## 4 Calculating Dynamic Inductance Gradient
### The Inductance Gradient of the Inner Rail
\begin{array}{l}B_{6y}=\frac{h_{0}}{4\pi}\frac{I}{I_{S}}\int_{\
u_{1} ^{\prime}}^{w_{1}^{\prime}+\sigma_{r}^{\prime}}\int_{\frac{h-h_{1}^{\prime}} {2}}^{\frac{h+h_{1}}{2}}\frac{(\eta+z)k(l)}{(\eta+z)^{2}+(\xi-y)^{2}}\mathrm{d }\xi\mathrm{d}\eta\\ B_{7y}=\frac{h_{0}}{4\pi}\frac{I}{I_{S}}\int_{\
u_{1}^{\prime}}^{w^{\prime}+ \sigma_{r}^{\prime}}\int_{\frac{h-h_{1}^{\prime}}{2}}^{\frac{h+h_{1}^{\prime}} {2}}\frac{-\sigma_{r}^{\prime}}{(\eta+z)^{2}+(\xi-y)^{2}}\mathrm{d}\xi\mathrm{d }\eta\\ B_{8y}=\frac{h_{0}}{4\pi}\frac{I}{I_{S}}\int_{w-\sigma_{r}}^{w}\int_{\frac{h-h _{1}^{\prime}}{2}}^{\frac{h+h_{1}^{\prime}}{2}}\frac{(\eta+z)k(l)}{(\eta+z)^{ 2}+(\xi-y)^{2}}\mathrm{d}\xi\mathrm{d}\eta\\ B_{9y}=\frac{h_{0}}{4\pi}\frac{I}{I_{S}}\int_{w^{\prime}_{1}+w_{1}-\sigma_{r}^{ \prime}}^{w_{1}^{\prime}+w_{1}^{\prime}+w_{1}^{\prime}-\sigma_{r}^{\prime}} \int_{\frac{h-h_{1}^{\prime}}{2}}^{\frac{h+h_{1}}{2}}\frac{(\eta+z)k(l)}{(\eta+ z)^{2}+(\xi-y)^{2}}\mathrm{d}\xi\mathrm{d}\eta\end{array} \right. \tag{19}\]  
where \(\delta_{\mathrm{r}^{\prime}}\) is the skin depth of the region K and w\({}_{1}\)\({}^{\prime}\) = (w+w)\({}^{\prime}\))/2 is the area of the current region. According to Equations (15) and (17), as well as symmetry, the expression for the induced gradient for each inner rail (L\({}^{\prime}\)\({}_{\mathrm{c}i}\), l = 1,2,3,4,5,6) is obtained as  
\[L^{\prime}_{{}_{\mathrm{c}i}}=\frac{4I}{h_{a}}\int_{0}^{s}\int_{\frac{h-h_{1} ^{\prime}}{2}}^{\frac{h+h_{1}^{\prime}}{2}}B_{iy}\mathrm{d}y\mathrm{d}z \tag{20}\]  
According to the size of the electromagnetic launcher in Table 1 and Equation (20), the variation law of L\({}^{\prime}\)\({}_{\mathrm{c}i}\) with the skin depth can be obtained, as shown in Figure 11.  
According to each equivalent inner rail in Figure 9, it is known that the areas enclosed in rails 2-4 increase with increasing the skin depth, which increases the magnetic chain generated at the same current. Because \(h-h^{\prime}\) = 0.001 mm, \(L^{\prime}\)\({}_{\mathrm{c}1}\) decreases with increasing the skin depth. However, the area enclosed by rail 5 decreases with increasing the skin depth, which is because the current region of rail 5 is affected by both rails 2 and 3. This results in a decrease in the area enclosed between the rail, causing a reduction in the magnetic chain generated at the same current. Considering the engineering reality, when the skin depth exceeds 3 mm, the value of \(L^{\prime}\)\({}_{\mathrm{c}5}\) becomes 0. In addition, the values of \(L^{\prime}\)\({}_{\mathrm{c}1}\), \(L^{\prime}\)\({}_{\mathrm{c}2}\), \(L^{\prime}\)\({}_{\mathrm{c}3}\), and \(L^{\prime}\)\({}_{\mathrm{c}4}\) increase rapidly with the increase of the skin depth because the Is increases slowly with the skin depth. Because the change in \(L^{\prime}{}_{\text{c}\text{f}}\), \(L^{\prime}{}_{\text{c}\text{f}}\), \(L^{\prime}{}_{\text{s}}\), and \(L^{\prime}{}_{\text{c}\text{g}}\) are very small, they are can be seen as part of \(L^{\prime}{}_{\text{c}\text{l}}\), \(L^{\prime}{}_{\text{c}\text{l}}\), \(L^{\prime}{}_{\text{c}\text{d}}\), \(L^{\prime}{}_{\text{c}\text{f}}\) and not represented in Figure 11.