Coupled magnetic-thermal fields analysis using an adaptive single-mesh method

Abstract

This paper proposes a coupled finite element analysis of magnetic-thermal fields using an adaptive single-mesh method. To avoid errors from the data transfer and the mesh reconstruction due to different mesh requirements for the magnetic and thermal analysis, this proposed method built single mesh for both analyses. The results of the coupled parameters were transmitted through the single mesh between magnetic and thermal fields. According to the characteristics of the magnetic and the thermal distribution, the master-slave node method was employed to simplify the original element matrix into the magnetic and the thermal matrices, where slave nodes were no longer solved in the solving process. Results of the slave nodes were expressed by their mater nodes using weight coefficients to regain the results of the original element matrix. A permanent magnet machine was introduced to validate the proposed method with both the commercial software and the experimental tests. Compared to the commercial software, the time cost was reduced by more than 8.0% when a higher accuracy of the thermal analysis was obtained.

Keywords

Coupled analysis finite element analysis master-slave method single mesh thermal analysis

1. Introduction

Recently, requirements of miniaturization, energy efficiency, and cost reduction have been claimed for electric machines with extensive applications of the military, transportation, aeronautics, and astronautics, etc. Taking advantages of superior power density and efficiency, low noise, and low maintenance costs [1–3], permanent magnet (PM) machines have been widely used for these applications, where the electromagnetic performance of electrical machines has been pushed to the extreme. While the materials, such as winding insulation and permanent magnets, are subjected to their temperature limits, it is necessary to analyze the thermal field to the same extent as the electromagnetic design [4]. The thermal analysis for electrical machines needs comprehensive consideration of magnetic, fluid, and thermal fields. Moreover, the problems of iron losses and thermal coefficients calculation remain unresolved. Lumped parameters (LP) [5–7], finite element analysis (FEA) [8,9], and computational fluid dynamics (CFD) [2] have been widely used for the thermal analysis in electrical machines. Normally, losses of each component in machines are calculated by the magnetic analysis and then imported as heat sources into the foregoing thermal analysis models [10,11]. The LP method is preferred for the preliminary and optimization designs due to its low computation need and incapability of field distribution calculation. Although the CFD method improves the thermal analysis by considering the influence of fluid dynamics, the demand of the computing effort is quite large with the fluid domain modeling, especially for the transient analysis. Various researches have proved that the FEA method can achieve reasonable results for both the magnetic and the thermal analysis with an acceptable efficiency using an adaptive method, shape optimization, etc. [10–15]. However, there are still lots of problems during the coupled analysis using commercial FEA software, such as massive computing effort, customized settings, iron loss calculation under non-sinusoidal magnetization, errors from the data transfer and the mesh reconstruction, etc. [16,17]. The meshing requirements of the magnetic and thermal analysis can be quite different according to the characteristics of the field distribution. For electrical machines, iron losses account for a large proportion of the total losses. Despite the heat sources from the iron losses are closely related to the flux density distribution, the thermal gradient is quite different from the magnetic gradient with specific cooling conditions. The emphasis of the mesh subdivision for the magnetic and thermal analysis is quite different. In many researches works, two meshes are separately built for the magnetic and thermal analysis, which results in errors from the data transfer and the mesh reconstruction.

This paper proposes an adaptive single-mesh method for the coupled magnetic-thermal fields analysis in machines. Firstly, a fine mesh is built for both the initial magnetic and the initial thermal analysis to avoid errors from the data transfer and the mesh reconstruction. Then, an adaptive meshing based on the master-slave method is employed to reduce the matrix order for the non-linear calculation process. Moreover, the object-oriented method is applied to optimize the programming architecture for a flexible customization in the coupled analysis [18]. Finally, a classical surface-mounted permanent magnet (PM) machine is implemented to compare the performance of the proposed method with ANSYS Workbench. Both the magnetic and the thermal results are validated by experimental tests.

