Temperature Field Calculation of the Hybrid Heat Pipe Cooled Permanent Magnet Synchronous Motor for Electric Vehicles Based on Equivalent Thermal Network Method

: A hybrid heat-pipe cooling structure for the permanent magnet synchronous motor for electric vehicles was analyzed in this paper to effectively equalize the axial temperature rise and reduce the average temperature of each heat-generating component in the motor. A temperature ﬁeld calculation method for the hybrid heat-pipe cooling permanent magnet synchronous motor based on an equivalent thermal network method was proposed in this paper to save computing resources in temperature ﬁeld analyses for hybrid heat-pipe cooling motors. The results were veriﬁed against the simulation results of the computational ﬂuid dynamics method under three common operating conditions for electric vehicles. The error was proven to be within 5%. The calculation time of the proposed method was compared with the computational ﬂuid dynamics method, which demonstrated that the calculation time of the proposed method was within 194 s.


Introduction
With the development of electric vehicle technology, its advantages, such as low energy consumption and no harmful emissions. have gradually made it a major transportation alternative to traditional internal combustion engine vehicles, providing an effective solution to the problem of human carbon neutrality [1]. Currently, common electric vehicle drive motors include asynchronous motors and permanent magnet synchronous motors (PMSM). With their high torque density and high efficiency, the PMSM has become the main traction motor used in electric vehicles such as the Tesla Model 3 and Porsche Taycan [2]. With the gradual maturity of electric vehicle technology, the permanent magnet synchronous motor as a drive motor is also developing in the direction of higher torque density and higher efficiency, which leads to a sharp increase in the temperature rise of the internal heating components of the motor, and the design of a reasonable cooling structure for it has become a hot research topic today.
In order to realize a rational design of the motor cooling system, an accurate calculation technique of the motor temperature field is an important tool in motor design work. Common temperature field calculation methods for permanent magnet synchronous motors mainly use commercial finite volume method temperature field calculation procedures based on computational fluid dynamics (CFD) [3], a main problem of which is that the calculation of complex motor cooling structures is time-consuming and requires a lot of computational resources. Such problems are particularly evident in the calculation of the motor temperature field of phase-change cooling structures [4], and it has become particularly important to improve the efficiency of the calculation of the motor temperature motor temperature field calculation was compared to the conventional CFD method in this paper.

Motor Model
The hybrid heat-pipe cooling structure adopted in the PMSM for EVs. The main features of the three-dimensional structure are shown in Figure 1. It can be seen that there are two forms of heat pipes in the hybrid heat-pipe cooling structure, one is rotating and the other is fixed. An additional heat path from the rotor and permanent magnet to the water-cooled hollow shaft of the motor was built using the Ushaped rotating heat pipe installed inside the rotor, so that the large amount of heat generated by the rotor and permanent magnet when the motor is in high-speed operating conditions can be rapidly transferred to the cooling water inside the hollow shaft. The problem of the high average temperature rise of the rotor and permanent magnet, and uneven axial temperature rise, was solved effectively [20]. The additional heat path from the end winding to the motor water-cooled shell was constructed by using the ring-shaped fixed heat pipe, so that a large amount of heat generated by the winding can be quickly transferred to the motor-cooling water. The problem of the high temperature rise at the end of the winding was solved [21]. To further enhance the heat-transfer efficiency from the motor winding to the annular fixed heat pipe, a high thermal conductivity potting material was used to pot the winding end to the annular fixed heat pipe.

The Equivalent Thermal Conductivity of Hybrid Heat Pipe
The parameters of the hybrid heat-pipe cooling PMSM researched in this paper are shown in Table 1. In order to obtain the different thermal conductivity of the hybrid heat pipe under different working conditions of the hybrid heat-pipe cooling PMSM, the thermal conductivity and the internal mass motion state of the fixed and rotating heat pipe under different operating conditions of the PMSM were simulated by the CFD method. The VOF two-phase flow model and the LEE phase-change model were used to simulate the phase-change process and the internal mass motion of the rotating heat pipe at different speeds of the motor. The main working medium used in the working process of the rotating heat pipe involved in this paper was distilled water, the gaseous working It can be seen that there are two forms of heat pipes in the hybrid heat-pipe cooling structure, one is rotating and the other is fixed. An additional heat path from the rotor and permanent magnet to the water-cooled hollow shaft of the motor was built using the U-shaped rotating heat pipe installed inside the rotor, so that the large amount of heat generated by the rotor and permanent magnet when the motor is in high-speed operating conditions can be rapidly transferred to the cooling water inside the hollow shaft. The problem of the high average temperature rise of the rotor and permanent magnet, and uneven axial temperature rise, was solved effectively [20]. The additional heat path from the end winding to the motor water-cooled shell was constructed by using the ringshaped fixed heat pipe, so that a large amount of heat generated by the winding can be quickly transferred to the motor-cooling water. The problem of the high temperature rise at the end of the winding was solved [21]. To further enhance the heat-transfer efficiency from the motor winding to the annular fixed heat pipe, a high thermal conductivity potting material was used to pot the winding end to the annular fixed heat pipe.

