## 3 Establishment of the Improved Electromagnetic-Thermal Coupling Model
### Prediction Function of the Armature Velocity  
Based on the data obtained from Figure 5, the armature velocity relations with the armature conductivity and track one at different times are obtained, which are shown in Figure 12.  
It can be seen that, at the same time point, the armature velocity gradually increases with the armature conductivity \(\sigma_{\text{d}}\), and gradually decreases with the track conductivity \(\sigma_{\text{g}}\), in which the ascending (descending) speed gradually slows down. So, the following  
Figure 11: Flow chart of the establishment of the improved model. Figure 12: The armature velocity relations with the conductivity at different times. (**a**) Armature velocity curves with the armature conductivity \(\sigma_{\text{d}}\); (**b**) armature velocity curves with the track conductivity \(\sigma_{\text{g}}\). equation can be used to approximately describe the armature velocity with the armature conductivity and track one. \[\left\{ \begin{array}{l}v_{1}=r_{1}\ln(\sigma_{\rm d})+s_{1}\\ v_{2}=r_{2}\ln(\sigma_{\rm g})+s_{2}\end{array} \right. \tag{19}\]  
where, \(v_{1}\) and \(v_{2}\) are the influence factors of the armature velocity under the influence of the armature conductivity and the track one, respectively. \(\sigma_{\rm d}\) and \(\sigma_{\rm g}\) are the armature conductivity and the track one (MS/m), respectively. \(r_{1}\), \(r_{2}\), \(s_{1}\) and \(s_{2}\) are the corresponding fitting coefficients, respectively. According to Equations (9), (11) and (15), the analytic armature velocity \(v_{0}\) can be calculated by:  
\[v_{0}=\left\{ \begin{array}{l}\frac{L_{0}L_{0}^{2}}{m_{0}}\cdot\left[\frac{1} {4}t-\frac{1}{10\pi}\sin\left(\frac{\pi t}{62}\right)\right]\quad 0\leq t\leq 0.2 \,\text{ms}\\ \frac{L_{0}L_{0}^{2}}{m_{0}}\cdot\left[\frac{1}{24t}\left(1-e^{-1.22(t-0.2)} \right)+0.05\right]\quad 0.2<t\leq 0.6\,\text{ms}\end{array} \right. \tag{20}\]  
where, \(L_{0}\) 1.0 uH/m. Then, the analytic relationship between the armature velocity \(v_{0}\) and time is shown in Figure 13[18]. From Figures 5 and 13, it can be seen that, with the same conductivity, the finite element calculated value of the armature velocity is similar to the analytic value of the armature velocity. Therefore, the finite element calculated value of the armature velocity \(v\) can be represented by multiplying the analytic armature velocity \(v_{0}\) by a coefficient \(k_{0}\), i.e.,  
\[v=k_{0}v_{0} \tag{21}\]  
where \(k_{0}\) can be obtained through interpolating the time and conductivity, which is shown in Figure 14.  
Figure 14: \(k_{0}\) curves of time and conductivity. Figure 13: The analytic relationship between the armature velocity \(v_{0}\) and time. It can be seen that \(k_{0}\) changes dynamically with time with the same conductivity. The curves rise rapidly from 0 to 0.2 ms and gradually stabilize from 0.2 to 0.6 ms. Likewise, \(k_{0}\) changes with the conductivity at the same time point. Therefore, when considering the dynamic conductivity, \(k_{0}\) can be expressed as a multiplication of a function only related to time \(k_{0}(t)\) and a function only related to conductivity (\(v_{1}+v_{2}\)), i.e.,  
\[k_{0}=k_{0}(t)\cdot\left[r_{1}\ln(\sigma_{\rm d})+r_{2}\ln(\sigma_{\rm g})+s_ {0}\right] \tag{22}\]  
where \(s_{0}=s_{1}+s_{2}\). Define \(k_{0}(t)\) as follows:  
\[k_{0}(t)=\sum\limits_{i=0}^{n}w_{i}\ t^{i} \tag{23}\]  
where \(w_{i}\) is the fitting coefficient, \(n\) is the degree of the fitting polynomial. \(k_{0}(t)\) is fitted by taking the average values of each group of curves in Figure 14 as the basis. The fitting coefficients are shown in Table 2. When \(n=6\), the \(R^{2}\) value is 0.9977, indicating a good degree of fit. The armature velocity prediction functions combined with Equations (21)-(23) are defined as follows. \[\left\{ \begin{array}{l}v=k_{0}v_{0}\\ k_{0}=k_{0}(t)\cdot\left[r_{1}\ln(\sigma_{\rm d})+r_{2}\ln(\sigma_{\rm g})+s_ {0}\right]\\ k_{0}(t)=\sum\limits_{i=0}^{6}w_{i}\ t^{i}\end{array} \right.