## 3 Electromagnetic and Thermal Finite Element (FE) Model
### Electromagnetic and Thermal Modelling of TPMA  
Both electromagnetic and thermal analyses are essential when designing a TPMA, because the thermal and electromagnetic characteristics are coupled [31]. Thus, a coupled-fields analysis model was applied. The results of the EM analysis were used as heat sources for the thermal analysis, and the results of the thermal analysis were then used to describe changes in the EM field caused by the heat source.  
Electromagnetic FE techniques provide an accurate means of determining the field distribution, including cases with flux leakage, fringing, magnetic saturation, and complex core shapes [32]. The EM analysis was performed using an FE model considering nonlinear magnetic and cylindrical symmetric conditions. A 3-D FE analysis is required to calculate the eddy current loss from permanent magnets accurately [33]. However, in the case of the proposed TPMA, the eddy current loss was as small as 0.25% of the copper loss because the servo bandwidth of the TPMA was 10 Hz and the core of the stator was composed of laminated silicon steel. Accordingly, time-effective two-dimensional (2-D) FE analysis was used for the design; the accuracy of the 2-D FE model was verified by comparison with the results of the 3-D FE model. The difference between the magnetic flux density in the air gap was less than 1%.  
The 2-D cylindrical FE model consisted of five pairs of three-phase coils, a laminated silicon steel core, and quasi-Halbach magnets. The aluminum housing and austenitic stainless steel shaft were excluded from the model because they do not affect the magnetic fields. Figure 8a shows the FE-predicted open-circuit flux distributions for zero axial displacements using the commercial software ANSYS Maxwell (V17.2.0, Ansys, Inc., Canonsburg, PA, USA). The TPMA thermal analysis model was built in three dimensions and applied to the boundary conditions and thermal critical parameters throughout the previous six-slot prototype experiment to find the temperature increase  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Conductivity (W/m-k)** & **Specific Heat (J/kg-k)** & **Density (kg/m\({}^{3}\))** \\ \hline Air & 0.0263 & 1007 & 1.1614 \\ Copper & 400.0 & 385.0 & 8933 \\ Silicon Steel & 24.2 (\(r\) direction) & 446.0 & 7650 \\ Aluminum & 210.0 & 900.0 & 2700 \\ NdFeB & 6.5 & 460.0 & 7600 \\ \(R_{out}\) _to cover_ & 0.1 & - & - \\ \(R_{start}\) _to never_ & 0.15 & - & - \\ \(R_{core}\) _to housing_ & 0.3 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: Thermal properties for thermal simulation.  
Figure 7: Saturation temperature of six-slot prototype model: (**a**) maximum temperature of the coil at various air velocities with a direct current of 30 A; (**b**) temperature distribution at three input currents with 8 m/s forced convection.  
and the maximum temperature due to losses from each part of the actuator. As shown in Figure 8b, the 3-D thermal FE model consisted of five pairs of three-phase coils, a laminated core, an aluminum cover, a magnet, a yoke, a shaft, and a housing. A one-sixth cylindrical model was used to reduce the analysis time. The maximum saturated temperature was 112 \({}^{\circ}\)C in the V phase coil with DC 80 A and 3 m/s forced convection using the commercial software ANSYS (V17.2.0, Ansys, Inc., Canonsburg, PA, USA); heat was transferred through conduction and convection.