The Equivalent Thermal Conductivity of Hybrid Heat Pipe
The parameters of the hybrid heat-pipe cooling PMSM researched in this paper are shown in Table 1. In order to obtain the different thermal conductivity of the hybrid heat pipe under different working conditions of the hybrid heat-pipe cooling PMSM, the thermal conductivity and the internal mass motion state of the fixed and rotating heat pipe under different operating conditions of the PMSM were simulated by the CFD method. The VOF two-phase flow model and the LEE phase-change model were used to simulate the phasechange process and the internal mass motion of the rotating heat pipe at different speeds of the motor. The main working medium used in the working process of the rotating heat pipe involved in this paper was distilled water, the gaseous working medium was water vapor, and the phase-transition temperature of the liquid working medium inside the heat pipe was 15 • C. The volume fraction of the internal mass of the rotating heat pipe under rated working condition is shown in Figure 2. It can be seen that the liquid mass is mainly distributed in the radial top of the rotating heat pipe under the centrifugal force of rotor rotation. medium was water vapor, and the phase-transition temperature of the liquid working medium inside the heat pipe was 15 °C. The volume fraction of the internal mass of the rotating heat pipe under rated working condition is shown in Figure 2. It can be seen that the liquid mass is mainly distributed in the radial top of the rotating heat pipe under the centrifugal force of rotor rotation. The temperature distribution cloud diagram of the rotating heat pipe at the rated working condition of the motor is shown in Figure 3. As shown in the diagram, the evaporation section of the heat pipe absorbs a large amount of heat generated by the rotor and permanent magnet, and the condensation section of the heat pipe effectively transfers the heat to the cooling water inside the hollow shaft of the motor. The equivalent thermal conductivity of the heat pipe calculated by the CFD method is shown in Table 2. It can be seen that when the PMSM for EVs is in the maximum torque working condition, that is, the speed is 600 r/min and the output torque is 380 N·m, the calculated thermal-resistance value is low. When the motor is at rated speed, due to the high motor speed, the fluid mass inside the rotating heat pipe is further accelerated back to the evaporation section by the centrifugal force of the rotor rotation, which further increases the equivalent thermal conductivity. The temperature distribution cloud diagram of the rotating heat pipe at the rated working condition of the motor is shown in Figure 3. As shown in the diagram, the evaporation section of the heat pipe absorbs a large amount of heat generated by the rotor and permanent magnet, and the condensation section of the heat pipe effectively transfers the heat to the cooling water inside the hollow shaft of the motor. medium was water vapor, and the phase-transition temperature of the liquid working medium inside the heat pipe was 15 °C. The volume fraction of the internal mass of the rotating heat pipe under rated working condition is shown in Figure 2. It can be seen that the liquid mass is mainly distributed in the radial top of the rotating heat pipe under the centrifugal force of rotor rotation. The temperature distribution cloud diagram of the rotating heat pipe at the rated working condition of the motor is shown in Figure 3. As shown in the diagram, the evaporation section of the heat pipe absorbs a large amount of heat generated by the rotor and permanent magnet, and the condensation section of the heat pipe effectively transfers the heat to the cooling water inside the hollow shaft of the motor. The equivalent thermal conductivity of the heat pipe calculated by the CFD method is shown in Table 2. It can be seen that when the PMSM for EVs is in the maximum torque working condition, that is, the speed is 600 r/min and the output torque is 380 N·m, the calculated thermal-resistance value is low. When the motor is at rated speed, due to the high motor speed, the fluid mass inside the rotating heat pipe is further accelerated back to the evaporation section by the centrifugal force of the rotor rotation, which further increases the equivalent thermal conductivity. The equivalent thermal conductivity of the heat pipe calculated by the CFD method is shown in Table 2. It can be seen that when the PMSM for EVs is in the maximum torque working condition, that is, the speed is 600 r/min and the output torque is 380 N·m, the calculated thermal-resistance value is low. When the motor is at rated speed, due to the high motor speed, the fluid mass inside the rotating heat pipe is further accelerated back to the evaporation section by the centrifugal force of the rotor rotation, which further increases the equivalent thermal conductivity.

