# A Study on the Electromagnetic-Temperature Coupled Analysis Method for In-Wheel Motors
## 3 Analysis of Coupling Factors
### Analysis of Coupling Factors
#### 3.2.2 Coupling Factors of the Temperature Field to the Electromagnetic Field  
The influence of temperature on electromagnetic characteristics is mainly reflected in the temperature of material properties. The following is a detailed analysis of the influence of temperature on different material properties in the motor, and the main influencing factors are obtained through analysis. * The influence of temperature on resistance The effect of temperature on the electric field is mainly manifested in the influence of temperature on resistance. The equation for calculating resistance is as follows:  
\[R=\frac{\rho_{\mathrm{r}}l}{A}. \tag{8}\]  
Equation (8) shows that the resistance depends mainly on the resistivity. For the resistivity of copper material, it increases with the increase in temperature in the low-temperature region, while the high-temperature working environment of the motor is still in the low-temperature region for copper. The resistivity at different temperatures can be obtained by Equation (9) [21]  
\[\rho_{T}=\rho_{0}(1+\alpha(T-T_{0})). \tag{9}\]  
Equation (9) shows that the resistivity of a material varies with temperature. Table 3 gives the resistivity and temperature coefficient values of various materials. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Permanent Magnet** & **Winding (Copper)** & **Rotor and Stator Core** \\  & **(Nd-Fe-B)** & **(DW465-50)** \\ \hline Resistivity at 20 \({}^{\circ}\)C (\(\Omega\cdot\)m) & \(1.6\times 10^{-6}\) & \(1.678\times 10^{-8}\) & \(5\times 10^{-7}\) \\ Temperature coefficient of resistance (\({}^{\circ}\)C\({}^{-1}\)) & 0.004 & 0.00393 & 0.004 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The resistivity and resistance temperature coefficient of materials. Figure 3 shows the resistivity curves of various materials vary with temperature calculated from Equation (9) and Table 3. To enable the comparison of different resistivity variations, the normalized calculation is carried out, and the normalized values can be obtained by  
\[\tau_{iT}=\frac{\rho_{iT}}{\rho_{iT\max}}. \tag{10}\]  
Based on Equation (10), the normalized results of different resistivity variations can be obtained, as shown in Figure 4.  
Figures 3 and 4 show that the resistivity of the PM is relatively large in the full temperature range, followed by the iron core resistivity and winding resistivity. But the influence trend of temperature on different material resistivity is almost the same. * The effect of temperature on the remanence of permanent magnets. The increase in temperature will also affect the permanent magnet. In this paper, the permanent magnet material of the IWM is Nd-Fe-B material, and its maximum operating temperature is 180 \({}^{\circ}\)C. When it works in a high-temperature environment, permanent demagnetization easily occurs. At the same time, the remanence of the permanent magnet will decrease with the increase in temperature. The main relationship can be expressed by the following equation [21]:  
\[B_{r}=[1+(T-20)\alpha_{Br}](1-\frac{IL}{100})B_{r20}. \tag{11}\]  
Figure 4: The resistivity curve changes with temperature. Figure 3: The resistivity curve changes with temperature. _IL_ accounts for the irreversible demagnetization of permanent magnets due to temperature increase, here, _IL_ is 5%. According to Equation (11), the curve of remanence versus temperature can be obtained, as shown in Figure 5.  
Previous research shows that the maximum temperature of a permanent magnet is about 80 \({}^{\circ}\)C when the IWM works. From the curve of remanence with temperature in Figure 5, it can be observed that the remanence decreases by about 0.08 from 20 \({}^{\circ}\)C to 80 \({}^{\circ}\)C. This factor will be considered in the following bidirectional coupling analysis. * The effect of temperature on the permeability of silicon steel. According to the inquiry data, the change in relative permeability of silicon steel with magnetic flux density is different at different temperatures [22].