Thermal finite element modeling and simulation of a squirrel-cage induction machine

Abstract

Finite element models of electrical machines allow insights in electrothermal stresses which endanger the insulation system of the machine. This paper presents a thermal finite element model of a 3.7 kW squirrel-cage induction machine. The model resolves the conductors and the surrounding insulation materials in the stator slots. A set of transient thermal scenarios is defined and measured in the machine laboratory. These data are used to assess the finite element model.

Keywords

Finite element method electrical machines insulation system thermal stresses numerical field simulation

1. Introduction

The insulation system of an electrical machine is a complex arrangement of different insulating materials [1]. Its thermal and mechanical stability under adversary operating conditions defines the permissible operating range and determines the lifetime of a machine [2,3]. Currently, the reliability and lifetime of the insulation system during transient thermal loads are usually not considered during the design process [4]. As a consequence, many machines are overdimensioned, whereas others are operated beyond their thermal tolerances [5]. A detailed understanding of the thermal stresses inside the insulation system is, thus, of great importance for the improvement of the cost-effectiveness and robustness of electrical machines. Experimental investigations and diagnostic approaches for machine insulation systems are limited [6,7]. Measurement campaigns are expensive, cumbersome and slow down the design process. A thermal analysis of an electric machine can be performed using lumped-parameter thermal-networks (LPTNs), finite element method (FEM) or computational fluid dynamics (CFD), with increasing computational effort [8]. The most common approach is the use of LPTNs e.g., [9–12]. However, the LPTNs approach does not resolve the geometry of the machine, so that no detailed information about the temperature distribution can be obtained. It is therefore not suitable for investigating thermal stresses within the insulation system. A promising alternative are two-dimensional (2D) [13–15] or three-dimensional (3D) [16] finite element (FE) simulations. Since transient 3D FE simulations can lead to prohibitively large computational costs, this work presents a complete transient 2D thermal simulation model of a 3.7 kW squirrel-cage induction machine, that considers 3D effects using boundary conditions and effective thermal conductivities. This includes the axial heat transfer from the rotor via the shaft, which is often neglected.

The 2D FE model is implemented in the open-source framework Pyrit [17]. Since the interturn insulation is considered to be the weakest part of the insulation system [18], the model explicitly resolves the arrangement of conductors and insulating components within the stator slots. In the current literature, these details have not been thoroughly resolved in the thermal models that have been proposed, and experimental validation of such resolution is lacking even the more. At the same time, at large, electric machine designers are not aware of the impact of the distribution of the thermal stresses and neglect it in most analyses. With our work, we seek to contribute to closing this gap by providing both the resolution and the experimental investigation so as to quantify the thermal stress resulting from transient operations that bring thermal cycling of local components.

A series of thermal and electrothermal measurements of a 3.7 kW induction machine has been conducted at the Electric Drives and Machines Laboratory at the TU Graz [19]. The simulation model is successfully validated by comparing the measured and simulated thermal time constants within different areas of the machine. Furthermore, the measured and the simulated transient temperature distributions during a load cycle have been compared. The presented model allows the investigation of various modes of machine operation, such as on-off cycling, switch-on and operation at peak load, and thereby adds to a deeper understanding of thermal stresses inside the machine’s insulation.

Fig. 1.

Stator and cooling jacket of the squirrel-cage induction machine, before assembly. Each of the white wires is connected to a temperature sensor. Once assembled, an additionally heat-insulating layer is wrapped around the cooling jacket of the machine.

Table 1

Data of example-case squirrel-cage induction machine (Y-connection)

Nominal data	Value
Nominal power	3.7 kW
Nominal voltage RMS	400 V
Nominal current RMS	6.9 A
Nominal speed	1430 rpm
Nominal frequency	50 Hz
Number of pole pairs	2
Moment of inertia	0.0195 kgm²

Fig. 2.