Thermal Network Model Construction
The specific process of calculating the temperature field of the hybrid heat-pipe cooling motor by using the equivalent thermal network method is shown in Figure 4. Before the thermal network construction of the motor, the components with little influence on the temperature field calculation in the motor, such as the junction box, are generally simplified. After simplifying the motor model, the equivalent thermal network was established by treating the heating parts in the motor as nodes and the heat-transfer path as thermal resistance, and the thermal resistance of each heat-transfer path in the motor was calculated according to different working conditions and different speeds of the motor. The thermal resistance mainly included contact and convective thermal resistance between different materials. Then, the core loss of the stator and rotor, the eddy current loss of the permanent magnet, the copper loss of the winding, and the mechanical friction loss under three typical working conditions of the motor were calculated using FEA software and an empirical formula. These losses were taken as input values into the established equivalent thermal network. The node voltage method in the analogy circuit used the temperature of the external coolant as the reference node to obtain the cooling matrix. Finally, the cooling matrix was calculated and iteratively corrected until the temperature of each node, satisfying the residual conditions, was obtained.

Thermal Network Model Construction
The specific process of calculating the temperature field of the hybrid heat-pipe cooling motor by using the equivalent thermal network method is shown in Figure 4. Before the thermal network construction of the motor, the components with little influence on the temperature field calculation in the motor, such as the junction box, are generally simplified. After simplifying the motor model, the equivalent thermal network was established by treating the heating parts in the motor as nodes and the heat-transfer path as thermal resistance, and the thermal resistance of each heat-transfer path in the motor was calculated according to different working conditions and different speeds of the motor. The thermal resistance mainly included contact and convective thermal resistance between different materials. Then, the core loss of the stator and rotor, the eddy current loss of the permanent magnet, the copper loss of the winding, and the mechanical friction loss under three typical working conditions of the motor were calculated using FEA software and an empirical formula. These losses were taken as input values into the established equivalent thermal network. The node voltage method in the analogy circuit used the temperature of the external coolant as the reference node to obtain the cooling matrix. Finally, the cooling matrix was calculated and iteratively corrected until the temperature of each node, satisfying the residual conditions, was obtained. Firstly, the hybrid heat-pipe cooling motor is divided into thermal network nodes according to the materials of each heating component, and different nodes are connected by thermal resistance according to heat conduction and heat convection. In order to improve the calculation efficiency, the following assumptions are proposed under the premise of ensuring calculation accuracy [22,23]: Firstly, the hybrid heat-pipe cooling motor is divided into thermal network nodes according to the materials of each heating component, and different nodes are connected by thermal resistance according to heat conduction and heat convection. In order to improve the calculation efficiency, the following assumptions are proposed under the premise of ensuring calculation accuracy [22,23]: The motor's cooling effect and temperature distribution are circumferentially symmetrical; 1. The motor's cooling effect and temperature distribution are circumferentially symmetrical.; 2. All the heat generated in the motor is regarded as all derived from the coolant, and the coolant temperature is regarded as the external node; 3. Due to the heat pipe's relatively large thermal conductivity, heat pipe's additional heat path is regarded as heat conduction only through the heat pipe. 4. All the losses of the rotor and the permanent magnet are transmitted from the heat pipe to the coolant in the hollow shaft.
Based on the above settings, the motor can be divided into stator yoke, stator teeth, winding, end winding heat pipe, thermal conductive potting material, rotor boot, permanent magnet, rotor yoke, rotor heat pipe, hollow shaft, bearing, end cover and shell according to the material and structure. Considering that the thermal conductivity of stator core and rotor core is low due to the stacking of silicon steel sheets in the axial direction, the stator core, rotor core, and permanent magnet are divided into three parts along the axial direction. Finally, the hybrid heat-pipe cooling motor is simplified to the equivalent node distribution diagram shown in Figure 5. According to the equivalent node diagram of the motor, the thermal resistance of heat conduction or thermal convection type can be added between the nodes, and the equivalent thermal network of the hybrid heat-pipe cooling permanent magnet synchronous motor for electric vehicle can be obtained. Among them, nodes 1-3 are stator yokes, nodes 7-11 are windings, nodes 4-6 are stator teeth, nodes 12-14 are rotor boots, nodes 18-20 are rotor yokes, nodes 15-17 are permanent magnets, nodes 21-22 are bearings, nodes 28 are hollow shafts, nodes 29-30 are rotor rotating heat pipes, nodes 25-27 are air domains, and nodes 23-24 are motor end covers. Node 31-32 is the heat pipe at the end winding, and node 33 is the water-cooled shell.

