A Study on the Electromagnetic–Temperature Coupled Analysis Method for In-Wheel Motors

Abstract

As the core component of in-wheel motor-driven electric vehicles, the in-wheel motor (IWM) directly affects the driving/braking performance of each driving wheel and the driving performance of the vehicle. The IWM operation involves a coupling of multi-fields, including the electromagnetic, temperature, flow, and mechanical fields, which influence each other. It is necessary to study coupling analysis methods to obtain accurate and consistent results. In this paper, a 15 kW in-wheel motor is taken as the research object. Based on the finite element model of the IWM, the coupling factors between the electromagnetic and temperature field, and the influence trend of coupling factors on the two fields are investigated. On this basis, considering the strong coupling factors obtained from the above analysis, the unidirectional coupling and bidirectional coupling analysis methods are used to analyze the electromagnetic–temperature characteristics of the IWM, and the comparative results between the two methods are discussed. It was found that the results showed the temperature of the IWM calculated by the bidirectional coupling method was higher than that obtained by the unidirectional coupling analysis method. The maximum temperature of stator windings calculated by bidirectional coupling was 7.1% higher than that calculated by unidirectional coupling analysis, and the effect on the relative difference of torque could reach 7.4%. Bidirectional coupling can more accurately reflect the variation of variables in the fields and the prediction of motor performance in the process of motor operation. The progress made in the electromagnetic–temperature coupled analysis method can provide a theoretical basis and useful ideas for the multi-fields coupling analysis of IWMs.

1. Introduction

The in-wheel motor (IWM) drive is one of the key technologies of the next generation ‘pure electric drive’ electric vehicle, which eliminates power transmission devices, such as the transmission and differential. It not only improves the transmission efficiency of the system but also makes the chassis structure simpler and more flexible, and each wheel motor can be controlled independently, which can meet the power demand of the vehicle well [1,2,3,4].

As the core component of IWM driven electric vehicles, the structure and performance of the IWM directly affect the driving/braking performance of each driving wheel and the driving performance of the vehicle. It is necessary to analyze its performance accurately in the design. During operation, the coupling of electromagnetic, temperature, and other fields in the motor takes place [5,6]. However, most of the current studies on IWMs adopt a decoupling simplification method to separate the coupled fields, which not only affects the accuracy of the solution results but also affects the design results and performance prediction. Therefore, it is necessary to study the coupling analysis method of the electromagnetic and temperature fields of an IWM to improve the accuracy of the calculation and to accurately predict the performance of the IWM.

In the coupling analysis of the electromagnetic and temperature field of an IWM, Li W. et al. studied the optimization of the magnetic circuit of the motor. The basic parameters, such as torque and loss, are calculated by the two-dimensional transient electromagnetic field of the optimized structure, and then the temperature distribution using the results of electromagnetic calculation was analyzed [7]. Vese I. C. et al. simulated the electromagnetic and temperature characteristics of the motor. First, the electromagnetic field of the motor was analyzed by the finite element method, and the loss value of the motor was calculated. The loss value was taken as the main thermal source in the temperature field. The finite element analysis of the temperature distribution of the motor was carried out [8]. Alberti L. et al. analyzed the electromagnetic–thermal coupling of a three-phase induction motor. First, the parameters of the equivalent circuit of the motor were obtained by establishing the finite element model, and the loss was calculated as the heat source of thermal analysis. Then the lumped parameter temperature model was established. At the same time, the parameters of the lumped parameter thermal model grid were calculated accurately by using the finite element method, then the temperature rise of different components was calculated by the lumped parameter thermal network model, and fed back to the electromagnetic analysis. The parameters of the electromagnetic analysis were updated and calculated until convergence [9]. In summary, in reviewing the research on the electromagnetic and temperature fields of an IWM, we find most of them are unidirectional coupling analyses, that is, the final results of loss calculated by the electromagnetic field are taken as the initial condition of the temperature field, and the temperature distribution is analyzed with little consideration given to the influence of temperature on electromagnetic characteristics [10,11,12,13,14,15]. In fact, the components of an IWM adopt different materials, and the change in temperature varies the material properties of each component, which can lead to change in important electromagnetic parameters and affect the electromagnetic characteristics. The change in electromagnetic characteristics can further affect the temperature characteristics of each component. This is a process of mutual influence and interaction. Different coupling methods used in the calculation may obtain different results. The analysis of the influence of different coupling methods on the calculation results deserves further discussion.

In this paper, a 15 kW IWM is taken as the research object. Based on establishing the finite element model of the IWM, the coupling factors between the electromagnetic and temperature fields, and the influence trend of the two fields are analyzed. According to the analysis, ignoring the influence factors of weak coupling, the coupling calculation of the electromagnetic field and temperature field of an IWM is carried out by using the unidirectional coupling method and bidirectional coupling method, respectively, and the calculation errors of different coupling analysis methods are predicted by comparing and analyzing the results.