2. Adaptive single-mesh method for coupled FEA

2.1. Thermal equations based on Galerkin method

According to the principle of heat transfers, the differential equation of the transient heat conduction with an internal heat source can be deduced as Eq. ((1)). $\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{k}{{\rho}c_{p}}}\left({\displaystyle \frac{\partial ^{2}T}{\partial x^{2}}}+{\displaystyle \frac{\partial ^{2}T}{\partial y^{2}}}+{\displaystyle \frac{q_{v}}{k}}\right) & & \displaystyle\end{eqnarray}$ (1) where T denotes the temperature, t denotes the time, k denotes the heat transfer coefficient, 𝜌 denotes the density, c _p denotes the specific heat capacity, and q _v denotes the volumetric heat source.

The first-class boundary condition, where the temperature is a constant value or a function of the position and time, can be treated as Eq. ((2)). The second-class boundary condition, where the heat flux is specific, is expressed as Eq. ((3)). For third-class boundary condition, where the temperature and heat convection coefficient is known, the formula is established as Eq. ((4)). $\begin{eqnarray}\displaystyle & \displaystyle T|_{{\Gamma}}=T_{w}\quad \text{or}\quad T|_{{\Gamma}}=f(x,y,t) & \displaystyle\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle & \displaystyle -k\left.{\displaystyle \frac{\partial T}{\partial n}}\right|_{{\Gamma}}=q_{c}\quad \text{or}\quad -k\left.{\displaystyle \frac{\partial T}{\partial n}}\right|_{{\Gamma}}=g(x,y,t) & \displaystyle\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle & \displaystyle -k\left.{\displaystyle \frac{\partial T}{\partial n}}\right|_{{\Gamma}}={\alpha}(T-T_{f})|_{{\Gamma}} & \displaystyle\end{eqnarray}$ (4) where 𝛼 represents the heat convection coefficient, and T _f represents the temperature at interface between solid and fluid.

To simplify the problem of solving differential equations into the problem of solving linear algebraic equations, the Galerkin method is employed. The weight function can be written as Eq. (5), so Eq. (1) can be deduced as Eq. (6). $\begin{eqnarray}\displaystyle & \displaystyle W_{l}={\displaystyle \frac{\partial \widetilde{T}}{\partial T_{l}}}\quad (l=1,2,\ldots ,n) & \displaystyle\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle & \displaystyle \iint _{D}W_{l}\left[k\left({\displaystyle \frac{\partial ^{2}\widetilde{T}}{\partial x^{2}}}+{\displaystyle \frac{\partial ^{2}\widetilde{T}}{\partial y^{2}}}\right)+q_{v}-{\rho}c_{p}{\displaystyle \frac{\partial \widetilde{T}}{\partial t}}\right]∼\text{d}x∼\text{d}y=0. & \displaystyle\end{eqnarray}$ (6)

According to Green’s theorem and the differential relation between coordinates and boundary elements on the boundary in Eq. (7), Eq. (6) can be deduced as Eq. (8). $\begin{eqnarray}\displaystyle & \displaystyle -{\displaystyle \frac{\partial T}{\partial y}}∼\text{d}x+{\displaystyle \frac{\partial T}{\partial x}}∼\text{d}y={\displaystyle \frac{\partial T}{\partial n}}∼\text{d}s & \displaystyle\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle & \displaystyle \iint _{D}\left[k\left({\displaystyle \frac{\partial W_{l}}{\partial x}}\cdot {\displaystyle \frac{\partial \widetilde{T}}{\partial x}}+{\displaystyle \frac{\partial W_{l}}{\partial y}}\cdot {\displaystyle \frac{\partial \widetilde{T}}{\partial y}}\right)-q_{v}W_{l}+{\rho}c_{p}W_{l}{\displaystyle \frac{\partial \widetilde{T}}{\partial t}}\right]∼\text{d}x∼\text{d}y-\oint _{{\Gamma}}kW_{l}{\displaystyle \frac{\partial \widetilde{T}}{\partial n}}∼\text{d}s=0. & \displaystyle\end{eqnarray}$ (8)