Thermal Resistance Calculation of Hybrid Heat-Pipe Cooling Structure
The heat transfer modes involved in the internal heating components of permanent magnet synchronous motor for electric vehicles include heat conduction, convective heat transfer and radiation heat transfer. This paper mainly analyzes the heat conduction and convective heat transfer inside the motor. According to the equivalent node diagram of the motor, the thermal resistance of heat conduction or thermal convection type can be added between the nodes, and the equivalent thermal network of the hybrid heat-pipe cooling permanent magnet synchronous motor for electric vehicle can be obtained. Among them, nodes 1-3 are stator yokes, nodes 7-11 are windings, nodes 4-6 are stator teeth, nodes 12-14 are rotor boots, nodes 18-20 are rotor yokes, nodes 15-17 are permanent magnets, nodes 21-22 are bearings, nodes 28 are hollow shafts, nodes 29-30 are rotor rotating heat pipes, nodes 25-27 are air domains, and nodes 23-24 are motor end covers. Node 31-32 is the heat pipe at the end winding, and node 33 is the water-cooled shell.

Thermal Resistance Calculation of Hybrid Heat-Pipe Cooling Structure
The heat transfer modes involved in the internal heating components of permanent magnet synchronous motor for electric vehicles include heat conduction, convective heat transfer and radiation heat transfer. This paper mainly analyzes the heat conduction and convective heat transfer inside the motor.
The form of heat conduction of permanent magnet synchronous motor for electric vehicles is mainly divided into plate heat conduction and cylinder heat conduction [24]. Most of the heat conduction in the motor is cylindrical heat conduction. The schematic diagram of the heat conduction principle on the cylinder wall is shown in Figure 6. For example, the stator yoke, the rotor yoke, the annular heat pipe and the rotating heat pipe can be equivalent to the heat conduction of the cylinder wall. The thermal resistance calculation method is as follows: where, l is the equivalent length of the cylinder parts such as the stator and rotor and the annular heat pipe, λ 1 is the thermal conductivity of the contact part 1, λ 2 is the thermal conductivity of the contact part 2, r 1 is the inner meridian of the cylinder, r 2 is the interface radius of the two-cylinder parts, and r 3 is the outer diameter of the cylinder.
The form of heat conduction of permanent magnet synchronous motor for electric vehicles is mainly divided into plate heat conduction and cylinder heat conduction [24]. Most of the heat conduction in the motor is cylindrical heat conduction. The schematic diagram of the heat conduction principle on the cylinder wall is shown in Figure 6. For example, the stator yoke, the rotor yoke, the annular heat pipe and the rotating heat pipe can be equivalent to the heat conduction of the cylinder wall. The thermal resistance calculation method is as follows: Where, l is the equivalent length of the cylinder parts such as the stator and rotor and the annular heat pipe, λ1 is the thermal conductivity of the contact part 1, λ2 is the thermal conductivity of the contact part 2, r1 is the inner meridian of the cylinder, r2 is the interface radius of the two-cylinder parts, and r3 is the outer diameter of the cylinder. The air and cooling water in the motor belong to the fluid, and the fluid is mainly divided into laminar flow and turbulent flow according to its flow state [25,26]. In the motor, the critical Reynolds number is mainly used to judge the flow state of the fluid. It is generally believed that the fluid is laminar when the Reynolds number is less than 2200, and the fluid is turbulent when the Reynolds number is greater than 2200. The air in the air gap of the motor studied in this paper is in a turbulent state under the rated and maximum speed conditions, and the fluid in the air gap is in a laminar state under the maximum torque condition. As shown in Formula (7), the equivalent Reynolds number of the fluid in the motor can be calculated: Where, D is the equivalent diameter of the cylinder, v is the kinematic viscosity coefficient of the fluid, and u is the velocity of the fluid.
According to the calculated equivalent Reynolds number, the Nusselt number of the fluid can be further calculated, and the convective heat transfer coefficient between the fluid and the solid can be finally obtained [27].  The air and cooling water in the motor belong to the fluid, and the fluid is mainly divided into laminar flow and turbulent flow according to its flow state [25,26]. In the motor, the critical Reynolds number is mainly used to judge the flow state of the fluid. It is generally believed that the fluid is laminar when the Reynolds number is less than 2200, and the fluid is turbulent when the Reynolds number is greater than 2200. The air in the air gap of the motor studied in this paper is in a turbulent state under the rated and maximum speed conditions, and the fluid in the air gap is in a laminar state under the maximum torque condition. As shown in Formula (7), the equivalent Reynolds number of the fluid in the motor can be calculated: where, D is the equivalent diameter of the cylinder, v is the kinematic viscosity coefficient of the fluid, and u is the velocity of the fluid. According to the calculated equivalent Reynolds number, the Nusselt number of the fluid can be further calculated, and the convective heat transfer coefficient between the fluid and the solid can be finally obtained [27].
where, λ is the thermal conductivity of the fluid, Nu is the Nusselt number of the fluid, Re is the Reynolds number of the fluid, Pr is the Prandtl constant, D is the equivalent diameter of the cylinder, L is the equivalent length of the cylinder, α is the convective heat transfer coefficient. According to the convection heat transfer coefficient of the fluid at different solid boundaries calculated above, the corresponding thermal convection thermal resistance can be obtained by substituting it into the following formula.
where, S is the heat transfer area of the fluid-solid boundary. The cooling structure of the hybrid heat pipe mainly includes the contact thermal resistance from the end winding to the annular fixed heat pipe, the convective thermal resistance from the rotor yoke to the cooling water in the hollow shaft through the U-shaped rotating heat pipe, and the equivalent thermal resistance of the annular heat pipe and the U-shaped rotating heat pipe itself. The three types of thermal resistance in the hybrid heat-pipe cooling structure are calculated below: 1.
The thermal resistance from the end winding to the annular fixed heat pipe is R 87 and R 1011 . Taking R 87 as an example: where, d cudb is the radial thickness of the end winding, and λ hp is the static thermal conductivity of the heat pipe.

