## 4 Calculating Dynamic Inductance Gradient
### Equivalent Inductance Gradient  
Since the inner and outer rails pass into the same directional current, the inner rail current is mainly concentrated on the inner surface, and the outer rail current is mainly concentrated on the outer surface.  
Then, the current distribution in each region of Figure 7 can be expressed as  
\[\left\{\begin{array}{l}I_{1}=c_{1}\frac{h^{\prime}l\
u_{t}}{l_{S}l}I=\frac{ c_{1}h^{\prime}l}{2(h+w-2\sigma_{r})}I\\ I_{2}=c_{1}\frac{(h-h^{\prime})l\sigma_{r}}{l_{S}l}I=\frac{c_{1}(h-h^{\prime}) l}{2(h+w-2\sigma_{r})}\\ I_{3}=c_{2}\frac{h\
u_{t}}{l_{S}l}I=\frac{c_{2}h^{\prime}l}{2(h+w-2\sigma_{r}) }\\ I_{4}=c_{1}\frac{\
u_{t}l\
u_{t}}{l_{S}l}I=\frac{c_{1}h^{\prime}l}{2(h+w-2\sigma _{r})}\\ I_{5}=c_{3}\frac{(w-w^{\prime}-2\sigma_{r})l\sigma_{r}}{l_{S}l}I=\frac{c_{3}(w -w^{\prime}-2\sigma_{r})l}{2(h+w-2\sigma_{r})}\\ I^{\prime}_{1}=c_{1}^{\prime}\frac{h\
u_{t}}{l_{S}l}I=c_{1}^{\prime}\frac{c_{ 1}h^{\prime}l}{2(h+w-2\sigma_{r})}\\ I^{\prime}_{2}=c_{2}^{\prime}\frac{(2w-w^{\prime}-d-2\sigma_{r})l\sigma_{r}}{l_ {S}l}I=\frac{c_{2}^{\prime}(2w-w^{\prime}-d-2\sigma_{r})l}{2(h+w-2\sigma_{r}) }\\ I^{\prime}_{4}=c_{4}^{\prime}\frac{h\
u_{t}}{l_{S}l}I=c_{1}^{\prime}\frac{c_{ 1}h1}{2(h+w-2\sigma_{r})}\\ I^{\prime}_{2}=I^{\prime}_{3}\end{array}\right. \tag{23}\]  
where \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c^{\prime}\)1, \(c^{\prime}\)2, and \(c^{\prime}\)3 are characterized as current distribution coefficients that satisfy the following conditions:  
\[\left\{\begin{array}{l}I=\sum\limits_{i=1}^{5}I_{i}\\ I=\sum\limits_{i=1}^{4}I^{\prime}_{i}\end{array}\right. \tag{24}\]  
Then, \({L_{c}}^{\prime}\) and \(M^{\prime}\) can be expressed as follows:  
\[\left\{\begin{array}{l}L^{\prime}_{c}=c_{1}({L_{c}}^{\prime}+{L_{c}}^{ \prime}+{L_{c}}^{\prime})+c_{2}{L_{c}}^{\prime}+c_{3}{L_{c}}^{\prime}\\ M^{\prime}=c_{1}^{\prime}{M_{1}^{\prime}}+c_{2}^{\prime}{M_{2}^{\prime}}+c_{3}^{ \prime}({M_{3}^{\prime}}+{M_{4}^{\prime}})\end{array}\right. \tag{25}\]  
Figure 12: The variation law of \({\rm M^{{}^{\prime}}_{i}}^{\prime}\) with the skin depth.  
In the literature [15], the current distribution coefficients on the rail are \(c_{1}=1.5\) and \(c_{2}\) = 0.5. According to the current distribution in the simulation results, this paper takes \(c^{\prime}{}_{1}=0.4\) and \(c^{\prime}{}_{2}=1.8\). Combining this with Equation (24), we obtain:  
\[\left\{\begin{array}{l}c_{3}=\frac{(w-w^{\prime}-2\sigma_{r})}{((w-w^{\prime }-2\sigma_{r})+(c_{1}-1)w^{\prime})}\\ c^{\prime}_{3}=\frac{(2w-w^{\prime}-d-2\sigma_{r})}{((2w-w^{\prime}-d-2\sigma_{ r})+(c^{\prime}_{1}+c^{\prime}_{2}-2)h)}\end{array}\right. \tag{26}\]  
Substituting these four parameters into Equations (13) and (25), we can obtain the variation law of the inner rail inductance gradient, outer rail mutual inductance, and rail equivalent inductance gradient with the skin depth, as shown in Figure 13.  
In Figure 13, we observe that the inner rail inductance gradient, outer rail mutual inductance, and rail equivalent inductance gradient all increase with the increase of skin depth. When the skin depth is less than 3 mm, the inner rail inductance gradient, the outer rail mutual inductance, and the rail equivalent inductance gradient increase with the increase of skin depth. However, when the skin depth exceeds 3 mm, the outer rail mutual inductance remains constant with a value of about 0.2042 uH/m. In this stage, the inner rail inductance gradient and the rail equivalent both increase suddenly with the increase of skin depth because Is increases slowly with the skin depth.