For the thermal FEA with triangular elements, the three-vertex linear interpolation method is employed like the magnetic analysis. The assembled algebraic equations can be deduced as Eq. (9), where the coefficient matrices [K], [N], and [P] are listed in the Appendix. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}c@{}}{\displaystyle \frac{\partial J_{e}}{\partial T_{i}}}\\[11.0pt] {\displaystyle \frac{\partial J_{e}}{\partial T_{j}}}\\[11.0pt] {\displaystyle \frac{\partial J_{e}}{\partial T_{m}}}\end{array}\right\}=\left[\begin{array}{@{}ccc@{}}K_{ii} & K_{ij} & K_{im}\\ K_{ji} & K_{jj} & K_{jm}\\ K_{mi} & K_{mj} & K_{mm}\end{array}\right]\left\{\begin{array}{@{}c@{}}T_{i}\\ T_{j}\\ T_{m}\end{array}\right\}+\left[\begin{array}{@{}ccc@{}}N_{ii} & N_{ij} & N_{im}\\ N_{ji} & N_{jj} & N_{jm}\\ N_{mi} & N_{mj} & N_{mm}\end{array}\right]\left\{\begin{array}{@{}c@{}}{\displaystyle \frac{\partial T_{i}}{\partial t}}\\[11.0pt] {\displaystyle \frac{\partial T_{j}}{\partial t}}\\[11.0pt] {\displaystyle \frac{\partial T_{m}}{\partial t}}\end{array}\right\}-\left\{\begin{array}{@{}c@{}}P_{i}\\ P_{j}\\ P_{m}\end{array}\right\} & & \displaystyle\end{eqnarray}$ (9)

2.2. Adaptive single-mesh based on master-slave method

To avoid errors from the data transfer and the mesh reconstruction in the coupling process, a fine mesh was built considering the characteristics of both the magnetic and the thermal field distribution. By applying an adaptive meshing method, the order of the original element matrix (OEM) was reduced into the magnetic element matrix (MEM) and thermal element matrix (TEM) by eliminating certain nodes according to characteristics of the magnetic and thermal fields, respectively. The magnetic and thermal problems were individually solved by the MEM and TEM, while results could be mutually exchanged through OEM by expressing slave nodes with their master nodes.

The realization of the adaptive meshing method was illustrated by taking a solution of a non-linear field as an example. According to the OEM, the zero-order approximate solution was calculated and posterior errors were then estimated according to the distribution. A diagram of the master-slave definition is illustrated in Fig. 1. Node 5 could be defined as a slave node when $\begin{eqnarray}\displaystyle |A_{5}^{(0)}-(m_{25}A_{2}^{(0)}+m_{35}A_{3}^{(0)})|<{\xi} & & \displaystyle\end{eqnarray}$ (10) where $A_{i}^{(0)}$ was the zero-order approximate solution of node i, 𝜉 was the upper limit of error, and m _jm was the weight coefficient of node j on node m. The size of coefficients was related to the geometric position of nodes.

Fig. 1.

Diagram of the master-slave definition.

Since node 5 was a slave node to node 2 and 3, the solution of node 5 could be expressed as $\begin{eqnarray}\displaystyle A_{5}^{(0)}=m_{25}A_{2}^{(0)}+m_{35}A_{3}^{(0)}. & & \displaystyle\end{eqnarray}$ (11) In this way, node 5 was no longer solved in the following process. The order of the OEM could be reduced by substituting Eq. (11). $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}cccc@{}}a_{11} & a_{12}+m_{25}a_{15} & a_{13}+m_{35}a_{15} & a_{14}\\ a_{21} & a_{22}+m_{25}a_{25} & a_{23}+m_{35}a_{25} & a_{24}\\ a_{31} & a_{32}+m_{25}a_{35} & a_{33}+m_{35}a_{35} & a_{34}\\ a_{41} & a_{42}+m_{25}a_{45} & a_{43}+m_{35}a_{45} & a_{44}\end{array}\right]\left(\begin{array}{@{}c@{}}s_{1}\\ s_{2}\\ s_{3}\\ s_{4}\end{array}\right)=\left(\begin{array}{@{}c@{}}f_{1}\\ f_{2}\\ f_{3}\\ f_{4}\end{array}\right). & & \displaystyle\end{eqnarray}$ (12)

To achieve the programming design of the adaptive mesh method, the reassembled coefficients of each slave node should be correspondingly applied to the original coefficient matrix and the source matrix. Thus, the definition and weight coefficients of master-slave nodes were added into the original node classes, as shown in Table 1.

