## 3 Deriving the Rail Skin Depth under the U-Shaped Armature  
Because of the skin effect, the currents on both the rail and the armature tend to be surface-distributed. When the velocity skin effect is significant, there is \(\lambda_{v}<<\lambda\), which is reflected in the literature [12]. Then, the average current density along \(x\)- and \(y\)-directions (\(\mathrm{J_{rx}}\) and \(\mathrm{J_{ry}}\), respectively) in the inner rail side is taken as \(\lambda_{v}\). Combining the geometric relations in Figure 2 yields.  
\[\tan\theta=-\overline{J_{ry}}/\overline{J_{rx}}=\int_{0}^{\Delta x}\frac{ \partial\mathbf{B_{r}}}{\partial x}\Big{|}_{y=0^{+}}d(\Delta x)/\int_{0}^{ \Delta x}\frac{\partial\mathbf{B_{r}}}{\partial y}\Big{|}_{y=0^{+}}d(\Delta x) \tag{8}\]  
The solution to the type of equation given by Equation (3) is  
\[B_{r(\Delta x,y)}=\int_{0}^{\Delta x}\frac{d\mathbf{B_{i}}(\xi)}{d\xi}\mathrm{ erfc}\Bigg{\{}\frac{y}{2[\mu_{0}\delta_{r}v(\Delta x-\xi)]^{1/2}}\Bigg{\}}d\xi \tag{9}\]  
From the same geometric relationship in Figure 2, we obtain  
\[\tan\theta=\frac{\lambda_{v}}{\sigma_{r}} \tag{10}\]  
Combining Equations (8)-(10), we can obtain  
\[\Bigg{\{}\begin{array}{l}\delta_{v}=\frac{1}{1.21^{2}\tan\theta}\cdot\frac{ 1}{\mu_{0}\sigma_{r}\gamma}\\ \tan\theta=1.21\sqrt{\frac{1}{\mu_{0}\sigma_{r}v\lambda_{v}}}\end{array} \tag{11}\]  
Figure 4 shows the actual current waveform in the experimental setup obtained by the current acquisition system, which indicates that the armature discharge time is about 1.531 ms. The actual current waveform can be approximated as a sawtooth wave, and its frequency can be regarded as \(1/2\) of the sinusoidal waveform according to the literature [13].  
The skin depth of the rail with the sinusoidal current can be obtained as  
\[\delta_{\mathrm{sin}}=\sqrt{\frac{1}{\pi\mu_{0}\sigma_{r}f}} \tag{12}\]  
where \(f=\frac{1}{2\times 1.531}\mathrm{kHz}\) is the current frequency according to Figure 4.  
The literature [14] proved that when the armature speed is low, the current skin effect dominates. When the armature speed exceeds the critical speed \(\mathrm{v_{c}}\), the velocity skin effect starts to be significant, so the equivalent skin depth can be obtained as  
\[\begin{cases}\delta_{r}=\delta_{\mathrm{sin}},&v\leq v_{c}\\ \delta_{r}=\sqrt{\frac{1}{\pi\mu_{0}\sigma_{r}}\cdot\frac{\mathrm{s_{v}}}{v}},&v> v_{c}\end{cases} \tag{13}\]  
where \(\mathrm{s_{v}}=\mathrm{k}\lambda\) is the effective length of the armature contact current and \(\mathrm{k}\) is the scale factor.  
From Equations (11) and (13), we can obtain k = 2.42\({}^{2}\), and according to the literature [14], v\({}_{\rm c}\) = 150 m/s.  
Assuming that v \(\leq\) v\({}_{\rm c}\) and substituting \(\sigma_{\rm r}\) = 5.998 \(\times\) 10\({}^{7}\) S/m into Equation (13), the skin depth is about 3.6 mm, so the effect of skin depth should not be neglected for a rail width of 10 mm.  
It can also be seen from Equation (13) that, during the launching process, the current skin effect plays a major role when the armature speed is less than the critical speed. After the armature speed exceeds the critical speed, the effect of the velocity skin effect is much larger than the current skin effect. As shown in Figure 5, obtaining the current density cloud map in the simulation software LSDYNA student version when the speed of the armature is 30 m/s, the current distribution in the rail is more uniform, and the skin depth is deeper. However, at a speed of 200 m/s of the armature, the current in the inner rail tends to be distributed on the surface, and the skin collection depth becomes shallow.