2. Structure of In-Wheel Motor

In this paper, the IWM was a 15 kW interior permanent magnet (PM) synchronous motor. It adopted the structure of an inner rotor and outer stator, 72 slots and 8 pairs of poles. The specific structure is shown in Figure 1. The basic structure parameters are shown in

Table 1. Structural parameters of the in-wheel motor (IWM).

Name	Value
Stator Diameter	310 mm
Stator Bore	240 mm
Stator length	69 mm
Rotor inner diameter	200 mm
Air gap length	0.9 mm
Rotor length	71 mm

, and the basic performance parameters are shown in

Table 2. Basic parameters of IWM.

Parameter Name	Value
Rated power	15 kW
Phase number	3
Rated line voltage	330 V
Winding form	Y
Rated torque	143.25 Nm
Rated speed	1000 r/min
Pole number	8

.

3. Analysis of Coupling Factors

To analyze the coupling of the electromagnetic and temperature field of an IWM more accurately, the coupling factors of electromagnetic and temperature characteristics were analyzed first, and the coupling relationship between the two fields was further clarified.

3.1. Coupling Relationship Between Electromagnetic and Temperature Field

The following Equation (1) shows Maxwell’s equations in the electromagnetic field [16]:

$$\{\begin{array}{l}\nabla \times H=J+\partial D/\partial t\\ \nabla \times E=-\partial B/\partial t\\ \nabla \cdot D=\rho \\ \nabla \cdot B=0\end{array}.$$

(1)

Equation (1), in turn, represents Ampere’s loop law, Faraday’s law of electromagnetic induction, Gauss’s law of electric current, and Gauss’s law of magnetic flux.

The heat conduction equation of the temperature field is shown as [17]

$${\rho }_{m}c\frac{\partial T}{\partial t}-k{\nabla }^{2}T=Q.$$

(2)

The above equation expresses that the heat required for heating is balanced with the incoming heat and the heat generated by the heat source itself.

Equations (1) and (2) show that the temperature of the IWM is affected by parameters such as heat source, heat conduction coefficient, material density, etc. The heat source is produced by the Joule heat and magnetocaloric effect converted from the loss caused by the electromagnetic field. At the same time, the change in temperature affects the material properties of the motor components, such as conductivity, permeability, and other parameters. Further, it will affect the magnetic field vector, flux density vector, and other electromagnetic parameters of the IWM, and then affect the electromagnetic characteristics of the IWM. During the operation of the IWM, the electromagnetic field and temperature field always interact with each other. Figure 2 shows the coupling relationship between the electromagnetic field and temperature field.

3.2. Analysis of Coupling Factors

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]:

$$P_{iron}=P_h+P_c+P_e=K_{h}f{B}_m^x+K_c{(f{B}_m)}^2+K_e{(f{B}_m)}^{1.5},$$

(3)

$$K_c={\pi }^2\cdot \sigma \cdot d^2/6.$$

(4)

• 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)

$$P_{copper}=m{I}^{2}R.$$

(5)

• Eddy current loss of permanent magnet

The main loss in permanent magnets is eddy current loss. The equation is as follow [20]:

$$P_{magnet}={\displaystyle {\int }_{V}E·J_{\mathrm{s}}}dv.$$

(6)

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:

$$q=P_{loss}/V.$$

(7)

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:

$$R=\frac{{\rho }_{\mathrm{r}}l}{A}.$$

(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]

$${\rho }_T={\rho }_0(1+\alpha (T-T_0)).$$

(9)

Equation (9) shows that the resistivity of a material varies with temperature.

Table 3. The resistivity and resistance temperature coefficient of materials.

	Permanent Magnet (Nd-Fe-B)	Winding (Copper)	Rotor and Stator Core (DW465-50)
Resistivity at 20 °C (Ω·m)	$1.6\times {10}^{-6}$	$1.678\times {10}^{-8}$	$5\times {10}^{-7}$
Temperature coefficient of resistance (°C−1)	0.004	0.00393	0.004

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

$${\tau }_{iT}=\frac{{\rho }_{iT}}{{\rho }_{iT\max }}.$$

(10)

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]:

$$B_r=[1+(T-20){\alpha }_{Br}](1-\raisebox{1ex}{$IL$}\!\left/ \!\raisebox{-1ex}{$100$}\right.)B_{r20}.$$

(11)

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.