Schematic of the 2D machine model. The machine shaft is located in the center of the rotor and colored in white. The rotor yoke is displayed in green. The squirrel cage is shown in yellow. The stator yoke is shown in light blue, and is surrounded by the cooling jacket (dark blue). The main area of interest in this work is the slot area filled with a dual layer copper winding that is insulated with epoxy resin. The upper and lower winding segments are displayed in two shades of pink. For clarity, the conductor and insulation system arrangement are not shown in this figure.

First, Section 2 introduces the induction machine specimen as well as the experimental setup. Section 3 presents the modeling of the transient thermal behavior of the machine. In Section 4 discusses the incorporation of 3D effects into a 2D FE model as well as the results, i.e., the successful validation of the previously presented model against measurement data. Thereby this paper presents a validated transient 2D FE thermal model suitable for the investigation of the machine’s insulation system.

2. Induction machine and measurement system

The investigated four-pole, squirrel-cage induction machine is designed for a nominal power of 3.7 kW at a nominal speed of 1430 rpm. The stator, shown in Fig. 1 has a distributed dual layer winding with three slots per pole and phase and a 7/9 pitching. There are 36 slots in the stator, with an area of 107.7 mm² each. Each winding is comprised of 18 conductors with a radius of 0.75 mm. This results in a copper filling factor of 59%. The rotor and stator have an outer diameter of 125 mm and 200 mm, respectively. Further parameters are summarized in Table 1 and a schematic of the machine’s cross section is shown in Fig. 2.

Fig. 3.

Positions of the temperature sensors inside the stator at the axial center. Sensors in the stator slot are red, sensors in the stator teeth are green and sensors in the stator yoke are yellow.

Fig. 4.

Position of the temperature sensors inside the stator slots at the axial center of the upper and lower winding. For simplification not all windings are displayed.

The machine is mounted on a test bench at the Electric Drives and Machines Laboratory of TU Graz [19,20]. The measurement setup includes the machine under test and a connected heating device. The machine is fully equipped with a temperature measurement system, including rotor temperature detection using type-T thermocouples via a telemetry system. Even though the machine is rated for air cooling, it is constructed with a cooling jacket. Thermally insulating materials cover the cooling jacket, the end plates, and the stator flange (International Mounting B5) in order to reduce heat flows which are not monitored by the sensors. A flow meter and platinum resistance detectors (Pt100) are mounted at the inlet and outlet tube of the cooling jacket to monitor the cooling performance. The temperature sensors are positioned as follows: 18 temperature sensors are inserted in the middle of the lower layer and the upper layer of the stator slots. The sensors of the lower and upper winding layer are located close to the slot separators and the wedges, respectively (see Fig. 4). Additionally, 12 sensors are positioned inside and on the surface of the winding overhang. The temperature sensors of the slots and the winding overhang are platinum resistance thin film temperature detectors (Pt1000, 2 × 10 mm) according to DIN EN 60751 [21]. Temperature sensors (Pt1000, 2 × 4 mm) monitor the temperature of the stator yoke, and are positioned at the axial center of the yoke. 12 sensors are positioned in small holes drilled in the center of the stator teeth and the stator yoke (see Fig. 3). Finally, five sensors monitor thermal transients in the rotor and the shaft. Mulitplexed Keithley 2000 and 2700 multimeters are used to record the output data of the temperature sensors. The rotor temperature signals are additionally transmitted by a Datatel dt3008T-T telemetry transmitter module. The mean voltage, current and power in the winding can be monitored using the Fluke Norma 5000 equipped with PP51 power phase.

Fig. 5.

Geometry of the 2D model with the corresponding boundary conditions for the cooling jacket, the motor shaft and the rotation symmetry. The wires surrounded by homogeneous insulation material are shown in the slots.

