}
\end{equation}
%
where the transverse electric field contribution for the incident field expression given in equation \ref{eq:Einc_freq} for conductor k is 

\begin{multline} \label{eq:Spice_Einc_ET_1}
E_T\left(z\right)_k  =  \int_{ref}^{conductor i} E^{inc}(\omega) \cdot dl \\[10pt]
  =  \int_{0}^{d_{k}} E_{inc}( \omega ) \left( e_x \frac{x_k-x_0}{d_i} + e_y \frac{y_k-y_0}{d_k} \right)e^{-j k_z z}
    e^{-j \left( k_x \frac{x s-x_0}{d_k} + k_y \frac{y s-x_0}{d_k}\right) } dl   \\[10pt]
  =  E_{inc}( \omega ) \left( e_x \frac{x_k-x_0}{d_k} + e_y \frac{y_k-y_0}{d_k} \right)e^{-j k_z z}
    e^{j \left( k_x \frac{x_0}{d_k} + k_y \frac{x_0}{d_k}\right) }
    \int_{0}^{d_{k}} e^{-j \left( k_x \frac{x}{d_k} + k_y \frac{y}{d_k}\right)s } dl  \\[10pt]
  =   E_{inc}( \omega )  \left( e_x \left(x_k-x_0\right) + e_y \left(y_k-y_0\right) \right)e^{-j k_z z}
    e^{j \left( k_x \frac{x_0}{d_k} + k_y \frac{x_0}{d_k}\right) }
    e^{-j\psi_k }\frac{\sin(\psi_k)}{\psi_k} 
\end{multline}

%\begin{equation} \label{eq:Spice_Einc_ET_1}
%\begin{array}{rcl}
%E_T\left(z\right)_k & = & \int_{ref}^{conductor i} E^{inc}(\omega) \cdot dl \\[10pt]
% & = & \int_{0}^{d_{k}} E_{inc}( \omega ) \left( e_x \frac{x_k-x_0}{d_i} + e_y \frac{y_k-y_0}{d_k} \right)e^{-j k_z z}
%    e^{-j \left( k_x \frac{x s-x_0}{d_k} + k_y \frac{y s-x_0}{d_k}\right) } dl   \\[10pt]
%
% & = & E_{inc}( \omega ) \left( e_x \frac{x_k-x_0}{d_k} + e_y \frac{y_k-y_0}{d_k} \right)e^{-j k_z z}
%    e^{j \left( k_x \frac{x_0}{d_k} + k_y \frac{x_0}{d_k}\right) }
%    \int_{0}^{d_{k}} e^{-j \left( k_x \frac{x}{d_k} + k_y \frac{y}{d_k}\right)s } dl  \\[10pt]
%    
% & = &  E_{inc}( \omega )  \left( e_x \left(x_k-x_0\right) + e_y \left(y_k-y_0\right) \right)e^{-j k_z z}
%    e^{j \left( k_x \frac{x_0}{d_k} + k_y \frac{x_0}{d_k}\right) }
%    e^{-j\psi_k }\frac{\sin(\psi_k)}{\psi_k} 
%\end{array}
%\end{equation}

where $(x_k, y_k)$ is the position of the kth conductor in the x-y plane,  $(x_0, y_0)$ is the position of the reference conductor in the x-y plane
and $d_k$ is the distance between these conductors and
\begin{equation} \label{eq:Spice_Einc_alpha}
\psi_k=\frac{ k_x x_k+k_y y_k}{2}
\end{equation}
%
If we assume that the origin in the x-y plane is at the position of the reference conductor then we can write
\begin{equation} \label{eq:Spice_Einc_ET_2}
\begin{array}{rcl}
E_T\left(z\right)_k & = &   E_{inc}( \omega )  \left( e_x x_k + e_y y_k \right)e^{-j k_z z}
                        e^{-j\psi_k }\frac{\sin(\psi_k)}{\psi_k}
\end{array}
\end{equation}
% \\[10pt]
The contribution from the longitudinal field is
%
\begin{equation} \label{eq:Spice_Einc_EL_1}
\begin{array}{lcl}
E_L\left(z\right)_i & = & E_{0}(\omega) \left( e^{-j\left( k_x x_k +k_y y_k +k_z z\right)}-  e^{-j\left( k_z z \right)}\right) \\[10pt]
                    & = & E_{0}(\omega) e^{-j k_z z} \left( e^{-j\left( k_x x_k +k_y y_k\right)}-  1\right) \\[10pt]
                    & = & -j \left( k_x x_k +k_y y_k \right)  E_{0}(\omega) e^{-j\psi_k } \frac{\sin(\psi_k)}{\psi_k}e^{-j k_z z}
\end{array}
\end{equation}
The shift of the origin of the cross section coordinate system will only have the effect of adding a timeshift to the incident field function.