4. Coupling Analysis of the Electromagnetic–Temperature Field

In the design process of an IWM, different coupling analysis methods will have different effects on the performance prediction accuracy of the IWM, which will directly affect the performance of the designed IWM and the applied vehicles. Aimed at the problem that the unidirectional coupling analysis method is often used to analyze IWMs at present, this Section will analyze the coupling characteristics of electromagnetic and temperature of the IWM by unidirectional coupling analysis and bidirectional coupling analysis, respectively, and compare the results.

Figure 7 below illustrates the two coupling methods and the coupling factors considered in the analysis.

Comsol software (Comsol 5.0, COMSOL company, Stockholm, Sweden) is used in the calculation, and the following simulations were performed in an open-loop fashion and refer to steady-state operation. The IWM speed and root-mean-square phase current were the input variables. Under the steady-state operation, the IWM speed was 1000 r/min, and the root-mean-square phase current of the stator winding was 27.91 A. Coordinate transformation and vector control were carried out by the software according to the basic theory [24].

4.1. Coupling Characteristic Analysis Based on Unidirectional Coupling Method

4.1.1. Establishment of the Finite Element Model

By comparison, the finite element method is high precision and widely used [25,26]. Using Comsol software, the finite element model was established according to the structure and parameters of the IWM in Section 2, and the dynamic mesh was set for the rotating part of the motor. In the process of mesh generation, the boundary meshes on both sides of the air gap were divided more strictly. On the two sides of the air gap, one side in the rotating region needed to be more refined than the one in the fixed region. Figure 8 shows the partial mesh partition of the two-dimensional IWM.

As can be observed from Figure 8, because the magnetic flux distribution was basically in the area surrounded by the stator and rotor cores, the mesh was small in the stator and rotor core, winding, and air gap, and large in the shell and motor shaft.

4.1.2. Establishment of the Electromagnetic Analysis Model

The establishment of the electromagnetic analysis model includes the definition of material, the determination of excitation source, and boundary conditions. Silicon steel sheet material DW465-50 was used as the core material of the motor stator and rotor, and the input BH curve was used as the constitutive relation of magnetic field; winding material was copper; rotor bracket, casing, and shaft material were steel; the PM material was NdFeB (material grades: N33UH), its magnetizing direction was set in the cylindrical coordinate system, and the constitutive relation of magnetic field was set as residual magnetic flux density. The insulation layer was made of insulating material. At the same time, the conductivity of copper winding material was the value of conductivity at 60 °C, while the conductivity of other components was the value of conductivity at 20 °C. The core region of the stator and rotor was bounded by Ampere’s law, while the other regions were bounded by the Gauss’ law. Then, the current of the stator winding was set up. Figure 9 shows the schematic diagram of a three-phase winding configuration.

According to Figure 9, the input and output boundaries of the winding current are set, and the winding turns, the conductor cross-section, and the current values under rated conditions are given.

4.1.3. Establishment of the Temperature Analysis Model

After the establishment of the electromagnetic analysis model, the temperature analysis model was established, which is mainly about the determination of heat source and boundary conditions of the temperature field.

• Heat Source for IWM

Using the established electromagnetic analysis model, the loss values of each part, including copper loss, iron loss of stator and rotor, and eddy current loss of PM, can be obtained by transient analysis under the rated working condition of the electromagnetic field.

Figure 10 and Figure 11 show the time-varying curves of core loss and PM loss of the stator and rotor, respectively.

From Figure 10 and Figure 11, it can be observed that the stator core loss was about 146.35 W; the rotor core loss was about 2.38 W; the PM loss was about 0.34 W; and the coil copper loss was a constant, 651.85 W. By substituting the calculated loss value into Equation (6), the heat generation rate of each part can be obtained and loaded into the temperature calculation model as a heat source for calculation.

Table 4. The heat generation rate of each part.

	Stator Core	Rotor Core	Winding	PM
Loss (W)	146.35	2.38	651.85	0.34
Volume (m3)	1.49 × 10−3	6.48 × 10−4	3.89 × 10−4	2.50 × 10−4
Heat Generation Rate (W/m3)	9.82 × 104	3.67 × 103	1.67 × 106	1.36 × 103

shows the volume and heating rate of the stator core, winding, and PM of the IWM.

• Determination of boundary conditions and material properties

As convection heat transfer generated on the outer surface of the casing circumferentially and on both sides of the casing axially, it is necessary to calculate the surface heat dissipation coefficient. Figure 12 shows the surfaces of the casing.

The equation for calculating the surface heat dissipation coefficient is as follows [27]:

$${\alpha }_{\mathrm{e}}={\alpha }_0\left(1+k_c\sqrt{v_{\mathrm{e}}}\right)\sqrt[3]{\frac{\theta }{25}}.$$

(12)

Through calculation, the heat dissipation coefficient of the outer surface of the casing circumferentially was 41.7 W/(m²·K), and that of the two side surface of the casing axially was 37.1 W/(m²·K). The air temperature on the outer surface was 22 °C.