2.
The thermal resistance of the rotor yoke through the rotating heat pipe to the cooling water inside the hollow shaft includes R 1828 , R 1928 and R 2028 . The thermal resistance R 1828 is analyzed as an example: where, α q is the convective heat transfer coefficient between the heat pipe and the cooling water, and S 1828 is the equivalent heat transfer area between the rotating heat pipe and the cooling water.

3.
The thermal resistance of the annular fixed heat pipe at the end of the winding and the rotating heat pipe at the rotor yoke can be obtained by the temperature difference between the evaporation section and the condensation section of the heat pipe obtained by the CFD simulation above, namely: where, T e is the steady-state temperature of the evaporation section of the heat pipe, T c is the steady-state temperature of the condensation section of the heat pipe, P is the heat load of the evaporation section of the heat pipe, and R is the equivalent thermal resistance of the heat pipe.

The Stator Yoke
According to the above thermal network division and node selection inside the motor, it can be seen that the stator yoke mainly contains five heat-transfer paths, which are the stator yoke to the winding, the stator yoke to the shell, the stator yoke to the potting glue, the stator yoke to the stator tooth, and the axial heat conduction of the stator yoke itself.

1.
The thermal resistance from the stator yoke to the winding in the slot is R 18 , R 29 and R 310 . Considering the consistency of thermal resistance, R 18 is taken as the analysis object: where, λ stj is the radial thermal conductivity of the silicon-steel sheet in the stator yoke, λ jy is the thermal conductivity of insulation in the slot, λ cu is the thermal conductivity of the winding"s copper wire, d de is the radial thickness of the stator yoke, d cu is the radial thickness of the winding, S 18 is the equivalent heat-transfer area between node 1 and node 8, and the same below.

2.
Taking the thermal resistance from the stator yoke to shell R 133 from node 1 to node 33 as an example: ln r sh r dw (10) where, λ Sh is the shell thermal conductivity, r den is the inner diameter of the stator yoke, r dw is the outer diameter of the stator yoke, and r sh is the outer diameter of the motor shell.

3.
The thermal resistance from the stator yoke to the potting material is R 131 and R 132 , taking R 131 as an example: where, R 12 is the axial thermal resistance of stator yoke, λ gf is the thermal conductivity of the potting adhesive, and S 131 is the equivalent heat-transfer area between the stator yoke and potting material.

