## 4 Performance Analysis and Design of the TPMA  
Figure 9 shows the design procedure of the TPMA. The gradient-based optimization method was used to maximize the thrust density and minimize the cogging force under the geometric and thermal constraints. The thrust density, maximum cogging force, and losses were determined by electromagnetic 2-D FE analysis of the initial models. The maximum cogging force was obtained by calculating the cogging force at each axial position without current. The temperature distribution was determined by thermal 3-D FE analysis considering forced convection using the thermal parameters determined through thermal experiments. Table 4 lists the design specifications, including the peak temperature, geometrical limitations, and design domains. Figure 8: Finite element (FE) model of the tubular permanent magnet actuator: (**a**) open-circuit flux distributions of the electromagnetic two-dimensional FE model; (**b**) temperature distribution with DC 80 A and 3 m/s forced convection. Figure 9: Design procedure of the tubular permanent magnet actuator for the active lateral secondary suspension of a railway vehicle. Figures 10 and 11 show the thrust density and the maximum temperature variation with radial variables using the electromagnetic and thermal FE analysis. To analyze the effect of each radial design variable on the thrust density and the maximum temperature, variables such as air gap, height of housing, radius of shaft and thickness of insulating paper were fixed. Although increasing the number of turns is the easiest technique for improving the thrust density, it engenders thermal complications, as shown in Figures 10d and 11d. When the height of the coil is larger than 17 mm, the thrust density becomes higher than 3.363 \(\times\) 10\({}^{5}\) N/m\({}^{3}\), but the maximum temperature of the actuator becomes higher than the allowable temperature. For this reason, the magnetic flux density must be improved to maximize the thrust force. However, at high magnetic flux densities, hysteresis and magnetic saturation effects occur in magnetic substances such as silicon steel [34]. \begin{table}
\begin{tabular}{c c} \hline
**Type** & **Values** \\ \hline Thermal constraints & Maximum temperature \(T_{p}\leq 130\) \\ \hline Geometric constraints & Stator outer radius \(R_{c}\leq 100\) (mm) \\ \hline Design domain & \(R_{1}=\{r_{m}|4<r_{m}<7\}\) \\  & \(R_{2}=\{r_{m}|12<r_{m}<18\}\) \\  & \(R_{3}=\{r_{m}|3<r_{c}<20\}\) \\  & \(R_{4}=\{r_{m}|7<r_{o}<13\}\) \\  & \(Z_{1}=\{\tau_{m}/\tau_{p}\) [\(0.5<r_{m}/\tau_{p}<0.75\}\) \\  & \(Z_{2}=\{l_{0}|2<l_{o}<16\}\) \\ \hline Other data & Current density J = 7.36 (A/mm\({}^{2}\)) \\  & Forced convection coefficient 38 (W/m\({}^{2}\)K) 1 \\ \hline \end{tabular}  
* \({}^{1}\) at 8 m/s forced convection.  
\end{table} 
Table 4: Design constraints and other data. Figure 10: Thrust density versus the radial variables: (**a**) magnet yoke; (**b**) coil yoke; (**c**) magnet; (**d**) coil. The heat generated in the coil is mainly transferred to the housing through the tooth and coil yoke, and the amount transferred to the mover is small due to the low thermal conductivity of the air. For this reason, the change in height of the magnet yoke and magnet does not significantly affect the maximum temperature of the actuator as shown in Figure 11a,c. However, as the magnet height increases, the magnetic flux density increases and the thrust density increases until magnetic saturation occurs as shown in Figure 10c. Figure 10a shows that there is an optimal magnet yoke height value to maximize the thrust density. As the height of the coil yoke becomes smaller than a certain value, the cooling effect by the forced convection is greatly improved, but at the cost of a reduced thrust density, as shown in Figures 10b and 11b. A Halbach array is a special arrangement of permanent magnets that augments the magnetic field on one side of the array while cancelling the field to near zero on the other side.