At the same time, as an object of heat conduction, the velocity of heat conduction is closely related to the thermal conductivity. Thermal conductivity is a parameter that characterizes the heat transfer of an object under a stable temperature. Components with different materials have different thermal conductivity, especially the stator and rotor cores with silicon steel sheets that have anisotropic thermal conductivity. The material properties of each part of IWM can be obtained, as shown in

Table 5. Material properties of components.

Component	Material	Thermal Conductivity (W/(m·°C)	Density (kg/m3)	Specific Heat Capacity (J/(kg·°C))
Stator and rotor core	DW465-50	40/40/0.95	7700	426
Winding	copper	379	8900	390
PM	NdFeB	6.16	7800	460
Insulation layer	Insulation material	0.3	1300	1340
Housing	45 steel	50.2	7850	480
Air	Air	0.0267	1.29	1000

.

4.1.4. Analysis of Unidirectional Coupling Results

After establishing the electromagnetic analysis model and the temperature analysis model, the unidirectional coupling analysis of the electromagnetic–temperature field can be carried out, and the changes in the two fields can be obtained. Figure 13 below shows the temperature distribution of the IWM at 12,000 s by unidirectional coupling calculation.

From Figure 13, it can be observed that the highest temperature appeared in the stator winding, and the lowest temperature distributed in the motor shaft. And the highest temperature of each part can be obtained, as shown in

Table 6. The highest temperature of each component.

	Stator Core	Rotor Core	Winding	PM
Highest temperature (°C)	111.2	85.5	117.8	85.4

.

4.2. 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. The highest temperature of each component.

	Stator Core	Rotor Core	Winding	PM
Highest temperature (°C)	117.95	89.8	126.2	89.8

.

4.3. Comparative Analysis of Different Coupling Methods

To better compare the results of unidirectional coupling and bidirectional coupling analysis, the transient analysis results of the temperature field under the two methods were compared. The temperature curves of the IWM windings, stator, and rotor cores and PMs are compared and analyzed, as shown in Figure 15, Figure 16, Figure 17 and Figure 18, respectively. The results of torque comparison under the two methods are shown in Figure 19.

It can be observed from Figure 15, Figure 16, Figure 17 and Figure 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. The highest temperature of each component.

	Maximum Temperature (°C)				Torque (Nm)
	Stator Core	Rotor Core	Winding	PM
Unidirectional coupling method	111.2	85.5	117.8	85.4	144.36
Bidirectional coupling method	117.95	89.8	126.2	89.8	133.62
Relative difference	6.1%	5.0%	7.1%	5.1%	7.4%

. And the relative difference is calculated as:

$$\epsilon =\frac{\left|x_{UCM}-x_{BCM}\right|}{x_{UCM}}\times 100\%.$$

(13)

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. Because the bidirectional coupling takes into account the mutual coupling between the two fields, the bidirectional coupling analysis was closer to the actual operation of IWM. Conditions can more accurately reflect the changes in variables in the field and the prediction of motor performance during the operation of the motor.

5. Conclusions

Based on the analysis of the coupling factors of the electromagnetic field and temperature field of a 15 kW IWM, the coupling characteristics of the electromagnetic–temperature field of the IWM were analyzed by different coupling methods. Through this study, the following useful conclusions are drawn:

• The electromagnetic field and temperature field are coupled and interact with each other in the IWM. The electromagnetic characteristic influences the temperature distribution of the motor through iron core loss, winding loss, and PM loss. By comparison, the winding loss is the greatest heat source of the IWM. Among the influencing factors of the temperature on the electromagnetic characteristics, the resistivity of the stator silicon steel material and the winding copper material is the primary factor.
• Because the coupling factors of the two fields mainly consider the above strong coupling factors, electromagnetic–temperature coupling analysis was carried out with the unidirectional coupling analysis and bidirectional coupling analysis methods. According to the comparison of the results, there were some differences in the calculation results of the two fields. The temperature of the stator windings, stator cores, rotor cores, and PMs calculated by the bidirectional coupling method was higher than that obtained by the unidirectional coupling analysis method. The maximum temperature of stator windings calculated by bidirectional coupling was 7.1% higher than that calculated by unidirectional coupling analysis, and the effect on the relative difference of torque reached 7.4%.
• The bidirectional coupling can more accurately reflect the variation of variables in the fields and the prediction of motor performance in the process of motor operation. Meanwhile, it should be pointed out that the coupling analysis of the two fields mainly considers the two strong coupling factors. If all the coupling factors of the two fields were considered in the coupling analysis, the results from the two methods would have a greater difference. This is worth considering in the coupling analysis.

The research work in this paper provides a theoretical basis and useful ideas for the multi-field coupling analysis of IWMs.