Table 1

Structure of the node classes for the adaptive meshing method

Global	M-S	Global	Master	Master	Master	Master
ID	flag	UID	ID 1	Weight 1	ID 2	Weight 2
1	0	1	N/A	N/A	N/A	N/A
2	1	2	N/A	N/A	N/A	N/A
3	1	3	N/A	N/A	N/A	N/A
4	0	4	N/A	N/A	N/A	N/A
5	−1	N/A	2	m ₂₅	3	m ₃₅

M-S flag denotes Master-Slave flag.

2.3. Coupling process

The data to be exchanged in the coupling process included losses and material properties, which were sensitive to the temperature. The material properties varied with the temperature, such as the copper resistivity, the coercivity of the PMs, and the B-H curves of the silicon steels. The variations of the material properties affected both the magnetic and the thermal performance. In this paper, the thermal modification of the copper resistivity and the coercivity of the PMs were considered in the coupling process. The heat sources for the thermal analysis were calculated according to the results of the magnetic analysis, especially the iron and copper losses. The process of the proposed coupled FEA was achieved as follows.

Step (1)

The zero-order approximate solution of the magnetic field was calculated according to the OEM. By applying the adaptive meshing method, the MEM was obtained by eliminating magnetic slave nodes from the OEM.

Step (2)

Results of magnetic master nodes along with the normal nodes were solved according to the MEM, and then applied for calculating results of magnetic slave nodes using the corresponding weight coefficients.

Step (3)

Heat sources were calculated in elements of the OEM. The zero-order approximate solution of the thermal field was solved to gain the TEM using the adaptive meshing method.

Step (4)

Results of the thermal field were obtained in the same way of Step 2.

Step (5)

The material properties, which were sensitive to the temperature, were re-defined according to the thermal results and imported to the magnetic analysis back to Step 2 until convergent results were achieved for both the magnetic and the thermal fields.

The diagram of the coupling process is illustrated in Fig. 2. The computation requirements were reduced by applying the simplified MEM and TEM, while the accuracy was improved by exchanging the coupled parameters in the single mesh OEM.

Fig. 2.

Process of coupled FEA, where 𝜌_{cu_0} is the initial value of the copper resistivity, and ϵ_T is the residual of the copper resistivity.

Table 2

Design specification of the PMSM

Parameter	Value
Rated speed	1500 r/min
Rated current	10 A
Stator outer diameter	120 mm
Stator inner diameter	75 mm
Air-gap length	0.5 mm
Slot number	24
Coil per slot	16
Magnet embrace	0.8
Magnet thickness	6 mm
Silicon steel model	DW465_50
PM model	NdFeB_30SH

3. Performance and validation

A classical three-phase 4-pole surface-mounted PM machine, as given in Table 2, was introduced to validate the proposed coupled FEA method. Software GID was employed for pre- and post- process during the coupled FEA. Triangular element type (maximum length of elements 2 mm) was applied to build the original fine mesh with 5907 nodes. For the transient magnetic analysis, the stop time and time step were set to 0.1 s and 0.002 s respectively, while the thermal problem was solved in the steady-state. The residual was set at 0.0001 and 0.01 for the magnetic and thermal analysis. Same simulation setup was used in the commercial software ANSYS. The thermal coefficient of the copper resistivity was considered in the later calculation of the copper loss, while the remanence and the coercivity of the PMs at different temperatures were obtained by the fitting curves given in Fig. 3. The remanence, shown in Fig. 3a, was a linear function the temperature. The relation between the coercivity and the temperature, shown in Fig. 3b, was non-linear. The thermal modification of the coercivity was achieved by piecewise linear fitting formulas. Both the remanence and the coercivity decreased with increasing temperature resulting in lower flux density as well as lower iron losses.

Fig. 3.

Remanence and coercivity of NdFeB_30SH at different temperatures.

