## 2 Simulation Model Establishment
### Mathematical Model of SICL  
The SICL mainly consists of an accelerating armature, various stages of drive coils, and an external power supply system. During the launching process, there is mutual inductance between each stage of drive coils and between the armature and drive coils. Under the influence of the coil pulse current, the current induced by the armature has non-uniform axial and radial distributions. A common analysis method is to discretize the armature by dividing its cross-section evenly along the axis and unevenly along the radial direction, turning a continuous armature into multiple independent ring armature slices. When the ring cross-section is small enough, it can be approximated that the distribution of induced current on each slice is uniform [12]. Assuming a synchronous induction coil launching device with n drive coils and m armature segments, when the k-th drive coil begins to conduct, the equivalent circuit model is as shown in Figure 5. The current through the drive coils is \(I_{\text{di}}\) (i = 1, 2, \(\ldots\), k); the resistance of the drive coil loop is \(R_{\text{i}}\) (i = 1, 2, \(\ldots\), k); the inductance of the drive coil loop is \(L_{\text{i}}\) (i = 1, 2, \(\ldots\), k); the resistance of the loop coil is \(R_{\text{di}}\) (i = 1, 2, \(\ldots\), k); the inductance of the loop coil is \(L_{\text{di}}\) (i = 1, 2, \(\ldots\), k); the mutual inductance between the drive coils is \(M_{\text{ij}}\) (i,j = 1, 2, \(\ldots\), k); the mutual inductance between the drive coils and armature segments is \(M_{\text{dij}}\) (i = 1, 2, \(\ldots\), k, j = 1, 2, \(\ldots\), m); the mutual inductance between armature segments is \(M_{\text{pij}}\) (i,j = 1, 2, \(\ldots\), m); the resistance of the armature segment loop is \(R_{\text{pi}}\) (i = 1, 2, \(\ldots\), m); the self-inductance of the armature loop is \(L_{\text{pi}}\) (i = 1, 2, \(\ldots\), m); and the current through the armature segment loop is \(I_{\text{pi}}\) (i = 1, 2, \(\ldots\), m). \begin{table}
\begin{tabular}{c c c c} \hline \hline Pulsed & Stage 1–2 & \begin{tabular}{c} Source capacitance (mF) \\ Charge voltage (kV) \\ \end{tabular} & 0.6 \\ \cline{2-3} power & Stage 3–4 & \
\begin{tabular}{c} Source capacitance (mF) \\ Charge voltage (kV) \\ \end{tabular} & 0.266 \\ \cline{2-3}  & &
\begin{tabular}{c} Charge voltage (kV) \\ \end{tabular} & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Power supply parameters of the external circuit. Figure 5: Equivalent circuit model of multi-stage coil launching device. When considering only the movement direction of the armature as the z-direction and neglecting other energy losses, the force acting on the armature at time t is:  
\[F(t)=\sum_{i=1}^{k}\sum_{j=1}^{m}\frac{\mathrm{d}M_{dij}(t)}{\mathrm{d}z}I_{di}( t)I_{pj}(t) \tag{7}\]  
From Equation (7), it can be seen that the force on the armature is related to the segments of the armature and the current of the driving coil as well as the mutual inductance gradient. The mutual inductance gradient depends on the relative position between the driving coil and the armature segments. When the armature segments pass through the centerline of the driving coil, the mutual inductance gradient between the driving coil and the armature segments is positive; otherwise, it is negative. Therefore, the magnitude and direction of the current in the armature segments directly determine the force on the armature. Due to the change in flux, induced currents are generated in the armature segments, and the current induced in the armature segment coil is:  
\[I_{pj}=\frac{1}{Z}\frac{\mathrm{d}\varphi}{\mathrm{d}t}=\frac{\mu}{Z}\int \frac{\mathrm{d}H}{\mathrm{d}t}\mathrm{d}A \tag{8}\]  
In the formula, \(\varphi\) represents the magnetic flux passing through the armature segment; Z is the impedance of the armature segment; and A is the effective area surrounded by the armature segment.