## 3 Analytical Model Calculation and Validation
### Validation of Calculation Results  
#### 3.3.1 Electromagnetic Forces Validation  
In a real system, the length of the conductive plate is approximately infinite compared with the PM. However, in analytical calculation and FEA calculation, it is necessary to reduce the length of the conductive plate properly to improve the calculation speed, but the error will increase if the length is set as too small. The variation in the analytical calculation result of the levitation force with the length of the conductive plate is shown in Figure 9. It can be seen that the levitation force changes very little when the length of the conductive plate is greater than \(1\,\mathrm{m}\), and the error is within \(2\%\) compared with that when \(l_{2}=2\,\mathrm{m}\). Therefore, in Table 1, the length of the conductive plate is set to \(1\,\mathrm{m}\) after considering the calculation complexity and accuracy. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(l\) & \(0.51\,\mathrm{m}\) & \(w_{2}\) & \(0.20\,\mathrm{m}\) \\ \(w\) & \(0.03\,\mathrm{m}\) & \(h_{2}\) & \(0.01\,\mathrm{m}\) \\ \(h\) & \(0.03\,\mathrm{m}\) & \(d\) & \(0.01\,\mathrm{m}\) \\ coercivity of PM \(B_{r}\) & \(1.08\times 10^{6}\,\mathrm{A}/\mathrm{m}\) & \(v_{\mathrm{x}}\) & \(30\,\mathrm{m}/\mathrm{s}\) \\ \(l_{2}\) & \(1\,\mathrm{m}\) & \(\sigma\) & \(3.5\times 10^{7}\,\mathrm{S}/\mathrm{m}\) \\ \hline \hline \end{tabular} When the PM rotation angle is \(\theta=15^{\circ}\), the 3-D simulation model and its mesh division, are shown in Figures 7 and 8, respectively.  
\end{table} 
Table 1: The parameters of the model. Figure 8: Mesh division of the model. Figure 7: Three-dimensional simulation model. In order to verify the validity of analytical equations, the variations in levitation, drag and lateral forces with varying deflection angles are calculated by the analytical method and FEA method, respectively, and the deflection angle \(\theta\) of PM ranges 0-20\({}^{\circ}\). The results are shown in Figure 10. It can be seen from Figure 10 that the levitation force on the PM first increases and then decreases with the increase in the deflection angle, and the maximum value of the deflection angle \(\theta\) is about 15\({}^{\circ}\), which shows that it is feasible to control the levitation force by adjusting the deflection angle of the PM. Both the drag force and the lateral force tend to increase with the increase in the deflection angle. Compared with the calculation results of the analytical method and the FEA method, the relative errors are shown in Figure 11. The relative error of the levitation force is about 1%, and the relative error of the drag and lateral force are both within 6%. The results of the lift-to-drag ratio and lift-to-weight ratio are shown in Figure 12. The maximum value of the lift-to-drag ratio is 3.849 when the deflection angle \(\theta\) is 7\({}^{\circ}\), and the maximum value of the lift-to-weight ratio is 11.30 when the deflection angle \(\theta\) is 15\({}^{\circ}\). Figure 10: Comparison of electromagnetic forces on the PM between the analytical and FEA methods. Figure 9: Variation in levitation force to the length of the conductive plate. In order to further study the characteristics of the PM-EDS system, the variations in the levitation and drag forces with varying speeds are calculated when the conductive plate is made of different materials; the results are shown in Figures 13 and 14, respectively. The parameters of the model are the same as in Table 1, and the deflection angle of the PM is fixed at 15\({}^{\circ}\) in order to maximize the levitation force. The conductivity of the copper plate, aluminum plate and stainless steel plate decreases in turn.