3.1. Heat sources and heat transfer coefficients

Heat sources were considered as the sum of copper losses, iron losses, and mechanical losses. The mechanical loss was composed of the bearing and the windage losses. Since the bearing losses could not be considered by 2-D analysis, only the windage loss, which was 1.5% of the rated power according to machine design experience from the manufacturer, was assumed in the following analysis. The copper losses P _cu were expressed by Eq. (13). $\begin{eqnarray}\displaystyle P_{\text{cu}}={\rho}_{\text{cu}}(T)\cdot J_{\text{rms}}^{2} & & \displaystyle\end{eqnarray}$ (13) where 𝜌_cu(T) and J _rms were the copper resistivity at the temperature T and the root mean square value of the current density, respectively. The relationship between 𝜌_cu(T) and T was calculated by Eq. (14) for the thermal modification in the coupling process. $\begin{eqnarray}\displaystyle {\rho}_{\text{cu}}(T)={\rho}_{\text{cu}\_20}\left(1+{\displaystyle \frac{T-20}{234.5}}\right) & & \displaystyle\end{eqnarray}$ (14) where the copper resistivity at 20 °C was 𝜌_{cu_20} = 1.724 × 10⁻⁶ Ω ⋅ cm.

Fig. 4.

Equivalent model for thermal conductivity calculation in slots.

Meanwhile, the iron losses were predicted according to Eq. (15). $\begin{eqnarray}\displaystyle P_{Fe}=K_{h}fB_{m}^{h}+K_{e}(fB_{m})^{2}+K_{a}(fB_{m})^{1.5} & & \displaystyle\end{eqnarray}$ (15) where B _m denoted the magnitude of flux density, f denoted the frequency, and h denoted the coefficient of flux density on hysteresis loss. K _h, K _e, and K _a denoted the coefficients of hysteresis, eddy-current, and additional losses, respectively. The coefficients for the iron losses, which were calculated in W∕m³, were given by the manufacturer as Eq. (16). $\begin{eqnarray}\displaystyle K_{h}=205.552,\quad K_{e}=1.404,\quad K_{a}=5.464e-3,\quad h=1.741 & & \displaystyle\end{eqnarray}$ (16)

In the thermal analysis, heat transfers by conduction and convection were discussed while the heat transfer by radiation was neglected. Thermal conductivities of the windings and silicon steels were calculated by Eqs ((17))–((19)), while convection coefficients of the air-gap and the housing were calculated by Eqs ((21)) and ((22)). The winding in each slot was equivalent to a concentrated winding enwrapped by the uniformly distributed insulation in the middle, shown in Fig. 4. The thermal conductivity of the windings was calculated as Eq. ((17)). $\begin{eqnarray}\displaystyle {\lambda}_{\text{wind.}}=\left.\mathop{\sum }_{i=1}^{n}d_{i}\right/\mathop{\sum }_{i=1}^{n}(d_{i}/{\lambda}_{i}) & & \displaystyle\end{eqnarray}$ (17) where d _i denoted the mean thicknesses of the air, insulation and copper while 𝜆_i denoted the corresponding thermal conductivities.

For thermal conductivity calculation in silicon steels, the axial thermal conduction was equivalent to a series connection of multi-layered flat walls while the radial and circumferential ones were equivalent to parallel connections of multi-layered flat walls. The thermal conductivities of the silicon steels were calculated as Eqs ((18))–((19)). $\begin{eqnarray}\displaystyle & \displaystyle {\lambda}_{z}={\displaystyle \frac{d_{\text{Fe}}+d_{0}}{{\displaystyle \frac{d_{\text{Fe}}}{{\lambda}_{\text{Fe}}}}+{\displaystyle \frac{d_{0}}{{\lambda}_{0}}}}}={\displaystyle \frac{1}{{\displaystyle \frac{K_{\text{Fe}}}{{\lambda}_{\text{Fe}}}}+{\displaystyle \frac{1-K_{\text{Fe}}}{{\lambda}_{0}}}}} & \displaystyle\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle & \displaystyle {\lambda}_{\text{rad.}}={\lambda}_{\text{cir.}}={\displaystyle \frac{d_{\text{Fe}}{\lambda}_{\text{Fe}}+d_{0}{\lambda}_{0}}{d_{\text{Fe}}+d_{0}}}=K_{\text{Fe}}{\lambda}_{\text{Fe}}+(1-K_{\text{Fe}}){\lambda}_{0} & \displaystyle\end{eqnarray}$ (19) where 𝜆_z, 𝜆_rad., and 𝜆_cir. were the thermal conductivities in axial, radial and circumferential directions, respectively. d _Fe was the thickness of the silicon steels, while d ₀ was the sum of the thickness of the insulation and the air. 𝜆_Fe and 𝜆₀ were the corresponding thermal conductivities where the conductivity of the air was treated as same as the one of the insulation. K _Fe was the stack factor of the iron cores.

