## 4 Calculating Dynamic Inductance Gradient
### The Gradient of Mutual Inductance of the Inner and Outer Rail  
In this paper, the mutual inductance gradients of each equivalent rail in the outer rail and the inner rail are obtained sequentially by applying the method in Section 4.1 from an energy point of view. Unlike the inner rail, the current circuit of the outer rail does not change with the armature position; then, we can obtain the magnetic induction produced by each commuted outer rail at point p:  
\[\left\{\begin{array}{l}B^{\prime}_{1y}=\frac{\mu_{0}}{4\pi}\frac{I}{I^{ \prime}_{c}}\int^{\sigma_{r}}_{0}\int^{w+d+\sigma}_{w+d+\tau}\int^{h}_{0}\frac {\left(\eta+z\right)\left[k(l)+k^{\prime}(l)\right]}{\left(\eta+z\right)^{2}+ \left(\xi-y\right)^{2}}\text{d}\xi\text{d}\eta\text{d}r\\ B^{\prime}_{2y}=\frac{\mu_{0}}{4\pi}\frac{I}{I^{\prime}_{c}}\int^{\sigma_{r}}_{ 0}\int^{w+d+\tau+w^{-\prime}-r}_{0}\int^{h-r}_{0}\frac{\left(\eta+z\right) \left[k(l)+k^{\prime}(l)\right]}{\left(\eta+z\right)^{2}+\left(\xi-y\right)^{2 }}\text{d}\xi\text{d}\eta\text{d}r\\ B^{\prime}_{3y}=\frac{\mu_{0}}{4\pi}\frac{I}{I^{\prime}_{c}}\int^{\sigma_{r}}_{ 0}\int^{w+d+\tau}_{0}\int^{w+d+\tau}_{0}\int^{w-d+\tau}_{r}\int^{h-r}_{0}\frac {\left(\eta+z\right)\left[k(l)+k^{\prime}(l)\right]}{\left(\eta+z\right)^{2}+ \left(\xi-y\right)^{2}}\text{d}\xi\text{d}\xi\text{d}\eta\text{d}r\\ B^{\prime}_{4y}=\frac{\mu_{0}}{4\pi}\frac{I}{I^{\prime}_{c}}\int^{\sigma_{r}}_{ 0}\int^{w+d+w-w^{\prime}}_{w+d+w-w^{\prime}}\int^{h}_{0}\frac{\left(\eta+z \right)\left[k(l)+k^{\prime}(l)\right]}{\left(\eta+z\right)^{2}+\left(\xi-y \right)^{2}}\text{d}\xi\text{d}\xi\text{d}\eta\text{d}r\end{array}\right. \tag{21}\]  
where \(I^{\prime}_{c}=2\sigma_{r}(h+w-2\sigma_{r})\), \(k^{\prime}(l)=(L-l)/\sqrt{\left(L-l\right)^{2}+\left(\eta+z\right)^{2}+\left( \xi-y\right)^{2}}\) and L is the length of the rail. Since L is much larger than the other parameters, k\({}^{\prime}\)(l) can be considered approximately 1. When 3 mm < \(\delta_{r}\leq\) 3.6 mm, B\({}^{\prime}{}_{\text{i}\text{j}}\) (i = 1,2,3,4) remains unchanged.  
Similarly, we obtain the expression for the gradient of the mutual inductance of each equivalent rail in the outer rail relative to the inner rail (M\({}^{\prime}{}_{\text{i}\text{j}}\), I = 1,2,3,4), which is  
\[M^{\prime}_{i^{\prime}}=\frac{2I}{h_{a}}\int^{s}_{0}\int^{\frac{h_{a}h_{a}}{2 }}_{\frac{h_{a}h_{a}}{2}}B_{\text{ay}}\text{d}\text{yd}z \tag{22}\]  
The variation of M\({}^{\prime}{}_{\text{i}\text{j}}\) with skin depth is the same as that of L\({}^{\prime}{}_{\text{c}\text{i}}\) with skin depth, as shown in Figure 12.  
Figure 11: The variation law of L\({}^{\prime}{}_{\text{c}\text{i}}\) with the skin depth.  
When the skin depth varies between 0.1 and 3 mm, \({\rm M^{{}^{\prime}}_{cl}}^{\prime}\), \({\rm M^{{}^{\prime}}_{c2}}^{\prime}\), and \({\rm M^{{}^{\prime}}_{c4}}^{\prime}\) increase with the increase of the skin depth. However, when the skin depth exceeds 3 mm, \({\rm M^{{}^{\prime}}_{c1}}^{\prime}\), \({\rm M^{{}^{\prime}}_{c2}}^{\prime}\), and \({\rm M^{{}^{\prime}}_{c4}}^{\prime}\) do not change anymore because \(w-w^{\prime}=6\)mm. During this period, \({\rm M^{{}^{\prime}}_{c2}}^{\prime}\) = 0. According to each equivalent outer rail in Figure 6, it is known that rail 2\({}^{\prime}\) and rail 3\({}^{\prime}\) have the same pattern of change, owing to symmetry. Therefore, only three curves \({\rm M^{{}^{\prime}}_{c1}}^{\prime}\), \({\rm M^{{}^{\prime}}_{c2}}^{\prime}\), and \({\rm M^{{}^{\prime}}_{c4}}^{\prime}\) are shown in Figure 12.