The cross section time delay is defined for conductor k as
%
\begin{equation} \label{eq:Spice_Einc_Txyk}
T_{xyk} = \frac{x_k-x_0}{v_x}+\frac{y_k-y_0}{v_y}
\end{equation}
%
The time delay for the incident field to travel the length of the transmission line is denoted $T_z$ and
the delay for mode i is denoted $T_i$ where
\begin{equation} \label{eq:Spice_Einc_Tz}
T_z = \frac{L}{v_z}
\end{equation}
and
\begin{equation} \label{eq:Spice_Einc_Tz}
T_i = \frac{L}{v_i} = \frac{\gamma_i L }{j\omega}
\end{equation}
%
Following \cite{PaulMTL} equations \ref{eq:Spice_Einc_modal_characteristics_sources_1} can be expressed for each mode i as
%
\begin{multline} \label{eq:Spice_Einc_modal_characteristics_sources_2a}
E_{inc0}\left( L \right)_i =   
E_{0}(\omega) \sum_{k=1}^{n} \left\{ \left[ \frac{\sin(\psi_k)}{\psi_k} \right] \left[T_{I}\right]^{t}_{i,k} \right. \\[10pt]
\left. \left[  e_z T_{xyk} L + \left( e_x x_k +e_y y_k \right) \left( T_i - T_z \right) \right]    
     \left[ \frac{e^{-j\omega \left(T_{xyk}/2 \right) } - e^{-j\omega \left(T_i+T_z+T_{xyk}/2 \right) }}{\left(T_i+T_z \right)}  \right] \right\} \\[10pt]
\end{multline}
%
\begin{multline} \label{eq:Spice_Einc_modal_characteristics_sources_2b}
E_{incL}\left( L \right)_i  =   
E_{0}(\omega)sum_{k=1}^{n} \left\{ \left[ \frac{\sin(\psi_k)}{\psi_k} \right] \left[T_{I}\right]^{t}_{i,k} \right. \\[10pt]
\left.   \left[  e_z T_{xyk} L + \left( e_x x_k +e_y y_k \right) \left( T_i - T_z \right) \right]                                      
   \left[ \frac{e^{-j\omega \left(T_i + T_{xyk}/2 \right) } - e^{-j\omega \left(T_z+T_{xyk}/2 \right) }}{\left(T_i-T_z \right)}  \right] \right\}
\end{multline}
%
The transformation to the time domain is again acheived by recognising the $e^{-j \omega T}$ terms as time delays to give

\begin{multline} \label{eq:Spice_Einc_modal_characteristics_sources_time}
E_{inc0}\left( t \right)_i  =  
\sum_{k=1}^{n} \left\{ \left[ e_z T_{xyk} L - \left( e_x x_k +e_y y_k \right)\left( T_i+T_z\right)  \right] \left[T_{I}\right]^{t}_{i,k} \right. \\[10pt]
\left.  p_k(t) * \frac{ E_0\left( t-T_{xyk}/2\right) - E_0\left( t-T_i - T_z - T_{xyk}/2\right) }{\left(T_i+T_z \right)}  \right\} \\[10pt]
E_{incL}\left( t \right)_i = 
\sum_{k=1}^{n} \left\{\left[ e_z T_{xyk} L + \left( e_x x_k +e_y y_k \right)\left( T_i-T_z\right)  \right] \left[T_{I}\right]^{t}_{i,k}  \right. \\[10pt]
\left.   p_k(t) * \frac{ E_0\left( t-T_i-T_{xyk}/2\right) - E_0\left( t - T_z - T_{xyk}/2\right) }{\left(T_i-T_z \right)}  \right\} \\[10pt]
\end{multline}
Where $p_k(t)$ is a pulse function for the kth conductor which is convolved with the delayed incident field functions. Provided that the transmission
line cross section is small it may be assumes that the pulse function is approximately an impulse function, we may also assmue that the
cross section delay time is negligable. Under this assumption the convolution is eliminated and the source terms are for mode i,
%
\begin{equation} \label{eq:Spice_Einc_modal_characteristics_sources_time_2}
\begin{array}{rcl}
E_{inc0}\left( t \right)_i & = &  \alpha_0 \left[ \frac{ E_0\left( t \right) - E_0\left( t - T_i - T_z \right) }{\left(T_i+T_z \right)}  \right] \\[10pt]
E_{incL}\left( t \right)_i & = &  \alpha_L \left[ \frac{ E_0\left( t -T_i\right) - E_0\left( t -  T_z \right) }{\left(T_i-T_z \right)}  \right] \\[10pt]
\end{array}
\end{equation}
%
where the vectors $\alpha_0$ and $\alpha_L$ are for mode i
\begin{equation} \label{eq:Spice_Einc_modal_characteristics_sources_alpha}
\begin{array}{rcl}
\alpha_{0,i} & = &  \sum_{k=1}^{n} \left\{ \left[ e_z T_{xyk} L - \left( e_x x_k +e_y y_k \right)\left( T_i+T_z\right)  \right] \left[T_{I}\right]^{t}_{i,k} \right\} \\[10pt]
\alpha_{L,i} & = &  \sum_{k=1}^{n} \left\{ \left[ e_z T_{xyk} L + \left( e_x x_k +e_y y_k \right)\left( T_i-T_z\right)  \right] \left[T_{I}\right]^{t}_{i,k} \right\} \\[10pt]
\end{array}
\end{equation}
%
The characteristic based expression for the incident field model in equation \ref{eq:Spice_Einc_modal_characteristics_2} may be written as
%
\begin{equation} \label{eq:Spice_Einc_modal_characteristics_3}
\begin{array}{lcl}
V^-_{m,0}(t) & = & V_{mi}\left( 0,t \right)- \left[Z_{mi}\right] I_{mi}\left( 0,t \right) \\[10pt]
 & = &        \left[V_{mi}\left( L,t-T_{mi} \right)- \left[Z_{mi}\right] I_{mi}\left( L,t-T_{mi}  \right) \right]  
    + \alpha_{0,i} \left[ \frac{ E_0\left( t \right) - E_0\left( t - T_i - T_z \right) }{\left(T_i+T_z \right)}  \right]     \\[10pt]
V^+_{m,0}(t) & = & V_{mi}\left( L,t \right)+ \left[Z_{mi}\right] I_{mi}\left( L,t \right) \\[10pt]
 & = &            \left[V_{mi}\left( 0,t -T_{mi} \right)+ \left[Z_{mi}\right] I_{mi}\left( 0,t-T_{mi}  \right) \right] 
  + \alpha_{L,i} \left[ \frac{ E_0\left( t -T_i\right) - E_0\left( t -  T_z \right) }{\left(T_i-T_z \right)}  \right]                                                
\end{array}
\end{equation}