The heat transfers between the stator and the rotor were accomplished through conduction and convection in the air-gap, which could not be fully considered by the third-class boundary. An equivalent convection coefficient 𝜆_e was introduced to the air-gap treatment [19]. The calculation of 𝜆_e was relevant to the fluid dynamics in the air-gap region. When the fluid in the air-gap was laminar flow, 𝜆_e was equal to the thermal conductivity of air. When the Taylor number, expressed by Eq. (20), in the air-gap was greater than the critical value 41.2, Taylor–Couette vortices were found in the air-gap region. $\begin{eqnarray}\displaystyle T_{a}={\displaystyle \frac{R_{m}^{0.5}{\delta}^{1.5}{\omega}}{{\mu}}} & & \displaystyle\end{eqnarray}$ (20) where R _m was the mean radius of the air-gap, δ was the length of the air-gap, 𝜔 was the angular velocity of the rotor, and 𝜇 was the kinematic viscosity. In this condition, 𝜆_e was calculated by the empirical formula as Eq. (21). $\begin{eqnarray}\displaystyle {\lambda}_{e}=0.23{\beta}\left({\displaystyle \frac{{\delta}}{R_{r}}}\right)^{0.25}R_{e}^{0.5}{\lambda}_{a} & & \displaystyle\end{eqnarray}$ (21) where 𝛽 was the empirical factor of the roughness of the rotor surface, which was set as 1.2. R _r was the outer diameter of the rotor, R _e was the Renolyds number, and 𝜆_a was the thermal conductivity of air.

The convection coefficient of the housing was expressed by the empirical formula as Eq. ((22)). $\begin{eqnarray}\displaystyle {\alpha}_{\text{hous.}}=14(1+0.5\sqrt{v_{k}})^{3}\sqrt{{\displaystyle \frac{T_{\text{amb.}}}{25}}} & & \displaystyle\end{eqnarray}$ (22) where v _k was the air velocity near the housing surface, and T _amb. was the ambient temperature.

3.2. Results

The time costs of the uncoupled magnetic and thermal analysis were 264.2 and 12.4 seconds. By applying the adaptive meshing method, 109 nodes (1.8%) and 121 nodes (2.0%) were defined as slave nodes for the magnetic and thermal analysis, respectively. Despite the massive computing efforts of the zero-order solution according to the OEM, the time costs of the non-linear process were correspondingly reduced by 21.1 s (8.0%) and 1.2 s (9.7%). The time cost of the coupled analysis with the proposed method was 51 minutes, 18 seconds, while the one with the commercial software ANSYS Workbench was 55 minutes, 52 seconds. The time cost was reduced by 274 s (8.2%) because of the matrix reduction and the lack of the mesh reconstruction by applying the adaptive single-mesh method. In addition, experimental tests were carried out to validate the proposed method. The magnetic and thermal analysis was verified under no-load and rated on-load operation. The test-rig shown in Fig. 5a was mainly composed of the PM machine, a companion machine, an inverter, and the temperature monitor. The temperatures of key components were measured by thermocouples, which were mounted in the surface of the stator, the middle of the stator yoke, and the winding in slots. The installation diagram of the thermocouples is shown in Fig. 5b, in which each Arabic numeral indicates four circumferential uniformly mounted thermocouples.

Fig. 5.

Experimental test setup. (a) Test-rig. (b) Installation diagram of the thermocouples.