4.
The thermal resistances from the stator yoke to stator teeth are R 14 , R 25 and R 36 . Taking R 14 as an example: (12) where, r dn is the inner diameter of the stator.

5.
The axial thermal resistance of the stator yoke is R 12 and R 23 . Taking R 12 as an example: (13) where, λ stz is the axial thermal conductivity of the stator silicon-steel sheet, and l is the axial length of the stator.

The Winding
According to the above division of the motor thermal network and the selection of nodes, it can be seen that there are three main heat-transfer paths of the winding, namely: winding to stator yoke, winding to stator teeth, winding axial thermal resistance.

1.
The thermal resistance from the winding to the stator yoke is, taking R 81 between node 8 and node 1 as an example: 2.
The thermal resistance from the winding to the stator teeth includes R 84 , R 95 , and R 106 , taking R 84 as an example: where, d dc is the radial thickness of stator teeth.

3.
The axial thermal resistance of the winding includes R 89 and R 910 , taking R 89 as an example:

The Stator Teeth
According to the above division of the thermal network inside the motor and the selection of nodes, it can be seen that the main heat-transfer paths of the stator teeth are divided into four: thermal resistance from the stator teeth to the stator yoke, thermal resistance from the stator teeth to the winding, axial thermal resistance from the stator teeth and the thermal resistance from the air-gap to the stator teeth.

1.
The thermal resistance from the stator teeth to the stator yoke includes R 41 , R 51 , and R 63 . Taking R 41 as an example for calculation: 2.
The thermal resistance from the stator teeth to the winding includes R 48 , R 59 , and R 610 , taking R 48 as an example: 3.
The axial thermal resistance of stator teeth includes R 45 and R 56 . Taking R 45 as an example: 4. The thermal resistance from stator teeth to air gap includes R 426 , R 526 and R 626 . Taking R 426 as an example: (20) where, α cq is the convective heat transfer coefficient between the stator teeth and the air-gap.

The Rotor Boot
There are four primary heat-transfer paths of the rotor boot: rotor boot to air-gap, rotor boot to permanent magnet, rotor boot to rotor yoke, and axial heat conduction of the rotor boot.

1.
The thermal resistance from the rotor boot to the air gap includes R 1226 , R 1326 and R 1426 . Taking R 1226 as an example: where, r rw is the outer diameter of the rotor, r re is the outer diameter of the rotor yoke, and α cq is the convective heat-transfer coefficient between the rotor boot and the air gap.

2.
The thermal resistance from rotor boot to the permanent magnet includes R 1215 , R 1316 and R 1417 . The thermal resistance R 1215 is analyzed as an example: ln r rw r re (22) where, λ pm is the thermal conductivity of the permanent magnet and d pm is the thickness of the permanent magnet.

3.
The thermal resistance from the rotor boot to the rotor yoke includes R 1218 , R 1319 and R 1420 . Taking R 1218 as an example for analysis: where, r m is the inner diameter of the rotor.

4.
The axial thermal resistance inside the rotor boot includes R 1213 and R 1314 . Taking R 1213 as an example for analysis:

The Permanent Magnet
There are three main heat-transfer paths of the permanent magnet in the motor: the permanent magnet to the rotor boot, the permanent magnet to rotor yoke and the axial heat conduction of the permanent magnet.

1.
The thermal resistance from permanent magnet to rotor boot includes R 1512 , R 1613 and R 1714 . Taking R 1512 as an example for analysis: 2.
The thermal resistance from the permanent magnet to the rotor yoke includes R 1518 , R 1619 and R 1720 . The thermal resistance R 1518 is analyzed as an example: 3.
The axial thermal resistance of permanent magnet includes R 1516 and R 1617 . Taking R 1516 as an example for analysis:

The Rotor Yoke
The primary heat-transfer path of the rotor yoke in the motor is divided into three: the rotor yoke to the rotor boot, the rotor yoke to the permanent magnet, and the rotor yoke axial heat conduction.