3. Thermal model

The thermal behavior of the induction machine is modeled by the transient heat conduction equation [22] in a 2D FE setting, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle c_{\text{v}}{\displaystyle \frac{\partial T}{\partial t}}-{\nabla}\cdot ({\lambda}{\nabla}T)=\dot{q}\quad \text{in }∼V,\end{eqnarray}$ (1a) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle T=T_{\text{iso}}\quad \text{on }∼{\Gamma}_{\text{iso}},\end{eqnarray}$ (1b) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle -{\lambda}{\displaystyle \frac{\partial T}{\partial \mathbf{n}}}=0\quad \text{on }∼{\Gamma}_{\text{adi}},\end{eqnarray}$ (1c) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\lambda}{\displaystyle \frac{\partial T}{\partial \mathbf{n}}}+h(T-T_{\text{iso}})=0\quad \text{on }∼{\Gamma}_{\text{rb}}.\end{eqnarray}$ (1d) Herein, T is the temperature, c_v is the volumetric heat capacity, 𝜆 is the thermal conductivity, h is the heat-transfer coefficient, and $\dot{q}$ is heat flux density. The computational domain is denoted as V , its boundary as 𝛤 and the normal vector at the boundary as n. The 2D modeling approach is motivated by the radial dominance of the temperature gradient. This predominance is attributed to two key factors: the presence of a cooling jacket and the anisotropic nature of the thermal conductivities due to laminating the stator and rotor yokes. Specifically, the radial thermal conductivities substantially exceed the axial thermal conductivities.

Additionally, the following boundary conditions are enforced: An isothermal boundary condition (1b) is prescribed on 𝛤_iso to account for the uniform temperature distribution of the cooling jacket. Due to the inherent symmetry of the machine, modeling a single quadrant suffices for accurate representation. Consequently, an adiabatic boundary condition is applied along the symmetry lines 𝛤_adi. The rotor shaft gives rise to axial heat transfer to the externally mounted equipment, the test bench, and the ambient air [15]. This modest yet significant 3D heat transfer is taken into account by the Robin-type boundary condition (1d), representing a weighted blend of Dirichlet and Neumann boundary conditions [23]. Robin boundary conditions are typically used to model the convection between solids and fluids [13,24]. However, in this work, it is used to incorporate axial 3D heat transfer. It is applied along the shaft surface line 𝛤_rb. Figure 5 shows the 2D model including the imposed boundary conditions.

Table 2

Thermal material parameters from literature

Domain	Parameter	Value	Unit
Stator yoke	𝜆_rho	40	[W/°C/m]
	𝜆_z	2.5	[W/°C/m]
	c _v	3.925 ⋅ 10⁶	[J/°C/m³]
Rotor yoke	𝜆_rho	40	[W/°C/m]
	𝜆_z	2.5	[W/°C/m]
	c _v	3.925 ⋅ 10⁶	[J/°C/m³]
Conductor	𝜆	398	[W/°C/m]
	c _v	3.435 ⋅ 10⁶	[J/°C/m³]
Insulation	𝜆	0.7	[W/°C/m]
	c _v	7.905 ⋅ 10⁶	[J/°C/m³]
Air gap	𝜆	0.026	[W/°C/m]
	c _v	1.210 ⋅ 10³	[J/°C/m³]
Shaft	𝜆	59.6	[W/°C/m]
	c _v	3.777 ⋅ 10⁶	[J/°C/m³]

Fig. 6.

Comparison of the cooldown process between measurements (solid lines), the simulations using literature values (dotted lines, Table 2) and the simulations considering the evaluated effective conductivities (dashed lines, Table 3). The begin temperature is 93 °C and the ambient temperature is 26 °C. The overshoot at t = 15 min of the stator curve is due to the control of the cooling jacket.

Table 3

Fitted thermal material parameters

Domain	Parameter	Value	Unit
Stator yoke	𝜆_eff	24	[W/°C/m]
Rotor yoke	𝜆_eff	16	[W/°C/m]
Air gap	𝜆_eff	0.052	[W/°C/m]
Shaft	𝜆_eff	59.6	[W/°C/m]
	h	0.235	[W/°C/m]

4. Simulation and measurement

4.1. Fitting parameters

Table 2 provides a comprehensive list of the thermal material parameters obtained from manufacturer specifications and established literature. Using those thermal material parameters, the dotted lines in Fig. 6 are obtained. Clearly, the temperature decline in the slot and in the stator is overestimated, while the temperature decline in the rotor is underestimated. This leads to the assumption that certain factors introduce systematic uncertainties into the model, defying a priori determination. These factors are:

Natural convection in the air gap: This effect is assumed to be minimal due to the small air gap. Notably, forced convection is absent as the rotor does not rotate.

