# A Study on the Electromagnetic-Temperature Coupled Analysis Method for In-Wheel Motors
## 4 Coupling Analysis of the Electromagnetic-Temperature Field
### Coupling Characteristic Analysis Based on the Bidirectional Coupling Method  
Based on the electromagnetic and the temperature analysis model, the bilateral coupled effect of the two fields was considered in the calculation and analysis. In each iterative calculation, the temperature was calculated by the loss result of the electromagnetic field. The temperature calculated from the temperature analysis was used to calculate the electromagnetic characteristics, and the two fields were calculated separately. So long as the iteration time step is short enough, the electromagnetic-temperature coupling analysis of the motor can be close to the actual situation [28]. By solving the temperature of the IWM under bi-directional coupling, the temperature distribution diagram of the bi-directional coupling transient analysis at 12,000 s is shown in Figure 14.  
As can be observed from Figure 14, the global maximum temperature in the transient state also appears in the stator winding. The highest temperature of each part can be obtained, as shown in Table 7. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Stator Core** & **Rotor Core** & **Winding** & **PM** \\ \hline Highest temperature (\({}^{\circ}\)C) & 111.2 & 85.5 & 117.8 & 85.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The highest temperature of each component. Figure 14: Temperature distribution with the bidirectional coupled method. Figure 13: Temperature distribution with unidirectional coupling method. It can be observed from Figures 15-18 that (1) the maximum temperature of each component was relatively small without considering the effect of temperature on material properties; (2) the temperature rise of each component was larger than that of unidirectional coupling when considering the temperature effect on winding conductivity and stator core material resistivity. In the process of bidirectional coupling, the electrical resistivity of the iron core, winding, and permanent magnet increased with the increase in temperature, as shown in Figure 3. And as a result, the copper loss of the winding, the iron loss of the stator and rotor, and the loss of permanent magnet increased with the increase in temperature, and the iron loss of stator all increased with the increase in temperature. Therefore, the temperature change in bidirectional coupling shows a higher trend than that of unidirectional coupling. From Figure 19, it can be observed that the torque under unidirectional coupling was basically stable at about 144.36 Nm, and the torque under bidirectional coupling decreased from about 144.36 Nm to about 133.62 Nm with the increase in temperature. The change in temperature has an influence on the torque. This is because the influence of temperature on the material properties of the IWM components is considered in the bidirectional coupling analysis, and the change in material properties will further affect the characteristics of the electromagnetic and temperature field. Based on the above analyses, the maximum temperature of each IWM component, stable torque, and the relative difference between unidirectional coupling and bidirectional coupling methods are listed in Table 8. And the relative difference is calculated as:  
\[\varepsilon=\frac{\left|x_{UCM}-x_{BCM}\right|}{x_{UCM}}\times 100\%. \tag{13}\]  
Figure 19: Torque evolutions. Figure 18: The temperature rise of the magnet. To illuminate the difference more intuitively, Figure 20 shows the comparison of the maximum temperature of each IWM component and the torque under bidirectional coupling and unidirectional coupling transient analysis. From Figure 20, the maximum temperature of each IWM component calculated by the bidirectional coupling and unidirectional coupling analysis method shows that the maximum temperature of the winding and stator had relatively large variations. The torque obtained from the two methods had relatively large variations also. It can be observed that there were some differences between the results of bidirectional coupling and unidirectional coupling analysis.