1.
The thermal resistance from the rotor yoke to the rotor boot includes R 1812 , R 1913 and R 2014 . The thermal resistance R 1812 is analyzed as an example: 2.
The thermal resistance from the rotor yoke to the permanent magnet includes R 1815 , R 1916 and R 2017 . The thermal resistance R 1815 is analyzed as an example: 3.
The axial thermal resistance of the rotor yoke includes R 1819 and R 1920 . Taking R 1819 as an example for analysis:

Solution of Equivalent Thermal Network Model
According to the principle introduced above, the heat conduction thermal resistance and heat convection thermal resistance between each node in the motor were calculated, respectively. Taking the loss of each active node as input, the outflow energy and inflow energy of each node in the motor are equal due to the conservation of energy. Therefore, the thermal balance equation for any node can be obtained by analogy with the node voltage method in the circuit.
where, G (i, i) is the self-thermal conductivity of node i, and the remaining thermal conductivity is the mutual thermal conductivity. Each node's heat-balance equations in the thermal network are listed to form a matrix.
that is: where, G is the thermal conductivity matrix, T is the temperature rise matrix, and P is the loss matrix. The cooling matrix was modeled, and the loss value of each node was substituted into the iterative solution to obtain the temperature rise of each node.

Hybrid Heat-Pipe Cooling PMSM for Evs
Based on the equivalent thermal network proposed in this paper, the temperature field calculation of a hybrid heat-pipe cooling PMSM for Evs was analyzed. The main parameters of the PMSM are shown in Table 1. The rated speed is 4000 rpm, the maximum speed is 12,000 rpm, and the maximum output torque is 380 Nm. The material properties of the potting glue used to enhance the heat-transfer efficiency of the winding end are shown in Table 3. The material properties of the main modeling areas involved in the motor are shown in Table 4.  This article analyzed the losses of various heating components in the motor before calculating the temperature field using an equivalent thermal network. The various losses in the motor mainly included copper loss, iron loss, permanent magnet eddy current loss, and mechanical loss. Due to the fact that the iron loss of the motor was mainly related to the harmonic magnetic field during actual operation, and, in order to make the subsequent calculation results of the motor's equivalent thermal network more accurate, this article used non-sinusoidal excitation collected from the actual inverter for loss analysis.
The corresponding losses of each node in the equivalent thermal network under nonsinusoidal excitation under three operating conditions are shown in Table 5. As is shown in Table 5, the losses of the motor under three typical working conditions were obtained by electromagnetic FEA and an empirical formula, respectively.

Equivalent Thermal Conductivity Experiment of Heat Pipe
The equivalent thermal conductivity of the heat pipe is obtained by designing a reasonable equivalent experiment to calculate the contact thermal resistance further accurately. The experimental platform is shown in Figure 7, including drive motor, PT100 thermocouple, rotating hollow shaft, DC heating wire, etc. This paper mainly analyzes the equivalent thermal conductivity of the rotating heat pipe at different speeds of the motor under the same thermal load of 200 W. The main principle of the experiment is to fix the U-shaped rotating heat pipe on the rotating shaft. By measuring the temperature difference between the evaporation section and the condensation section of the rotating heat pipe at different speeds, the equivalent thermal conductivity of the rotating heat pipe can be calculated by introducing the formula (34-35).
where, Φ is the heat flux transmitted through the heat pipe, T e is the steady-state temperature of the evaporation section of the heat pipe, T c is the steady-state temperature of the condensation section of the heat pipe, P is the heat load of the evaporation section of the heat pipe, ∆t represents the average temperature difference between the evaporation and condensation section, R is the equivalent thermal resistance of the heat pipe, k is the equivalent heat transfer coefficient of the heat pipe, and A is the equivalent contact area of the heat pipe. The experimental data of the equivalent thermal conductivity of the heat pipe are shown in Table 6. When the heat load of the rotor is constant, the thermal conductivity of the rotating heat pipe increases first, then decreases, and finally increases again with the increase of the rotor speed. The highest thermal conductivity can reach 76,804.9 W/m·k at 4000 rpm.  The experimental data of the equivalent thermal conductivity of the heat pipe are shown in Table 6. When the heat load of the rotor is constant, the thermal conductivity of the rotating heat pipe increases first, then decreases, and finally increases again with the increase of the rotor speed. The highest thermal conductivity can reach 76,804.9 W/m·k at 4000 rpm. As shown in Table 7, the equivalent thermal conductivity of the heat pipe obtained by the CFD simulation iwa compared with the conclusion of the equivalent experimental calculation designed in this paper. It can be seen that the thermal conductivity of the rotating heat pipe obtained by the CFD simulation is low, which is due to the fact that CFD cannot effectively simulate the liquid working medium provided by the wick structure inside the heat pipe to further accelerate the reflux effect when simulating the rotating heat pipe.

