## 2 Modeling Dynamic Electromagnetic Emissions  
The two-dimensional model of the electromagnetic rail launcher is shown in Figure 1. The pulse current I flows from the outer rails to the inner rail on one side. When the current passes through the armature, it flows back from the other inner rail. The pulse current generates the flux density **B**, which acts with the armature current to generate the electromagnetic thrust \(\mathrm{F_{a}}\) to drive the armature movement with velocity v.  
\(\lambda\) is the length of the armature, \(\theta\) the angle of current flow into the armature, \(\delta_{\mathrm{r}}\) the rail skin depth, \(\lambda_{\mathrm{v}}\) the range of current inflow in the \(x\)-direction at the armature-rail interface, and a is the point on the armature tail, whose horizontal coordinate range is \(\mathrm{x_{a}}\geq 0\). Figure 1: The electromagnetic rail launcher 2D model. Figure 2: Armature current line distribution. The magnetic diffusion equation for the inner rail is  
\[\frac{1}{\mu_{0}\sigma_{r}}(\frac{\partial^{2}\textbf{{B}_{\textbf{r}}}}{\partial x ^{2}}+\frac{\partial^{2}\textbf{{B}_{\textbf{r}}}}{\partial y^{2}})=\frac{ \partial\textbf{{B}_{\textbf{r}}}}{\partial t} \tag{1}\]  
where \(\mu_{0}\) is the vacuum magnetic permeability, \(\textbf{{B}_{\textbf{r}}}\) is the flux density on the inner rail, and \(\sigma_{r}\) the electrical conductivity of the inner rail material. Since \(\textbf{{B}_{\textbf{r}}}\) varies in the inner rail more rapidly in the \(y\)-direction than in the \(x\)-direction, we can simplify Equation (1) by neglecting \(\partial^{2}\textbf{{B}_{\textbf{r}}}/\partial x^{2}\) relative to \(\partial^{2}\textbf{{B}_{\textbf{r}}}/\partial y^{2}\) obtain a simple diffusion-like equation. \[\frac{1}{\mu_{0}\sigma_{r}}\frac{\partial^{2}\textbf{{B}_{\textbf{r}}}}{ \partial y^{2}}=\frac{\partial\textbf{{B}_{\textbf{r}}}}{\partial t} \tag{2}\]  
A section of displacement \(\Delta x\) (\(0\leq\Delta x\leq\lambda\)) is taken on the inner rail at the armature-rail interface (\(\text{x}_{\text{a}}\leq\text{x}\leq\text{x}_{\text{a}}+\lambda\)), as shown in Figure 3.  
Since the speed of the armature is very fast, when the armature goes through the displacement \(\Delta x\), the flux density on the inner rail can be treated as  
\[\frac{1}{\mu_{0}\sigma_{r}}\frac{\partial^{2}\textbf{{B}_{\textbf{r}}}}{ \partial y^{2}}=v\frac{\partial\textbf{{B}_{\textbf{r}}}}{\partial(\Delta x)} \tag{3}\]  
According to the boundary conditions of the armature in the conventional rail launcher in the literature [11], the boundary conditions of the armature of the electromagnetic rail launcher in this paper are set as follows. \[\left\{ \begin{array}{l}\textbf{{B}_{\textbf{a}}}(x_{a},y)=\textbf{{B}_{0}}+ \textbf{{B}^{\prime}}\\ \frac{\partial\textbf{{B}_{\textbf{a}}}}{\partial y}|_{y=-\frac{z}{2}}=0\\ \textbf{{B}_{\textbf{a}}}(x_{a}+\lambda,y)=\textbf{{B}^{\prime}}\end{array} \right. \tag{4}\]  
where \(\textbf{{B}_{\textbf{a}}}\) is the flux density on the armature, \(\textbf{{B}_{0}}\) is the flux density generated by the armature tail current density, and \(\textbf{{B}^{\prime}}\) is the flux density generated by the outer rail on the armature. Since the outer rail current loop does not vary with the armature displacement, which can be regarded as \(\textbf{{B}^{\prime}}\) independent of the armature position, \(\text{s}\) the distance between the two inner rails. Set the armature tail current density as \(\textbf{{J}_{0}}\); then, the current density decay of the armature along the \(y\)-direction at a distance \(\Delta x\) from the tail is  
\[J_{\text{av}}\Big{|}_{x=\Delta x}=-J_{0}e^{-\Delta x/\lambda_{y}} \tag{5}\]  
Figure 3: The armature–rail interface.