## 3 Numerical Model of the Single-Stage FLTD
### Simplified Model of Discharging Bricks  
The simplification process of film capacitors and gas switch in a discharging brick is based on two principles [14]: the equivalent capacitance, inductance, and resistance should stay the same; the magnetic field distribution outside the brick model should remain unchanged.  
Figure 6 shows the initial simplified model of discharging brick for finite element analysis, where a film capacitor is idealized as nine series-connected parallel-plate capacitors and the two electrodes of gas switch is connected by a thin cylindrical conductor. However, this model is still very complex in terms of number of DoFs and it needs further simplification for smooth transient EM field investigations on the FLTD cavity.  
According to equivalent circuit model and measured results, capacitance, inductance, and resistance of a main discharging brick are 50 nF, 160 nH, and 0.3 \(\Omega\), respectively. We construct the 3-D model of a discharging brick to represent only its equivalent inductance and resistance, while its capacitance and initial voltage is represented by a lumped element in the circuit model. As shown in Figure 6, through finite element analysis, we can determine inductance and resistance of the brick model to be 160 nH and 0.3 \(\Omega\), respectively. The field model and circuit model are coupled through the conduction path of gas switch, as illustrated in Figure 7. The equivalent capacitance of the brick is modeled in circuit and it should be exactly 50 nF (two series-connected 100 nF capacitors).  
To achieve the required electrical parameters and field distribution outside the brick, a genetic algorithm is used to optimize the shape and size of the internal equivalent conductor of the film capacitor. The final simplified model is shown in Figure 8. The main differences between the initial model and final model are the internal structure of film capacitor. However, they will exhibit the same brick discharge current due to the same electrical parameters. As can be observed from Figure 8, the current flow path and field distribution inside the capacitors will be different from those of the initial model. This is the tradeoff between accuracy and speed in our numerical calculation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Components**} & \multirow{2}{*}{**Materials**} & **Relative** & **Relative** & **Electrical Conductivity** \\  & & **Permitivity** & **Permeability** & **(S/m)** \\ \hline FLTD cavity and separate columns & Stainless-steel & 1.0 & 1.0 & \(6.99\times 10^{6}\) \\ Capacitor/switch conductors & Aluminum & 1.0 & 1.0 & \(3.56\times 10^{7}\) \\ Magnetic core & Metglas & 1.0 & 1000.0 & \(8.13\times 10^{-5}\) \\ Load cavity barrier & XLEE & 2.3 & 1.0 & 0 \\ Top/middle/bottom insulators & PTFE & 2.1 & 1.0 & 0 \\ Magnetic core outer casing & Epoxy resin & 4.0 & 1.0 & 0 \\ Deionized water filled cavity & Deionized water & 80.0 & 1.0 & \(5.55\times 10^{-6}\) \\
0.1 \(\Omega\) resistance dummy load & Water Solution & 78.5 & 1.0 & \(2.56\times 10^{1}\) \\ Background & Air & 1.0 & 1.0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Typical value of material characteristics in numerical calculations.  
Figure 6: Initial simplified model of the discharge brick. (**a**) Perspective view, (**b**) cut-away view.