Figure 6 compares the magnetic results from the proposed method and ANSYS Workbench. In Fig. 6, the distribution results of the magnetic flux lines, including the magnitudes, the tendencies, and the gradients, achieve a good correlation. Compared to the results from ANSYS, the main difference occurs in the four stator teeth between adjacent PMs marked by four eclipses in dash lines, which is the result that the proposed method lacks mesh refinement to consider the magnetic flux leakages near the edges of PMs. Figure 7 and Table 3 compare the simulation and experimental results of the no-load electromotive force (EMF) to validate the magnetic analysis. Partial data of the EMF waveform was chosen from the experimental test in steady operation to validate the simulation results. In Fig. 6, the values of the EMF from the proposed method and ANSYS are barely the same. Moreover, the simulation results achieve a good correlation with the experimental results. Since no skewing design was applied for the electromagnetic design, the EMF waveform had a slight distortion and noise. In Table 3, the simulation results of the no-load EMF are very similar, and the errors of the simulation results relative to the experimental results are less than 3%.

Fig. 6.

Transient magnetic results at 0.002 s. (a) Proposed method. (b) ANSYS Workbench.

Fig. 7.

Transient results of the proposed method, ANSYS, and the experimental test under no-load operation.

Table 3

Comparison of the reuslts of on-load EMF

Speed	Test	Proposed method		ANSYS
(r/min)	EMF (V)	EMF (V)	Error (%)	EMF (V)	Error (%)
500	47.24	46.15	2.31	46.19	2.22
1000	93.51	92.29	1.30	92.40	1.19
1500	140.47	138.43	1.45	138.51	1.40

Fig. 8.

Steady-state thermal results. (a) Proposed method. (b) ANSYS Workbench.

Figure 8 compare the thermal results obtained by the proposed method and ANSYS Workbench. In Fig. 8, the temperature of both FEA mainly varies in the radial direction and achieves a good correlation, while the temperature distribution from the proposed method has more details in the PM zones. Since the main heat transfers between the machine and the ambient air were realized by the convection between the housing surface and the ambient air, the temperature distribution decreased gradually along the radial direction towards the housing. The copper losses from the windings and the iron losses from the stator and the rotor accounted for a large proportion to the total loss. Since the heat generation density in the windings were greater than the one in the iron cores, the maximum temperatures were found in the slots with the values 415.85 K and 413.25 K using the proposed method and ANSYS, respectively. Table 4 gives the thermal results from the FEA methods and the experimental test at an ambient temperature of 25 °C. The experimental results are the maximum values from the thermocouples for each machine part except for the end-windings, whose temperature rises cannot be simulated by the 2-D analysis. It can be found in Table 4 that the maximum temperatures are measured from the winding thermocouples referred as Arabic number 3 in Fig. 5b. The errors of the ANSYS results are less than 8%, while the proposed method achieves more accurate results with errors less than 5%. According to the foregoing results, the adaptive single-mesh method for the coupled magnetic-thermal fields analysis was proved to achieve a more accurate simulation with lower computing efforts.

Table 4

Temperature rises of key components

	Test	Proposed method		ANSYS
Machine parts	Temperature rise (K)	Temperature rise (K)	Error (%)	Temperature rise (K)	Error (%)
Stator surface	107.2	103.4	3.54	100.8	5.97
Stator yoke	112.8	108.5	3.81	104.6	7.27
Winding	123.4	117.7	4.62	115.1	6.73
PM	N/A	111.5	N/A	108.2	N/A

4. Conclusion

In this paper, an adaptive single-mesh method is proposed for the coupled FEA of magnetic-thermal fields in electrical machines. Both fields are initially solved according to the same single fine triangular mesh while the order of computing element matrices is reduced by eliminating certain nodes with the master-slave definition according to the magnetic and thermal field characteristics, respectively. The thermal modification for the temperature sensitive materials is considered in the coupling process. By the experimental tests of a PM machine, the proposed method is validated to improve the accuracy of the thermal analysis in the commercial software ANSYS with the time cost reduced by more than 8.0%. Further research will focus on the influence of the temperature on the heat transfer coefficients and the applications of 3-D analysis.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 51607123 and 51577166) and the Fund of Wenzhou Municipal Science and Technology Bureau, China (No. ZG2017002).

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