## 3 Electromagnetic and Thermal Finite Element (FE) Model
### Thermal Model Parameters  
Although some thermal variables such as thermal contact resistance, the interface gap between components, and the forced convection heat transfer coefficient have a large effect on the temperature distribution of the TPMA, they are difficult to calculate without experiments or working experience of similar designs [24]. Thus, a six-slot prototype model, shown in Figure 6a, was fabricated to estimate the critical thermal parameters of the TPMA with initial and fixed design variables, as shown in Table 2. The six-slot prototype consisted of two pairs of three-phase coils, a laminated core, and an aluminum housing with fins. The air-cooling system was composed of annular fins circumferentially attached to the cylinder, and designed to enhance heat transfer from the surface to the surrounding air. The fin was designed in consideration of the manufacturing limitations, allowable stiffness, and cooling \begin{table}
\begin{tabular}{c c c c} \hline
**Initial Design Variables** & **Value** & **Fixed Variables** & **Value** \\ \hline Height of coil yoke \(r_{cy}\) & 10 mm & Height of housing \(r_{h}\) & 14 mm \\ Height of coil \(r_{c}\) & 15 mm & Radius of shaft \(r_{s}\) & 15 mm \\ Height of magnet \(r_{m}\) & 15 mm & Pole-pitch & 60 mm \\ Height of magnet yoke \(r_{my}\) & 10 mm & Air gap \(l_{g}\) & 2 mm \\ \(\tau_{mur}\) to \(\tau_{p}\) ratio & 0.5 & Thickness of insulating paper \(l_{p}\) & 0.5 mm \\ Slot opening \(l_{o}\) & 6 mm & Number of pole-pairs & 4 \\ - & - & Number of slots & 15 \\ \hline \end{tabular}
\end{table} 
Table 2: Initial and fixed design variables of the tubular permanent magnet actuator. Figure 4: Overall structure of the tubular permanent magnet actuator for the active lateral secondary suspension. Figure 5: Topology and design variables of the tubular permanent magnet actuator (axial cross-section). efficiency. The temperatures of eight points (on the outer surface of the mover, coils, and housing, and on the inner surface of the cores) were measured using thermocouples. There are five critical thermal parameters that can affect the thermal characteristics: the thermal contact resistance between the coil and the core, the core and the housing, and the stator and the mover; the thermal conductivity of the laminated core; and the coefficient of forced convection of the housing with fins, shown in Figure 6b. In general, it is difficult to estimate the air flow through a TPMA under a car body when the train is in motion. Therefore, to propose design guidelines for the air-cooling system, the flow through the TPMA must be simplified, and the cooling effect with respect to the air velocity must be verified experimentally. Figure 6c shows the setup for a wind-tunnel test of the six-slot prototype model. The external air flow in the direction normal to the axis of the TPMA and the thermal characteristics were validated from low to high speed (30 m/s). The thermal analysis was performed using 3-D FE analysis to ensure accurate simulation results [30]. Accordingly, to increase the reliability of the thermal FE model, the thermal experiments were carried out at three input direct current (DC) continuously induced in the three-phase coils. Figure 7a shows the results of the wind tunnel test at various air velocities. The maximum temperature in the coil decreased exponentially with air velocity. It was confirmed that air velocities from 3 m/s to 10 m/s substantially affected the thermal performance but enhancement of air cooling was markedly diminished at air velocities higher than 10 m/s. The coefficients of forced convection of the heat sink were determined for each wind speed. The measured temperature distributions agreed well with FE analysis using the determined thermal parameters, as shown in Figure 7b. Thus, the reliability of the five parameters was verified. The thermal properties used in the simulation are summarized in Table 3.  
Figure 6: Six-slot prototype model: (**a**) manufactured six-slot prototype; (**b**) critical thermal parameters; (**c**) wind tunnel test.