Equivalent Thermal Conductivity Experiment of Heat Pipe
Aimed at the cooling matrix equation proposed above, the iterative calculation of the temperature rise was solved in each modeling area under three working conditions of the rated condition, maximum speed and maximum torque. The iterative calculation time of the program was between 105-194 s under the three working conditions. The temperature rise calculation results of each modeling area under non-sinusoidal excitation are shown in Table 8. Subsequently, the CFD method was used to calculate the temperature field of the hybrid heat-pipe cooling motor to verify the effectiveness of the proposed calculation method based on the equivalent thermal network method. When the motor is at rated and maximum speed conditions, the k-epsilon classical turbulence model and the standard wall function are used to compensate for the grid-quality degradation caused by the complex shape inside the motor and the double-layer full y + processing to compensate for the grid-quality degradation caused by the complex shape of the motor during the simulation of the turbulence model. For the maximum torque condition, the fluid state in the air-gap can be judged as a laminar flow state according to the Reynolds number calculated by the air-gap, so the classical laminar flow model and the standard wall function were used for the simulation calculation. The radiation heat process was ignored because of its lower inference in terms of the temperature field calculation.
Based on the loss analysis and internal fluid state analysis of permanent magnet synchronous motors for hybrid heat-pipe cooled electric vehicles mentioned above, the CFD method was used to verify the accuracy of the proposed equivalent heat network method calculation results. The comparison is shown in Table 8. The temperature distribution nephogram of each heating component in the motor under three working conditions under non-sinusoidal excitation is shown in Figure 8. According to Table 8, it can be seen that the maximum error is within 3.2%, and the error mainly occurs at the end winding. This is mainly because the influence of the low thermal conductivity of the potting material was ignored in modeling the equivalent effect thermal network. After analysis, it can be considered that the error of the equivalent thermal network model established in this article is within an acceptable range.

Conclusions
In this paper, the equivalent thermal network method was used to analyze and calculate the temperature field of the hybrid heat-pipe cooling permanent magnet synchronous motor for electric vehicle with a rated power of 90 kw. The equivalent thermal conductivity of the rotating heat-pipe structure was obtained by the equivalent experiment. The temperature calculation results of each node under three typical conditions of rated condition, maximum torque condition, and maximum speed condition were compared with the simulation analysis results, and the following conclusions were obtained: 1. The equivalent thermal resistance of the rotating heat pipe decreases first, then increases, and finally decreases with the increase of the rotor rotation speed when the thermal load is constant. The thermal resistance is low at the rated speed of 4000 rpm, and the maximum equivalent thermal conductivity can reach 76,804.9W/m·k; 2. By establishing an equivalent thermal network for the motor, the temperature of each node was analyzed and calculated. When comparing the calculation results with the CFD simulation results, it was found that the temperature error of each node was within 5%, which is within the acceptable range. In view of the error, the analysis in this paper was mainly due to the complex shape of the cooling structure of the rotating heat pipe and the cooling structure of the heat pipe at the end winding, and the existence of potting glue between the end winding and the fixed heat pipe, which

Conclusions
In this paper, the equivalent thermal network method was used to analyze and calculate the temperature field of the hybrid heat-pipe cooling permanent magnet synchronous motor for electric vehicle with a rated power of 90 kw. The equivalent thermal conductivity of the rotating heat-pipe structure was obtained by the equivalent experiment. The temperature calculation results of each node under three typical conditions of rated condition, maximum torque condition, and maximum speed condition were compared with the simulation analysis results, and the following conclusions were obtained: 1.
The equivalent thermal resistance of the rotating heat pipe decreases first, then increases, and finally decreases with the increase of the rotor rotation speed when the thermal load is constant. The thermal resistance is low at the rated speed of 4000 rpm, and the maximum equivalent thermal conductivity can reach 76,804.9W/m·k; 2.
By establishing an equivalent thermal network for the motor, the temperature of each node was analyzed and calculated. When comparing the calculation results with the CFD simulation results, it was found that the temperature error of each node was within 5%, which is within the acceptable range. In view of the error, the analysis in this paper was mainly due to the complex shape of the cooling structure of the rotating heat pipe and the cooling structure of the heat pipe at the end winding, and the existence of potting glue between the end winding and the fixed heat pipe, which affects the accuracy of the thermal resistance calculation, so there is a certain error in the calculation results; 3.
In this paper, by comparing the temperature field calculation method of hybrid heatpipe cooling motor based on equivalent thermal network with the traditional CFD simulation time, it was found that the iteration time of the proposed calculation method is within 194 s, which greatly shortens the temperature field calculation time of the hybrid heat-pipe cooling PMSM for EVs.