These incident field source terms in this model are seen to consist of the time domain incident field function, $E_0\left( t \right) $ and 
delayed versions of this, each scaled by appropriate constants which are found in terms of the angle of incidence and polarisation of the incident plane wave (equations \ref{eq:Spice_Einc_modal_characteristics_sources_alpha}).

\subsection{Spice circuit for Incident field excitation}\label{Einc_circuit}
%
The equations \ref{eq:Spice_Einc_modal_characteristics_3} may be implemented in Spice using controlled sources and delay lines.
Two delay lines are required for each transmission line mode for the propagation model (delay $T_i$). In addition, for each mode, 
two delay lines with delays $T_i$ and $T_i+T_z$ are required for the incident field and a final delay line with delay $Tz$ is required. 
The structure of the Spice circuit model is shown in figure \ref{fig:spice_incident_field_cct}.

\begin{figure}[h]
\centering
\includegraphics[scale=0.75]{./Imgs/spice_incident_field_cct.eps}
\caption{Spice circuit for the simulation of a multi-conductor transmission line illuminated by a plane wave}
\label{fig:spice_incident_field_cct}
\end{figure}

If $T_i=T_z$ for any mode, the second term in equation \ref{eq:Spice_Einc_modal_characteristics_3} has a zero in the denominator
in this case the last term in square brackets may be rewritten as
%
\begin{equation} \label{eq:Spice_Einc_modal_characteristics_3b}
\left[ \frac{ E_0\left( t -T_i\right) - E_0\left( t -  T_z \right) }{\left(T_i-T_z \right)}  \right]_{T_i \Rightarrow T_z}=\frac{d}{dt}   E_0\left( t - T_i\right)
\end{equation}
%
The time derivative may be implemented by an inductor of 1H driven by a current source as shown in figure \ref{fig:spice_incident_field_cct_special_1}

\begin{figure}[h]
\centering
\includegraphics[scale=0.75]{./Imgs/spice_incident_field_cct_special_1.eps}
\caption{Spice circuit for the special case $T_i=T_z$ }
\label{fig:spice_incident_field_cct_special_1}
\end{figure}

A second practical issue is for the case when $Tz<0$ i.e. if $k_z < 0$ in this case the model would require negative time delays. This 
case is simply dealt with by reversing the transmission line terminations and $k_z$.

The Spice model for incident field excitation provides the incident field equivalent source in the form of the characteristic variables incident on the terminations of the transmission line i.e. 
$V_m\left(0\right)-Z_m I_m\left(0\right)$ and $V_m\left(L\right)-Z_m I_m\left(L\right)$. These waves require correction for the propagation losses in the transmission line. In order to include the 
propagation correction into the model these characteristic variables are multiplied by (convolved with) the appropriate mode propagation correction $H'(j \omega)$ giving the source terms on the transmission line terminations for mode i as
%
\begin{equation} \label{eq:Spice_Einc_modal_characteristics_3_corrected}
\begin{array}{rcl}
V_{Einc}^-(0,t) & = & H'_i(t)*\left( V_{mi}\left( 0,t \right)- \left[Z_{mi}\right] I_{mi}\left( 0,t \right) \right) \\[10pt]
V_{Einc}^+(L,t) & = & H'_i(t)*\left( V_{mi}\left( L,t \right)+ \left[Z_{mi}\right] I_{mi}\left( L,t \right) \right) \\[10pt]   \end{array}
\end{equation}

\subsection{Incident field excitation with a ground plane}\label{Einc_GP}

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. Working through the analysis above in this case shows \cite{PaulMTL} that the effect of a ground plane can be taken into account by simply doubling the $\alpha$ terms in equations \ref{eq:Spice_Einc_modal_characteristics_sources_alpha}.

\clearpage
