To better understand the coupling factors between the electromagnetic field and temperature field and their influence trend on the two fields, the coupling factors of the two fields will be analyzed in detail in this Section, which will lay a foundation for the subsequent calculation and analysis of electromagnetic–temperature coupling.
3.2.1. Coupling Factors of Electromagnetic Field to Temperature Field
From the above analysis, it can be observed that during the operation of the motor, due to the effect of the electromagnetic field, loss will be generated, and most of these losses will be converted into heat, thus affecting the temperature of the motor, which is the main factor affecting the temperature to the electromagnetic characteristics [
18]. The main losses of the motor are iron loss in the stator and rotor core, copper loss in the winding and eddy current loss of the PM.
• Iron loss
Iron loss includes hysteresis loss, eddy current loss, and remanence loss. The proportion of remanence loss is very small, about 0.1% to 0.2%. Therefore, it is generally ignored in analysis. Bertotti’s model of iron loss separation can be used to calculate the iron loss of silicon steel sheets. This method is widely used and has certain accuracy. The core loss of the stator and rotor can be calculated by the following Equation [
19]:
• Copper loss
The copper loss in the windings is related to the current and resistance of the windings. The copper loss of the coils can be calculated by Equation (5)
• Eddy current loss of permanent magnet
The main loss in permanent magnets is eddy current loss. The equation is as follow [
20]:
The losses mentioned above are eventually converted to heat, and the heat generation rate is used as the heat source load in the calculation of the temperature field. The relationship between loss and heat generation rate is as follows:
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:
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]
Equation (9) shows that the resistivity of a material varies with temperature.
Table 3 gives the resistivity and temperature coefficient values of various 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
Based on Equation (10), the normalized results of different resistivity variations can be obtained, as shown in
Figure 4.
Figure 3 and
Figure 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 °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]:
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 °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 °C to 80 °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].
Figure 6 shows the curve of relative permeability with magnetic flux density, and the data from reference [
22].
As shown in
Figure 6, during the operation of the motor, the silicon steel material basically works at a higher magnetic flux density, over 1.4 T, the change in relative permeability with temperature is also very small. At the same time, the magnetic properties of 35A360 annular materials at different temperatures were studied by Norio Takahashi et al., of Okayama University, Japan. When it works in the low-temperature region below 500 °C, the permeability does not change obviously with temperature. When it works in the high-temperature region above 500 °C, the permeability decreases obviously with the increase in temperature [
23]. In summary, in the process of motor operation, silicon steel works in high magnetic flux density and the low-temperature working area, and its basic physical properties of permeability can be regarded as not changing with temperature.
Through the analysis of the coupling factors mentioned above, it can be observed that the electromagnetic characteristics mainly influence the temperature distribution of the motor through the iron loss of the stator and rotor, copper loss of winding, and eddy current loss of the PM; among the influencing factors of the temperature on the electromagnetic characteristics, the resistivity of the stator material silicon steel and the winding material copper is the main factor. Therefore, in the subsequent electromagnetic–temperature coupling analysis, the coupling factors of the two fields mainly consider the strong coupling factors mentioned above.