## 2 Theoretical Analysis  
As mentioned in the Introduction, the eddy current effect will affect the suspension force of the maglev train. This section will mathematically describe this phenomenon through theoretical analysis. Thus, the suspension system model is established firstly. The single suspension magnet model can be regarded as the smallest unit of the suspension system ([PERSON] et al., 2018), whose three-dimensional structure and two-dimensional section is shown in Figures 1 and 2, respectively. Where the rail is a simplified U-shaped solid structure, and under the rail is a suspension magnet that is composed of the iron core and the coil. In order to calculate the eddy current effect conveniently, the single suspension magnet model shown in Figure 1 is equivalent to the model shown in Figure 3([PERSON] et al., 2018). The  
Ignoring the reluctance and magnetic leakage in the iron core, then  
\[R_{m}=\frac{2\delta}{\mu_{0}S} \tag{1}\]  
where, \(R_{m}\), \(\delta\), \(\mu_{0}\), and \(S\) are the reluctance, the suspension gap, the permeability in vacuum, and the pole area, respectively. As Table 1 shows, \(2a\) represents the rail width and \(L\) represents the length of the suspension magnet, then \(S=2a\times L\). Since the total magnetic potential in the magnetic circuit is \(NI\), where \(N\) represents the number of windings, \(I\) represents the suspension current. Let \(\Phi\) represent the magnetic flux in the circuit, and according to the [PERSON]'s law of magnetic circuit [28], the \(\Phi\) can be written as  
\[\Phi=\frac{NI}{R_{m}}=NI\frac{\mu_{0}S}{2\delta} \tag{2}\]  
Further, \(B\) is the magnetic flux density in the magnetic circuit and can be obtained as  
\[B=\frac{\Phi}{S}=NI\frac{\mu_{0}}{2\delta} \tag{3}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Unit** & **Meaning** & **Value** \\ \hline \(L\) & m & the length of the & 0.66 \(\times\) 2 \\ \(2a\) & m & the rail width & 0.0028 \\ \(2d\) & m & the rail height & 0.096 \\ \(S\) & m\({}^{2}\) & the polar area & 0.01848 \\ \(N\) & turns & the number of & 360 \\ \(\sigma\) & s/m & the rail conductivity & \(5\times 10^{6}\) \\ \(\mu_{0}\) & H/m & the permeability in & \(4\pi\times 10^{-7}\) \\  & & vacuum & \(4\pi\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Meanings, values, and units of parameters. Figure 3: The equivalent magnetic circuit model of the single suspension magnet model. According to Figure 1, the volume in the suspension gap is \(V=2\delta S\). Therefore, the magnetic field energy \(W_{m}\) in volume \(V\) can be obtained as  
\[W_{m}=\int\frac{BH}{2}dV=\frac{B^{2}\delta S}{\mu_{0}} \tag{4}\]  
where \(H\) is the magnetic field intensity in the suspension gap. Let \(F_{y}^{0}\) represent the suspension electromagnetic force without considering the eddy current effect. Then the \(F_{y}^{0}\) can be obtained as  
\[F_{y}^{0}=\frac{dW_{m}}{d\delta}=\frac{B^{2}S}{\mu_{0}}=\frac{\mu_{0}SN^{2}l^{ 2}}{4\delta^{2}} \tag{5}\]  
When there is an eddy current effect in the rail, the suspension force \(F_{y}\) is  
\[F_{y}=F_{y}^{0}\sum_{n=1}^{\infty}\frac{C_{n}^{2}}{2L}\bigg{(}L-\frac{1-e^{- \alpha_{n}L}}{\alpha_{n}}\bigg{)} \tag{6}\]  
where \(\alpha_{n}=\frac{-k+\sqrt{k^{2}+4\lambda_{n}^{2}}}{2}\), \(k=\frac{\
u\sigma\mu_{0}d}{\delta}\), \(\lambda_{n}=\frac{2n-1}{2a}\), \(C_{n}=\frac{4 sin[(2n-1)\pi/2]}{(2n-1)\pi}\), and \((n=1,2,3,...)\). Noting that Formula (6) only considers the fundamental part of the magnetic field and ignores the high-frequency component. As can be seen from Formula (6), the main parameters affecting the suspension force are: the permeability in air gap \(\mu_{0}\), the rail conductivity \(\sigma\), the rail height \(2d\), the rail width \(2a\), the length of the suspension magnet \(L\), the train running speed \(v\), the suspension air gap \(\delta\), and the ampere turns of the suspension electromagnet \(NI\).