Axial heat transfer along the shaft: This effect is parameterized by h and 𝜆 in the Robin boundary condition (1d).

Anisotropic heat transfer: Anisotropic thermal conductivities of the stator yoke and rotor lead to an anisotropic heat transfer.

Lamination: This characteristic arises from the anisotropic nature of the thermal conductivities in the lamination. To appropriately account for this effect within a two-dimensional model, an effective thermal conductivity is introduced.

These factors undergo a fitting process against the cool-down measurements of Section 4.2, enabling a refinement of their initial values.

Table 3 lists the fitted parameters of the machine model and the dashed lines in Fig. 6 shows the improved temperature behavior. The obtained effective thermal conductivities of 16 W/°C/m in the stator yoke and 24 W/°C/m in the rotor are in good accordance to the thermal conductivity suggested in [25]. The 2D FE model of the machine is simulated and analyzed with the open source Python-based FE framework Pyrit [17].

Table 4
Thermal time constants 𝜏 in (min)

Initial temperature Domain Measurement Simulation Rel. error

93 °C Slot 7.01 6.97 0.6%

Stator yoke 4.94 4.88 1.6%

83 °C Slot 6.81 6.81 0.02%

Stator yoke 4.82 4.75 1.4%

73 °C Slot 6.77 6.77 0.04%

Stator yoke 4.78 4.70 1.5%

63 °C Slot 7.24 7.18 0.6%

Stator yoke 5.19 5.29 1.9%

53 °C Slot 5.73 5.81 1.5%

Stator yoke 3.98 3.90 2.0%

45 °C Slot 6.17 6.21 0.7%

Stator yoke 4.27 4.33 1.4%

Initial temperature	Domain	Measurement	Simulation	Rel. error
93 °C	Slot	7.01	6.97	0.6%
	Stator yoke	4.94	4.88	1.6%
83 °C	Slot	6.81	6.81	0.02%
	Stator yoke	4.82	4.75	1.4%
73 °C	Slot	6.77	6.77	0.04%
	Stator yoke	4.78	4.70	1.5%
63 °C	Slot	7.24	7.18	0.6%
	Stator yoke	5.19	5.29	1.9%
53 °C	Slot	5.73	5.81	1.5%
	Stator yoke	3.98	3.90	2.0%
45 °C	Slot	6.17	6.21	0.7%
	Stator yoke	4.27	4.33	1.4%

Fig. 7.

Comparison between simulation and measurements for different steady-state target temperatures in the area of the slot. The error bars show the deviation between the measurements in all of the 36 slots.

4.2. Cooldown process scenario

In a first step, the cooling characteristics of the induction machine are analyzed by conducting a series of exclusively thermal experiments, i.e. experiments without any electric excitation. In these experiments a defined temperature profile is prescribed by the cooling jacket: First, the machine is heated until a steady-state is reached across all machine domains. Then, the cooling jacket temperature is reduced to ambient conditions of 26 °C and the cool-down process of the machine is recorded.

Fig. 8.

Steady-state temperature distribution in one quarter of the cross section of the machine for a power supply of 200 W.

Fig. 9.

The power of 200 W and 300 W are applied, to create a heat source in the slot. Then the power supply is set to 0 W in order to let the machine cool down again.

Fig. 10.

Load cycle scenario. In this case multiple measurements with a fixed ambient temperature are carried out. The compared positions are in the rotor yoke, stator yoke and slot.

