## 5 Experimental Verification  
Figure 13 shows the manufactured TPMA and the experimental setup. The TPMA was fabricated using the previously determined design variables. The axial length and outer diameter of the TPMA were 1 m and 200 mm, respectively, and its weight was 108 kg. A ball screw motor (APM-SE11 MC(G7)K1, Mecapion Co., Ltd., Daegu, Korea), connected in series with a load cell (LS-2, CAS Co., Ltd., Jeongeup, Korea) and the shaft of the mover, was installed to measure the cogging force and the thrust force. The data acquisition system was configured to obtain temperature data from the thermocouples embedded in the TPMA. Figure 13: Experimental setup of the tubular permanent magnet actuator. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **Initial** & **Optimized** \\ \hline Stator outer radius \(R_{o}\) & 100 mm & 100 mm \\ Height of housing \(r_{h}\) & 14 mm & 14 mm \\ Height of coil yoke \(r_{cy}\) & 10 mm & 12 mm \\ Height of coil \(r_{c}\) & 15 mm & 17 mm \\ Air gap \(l_{g}\) & 2 mm & 2 mm \\ Height of magnet \(r_{m}\) & 15 mm & 16 mm \\ Height of magnet yoke \(r_{my}\) & 10 mm & 6 mm \\ \(r_{m}\) to \(\tau_{p}\) ratio & 0.5 & 0.64 \\ Slot opening \(l_{o}\) & 6 mm & 13.4 mm \\ Radius of shaft \(r_{s}\) & 15 mm & 15 mm \\ Pole-pitch \(\tau_{p}\) & 60 mm & 60 mm \\ Thickness of insulating paper \(l_{p}\) & 0.5 mm & 0.5 mm \\ Number of pole-pairs & 4 & 4 \\ Number of slots & 15 & 15 \\ Current density \(J\) & 7.36 A/mm\({}^{2}\) & 7.36 A/mm\({}^{2}\) \\ Winding turns \(N\) & 117 & 133 \\ Saturation temperature & 93 \({}^{\circ}\)C & 130 \({}^{\circ}\)C \\ Thrust density & 2.252 \(\times\) 10\({}^{5}\) N/m\({}^{3}\) & 3.120 \(\times\) 10\({}^{5}\) N/m\({}^{3}\) \\ Maximum cogging force & 2216 N & 220 N \\ Thrust ripple & 28.8\% & 2.86\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Initial and optimized design variables. The thrust force was measured while increasing the current to a locked TPMA. As shown in Figure 14a, the measured thrust force increased linearly with the input current in both the positive and negative directions. The thrust force was 7700 N when the input current reached 96.3 A, and there was no saturation up to 102 A. The cogging force for a 40 mm period was measured while the mover slowly shifted using the ball screw motor. As shown in Figure 14b, the experimental results matched the simulation results well. The maximum cogging force was low compared with the maximum thrust force. Moreover, as the cogging force was constant at a given position, cogging force ripples could be compensated for via a feedforward control algorithm. The frequency response in Figure 15 shows that the bandwidth of the TPMA was greater than 10 Hz. Because the resonant frequency of the experimental setup was 24 Hz, the thrust constant increased by 6.3%. A phase shift of \(-7^{\circ}\) was caused by hysteresis of the load cell. Furthermore, a phase delay at frequencies above 10 Hz is related to the current amplifier dynamic. To measure the temperature of each component, the thermocouples were imbedded in the TPMA. Other thermocouples were used to measure the ambient and housing outer surface temperatures. Figure 16a shows the experimental results for an input current of 96.3 A, corresponding to 7700 N. The current was applied for at least 1 h and the maximum temperature was 130 \({}^{\circ}\)C. Convergence was  
Figure 14: Force characteristics of tubular permanent magnet actuator: (**a**) thrust; (**b**) cogging. Figure 15: Frequency response function of the tubular permanent magnet actuator: (**a**) thrust constant; (**b**) phase. achieved with a wind speed of 8 m/s at ambient temperature. This speed can be achieved using flow ducts without an additional cooling fan due to the high speed of the railway. As shown in Figure 16b, since well-defined thermal parameters were applied to the thermal FE model, the error between the experiment and the analysis was small, about 1.5%. Thermal experiments were performed to ensure robust thermal characteristics under harsh environmental conditions.