The cooling behavior inside the rotor, stator and slots starting from an initial temperature of 93 °C is shown as solid lines in Fig. 6. The temperature decline in the rotor is significantly slower compared to the stator due to the low thermal conductivity of the air gap (see Table 2). The simulation results in the stator domain are in very good agreement with the measurement data. A slightly larger deviation can be seen in the rotor domain (see Fig. 6). While the stator sensors are drilled into the longitudinal center of the machine, the rotor sensors are mounted onto the rotor surface. This might lead to additional unmonitored parasitic effects, e.g. the influence of the surrounding air.

Fig. 11.

Absolute error between measurement and simulation for the slot. The high peaks are due to the fast change in the temperature in the slot area, after switching of the power supply.

To assess the validity of the simulation results, the relative error of the thermal time constants is computed. The thermal time constant is defined as the time required for a temperature drop of 63% from the initial steady-state temperature. The relative error is computed as $\begin{eqnarray}{\varepsilon}_{\text{rel}}=\frac{|{\tau}_{\text{meas}}-{\tau}_{\text{sim}}|}{{\tau}_{\text{meas}}},\end{eqnarray}$ (2) where 𝜏_meas and 𝜏_sim are the thermal time constants corresponding to measurement and simulation, respectively.

Table 4 compares the measured and simulated time constants for different steady-state target temperatures, T_st, ranging from 45 °C to 93 °C. To account for measurement errors, the mean value of all temperature sensors in the slots and the stator yoke, respectively, is considered. Figure 7 shows the range of the temperature deviation between the sensors in the slot for three different steady-state target temperatures. Additionally, the resulting mean value of all temperature sensors and the simulation results are displayed. For the slots, the relative error lies within 0.02% and 1.5% and for the stator yoke within 1.4% and 2.0%. This relative error is smaller than the measurement tolerance of the sensors, therefore the model is successfully validated for the cooldown scenario.

4.3. Load cycle scenarios

The model is now applied to study a second heating scenario. Several load cycles with varying power levels are performed and the resulting temperature distributions are evaluated. The machine is energized with all windings series connected supplied by a DC current such that no movement of the rotor takes place. The power supply is ramped up slowly at a rate of 1 W/s such that the magnetic losses are negligible. Consequently, the supplied power is completely converted to heat by the Joule losses inside the stator windings. Additionally, the cooling jacket keeps a steady ambient temperature of 26 °C. Figure 9 shows the power supply during several load cycles with power levels of 200 W and 300 W, respectively. The resulting temperatures are shown in Fig. 10. Since the windings act as a primary heat source, the maximum temperature is located in the slot area (see Fig. 8) and therefore also close to the insulation system. The maximum temperatures in the slot area are 40 °C and 48 °C for the 200 W and 300 W load cycles, respectively.

Comparing the experimental and simulation data, also for this scenario, a good agreement can be observed. The absolute error of the simulated temperature in the slot area over time is displayed in Fig. 11. The absolute error fluctuates between approximately 0.3 °C during the plateau phases and 3 °C during the heating and cooling phases. The larger error during the transition phases are due to the fact that the time instant when the power supply changes is only known up to one minute, leading to slight differences in the switching times of measurement and simulation. The general good agreement is considered as a further validation of the model.

5. Conclusion

A thermal stress analysis of the insulation system is a crucial part for designing a robust machine. Presently, the design process often overlooks the insulation system’s reliability and longevity. This work presents a thermal model of a squirrel-cage induction machine. As the predominant temperature gradient is radial, a 2D FE model is both accurate as well as computationally cheap. Additional 3D effects are identified and included using boundary conditions and effective thermal conductivities. A series of transient thermal experiments is conducted and the results are used to assess the quality of the FE model. The comparison between measurements and simulations shows a very good accordance. In conclusion, the FE model is successfully validated.

Footnotes

Acknowledgements

This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 492661287 – TRR 361, the Athene Young Investigator Programme of TU Darmstadt and the Graduate School Computational Engineering at TU Darmstadt. We thank Greta Ruppert for the fruitful discussions.

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