*How much loss power can be extracted from the electric machine in dependency of the used soft magnetic material?*

Please note that this study does not aim to use a highly accurate thermal model of the machine. The model is kept as simple as possible to understand the fundamental correlations of the material choice. The following assumptions and boundary conditions are made: the rotor and stator of the machine are separated by an ideal thermal insulator, i.e., no heat is transferred through the air gap. The stator is equipped with a housing cooling. The rotor is equipped with a rotor shaft cooling. The stator notch cooling, as found in [5], is not considered. The geometry is simplified by a cylindrical shell model as depicted in Figure 9. The height of the stator yoke and the height of the rotor yoke are kept constant, i.e., the heat transfer path through this part of the lamination is kept constant, and the influence of the teeth is neglected. The model is two-dimensional, i.e., the influence of heat extraction in end windings, bearings and bearing shields is neglected. All stator losses are introduced in the shell of the stator winding, and all rotor losses are introduced in the shell of the rotor bar. The heat conduction of the stator winding shell and the rotor bar shell is infinite. All thermal interface resistances are neglected. The housing and the shaft have the same temperature as the cooling fluid *ϑ*fluid = 50 °C, i.e., the thermal resistance between the lamination and the housing/shaft, the convectional resistance between wall and fluid, and the heating up of the fluid are neglected. The bending of the shells is neglected, i.e., the shells are modeled as flat plates, utilizing the average diameter of the shell (*d*out + *d*i)/2. The thermal resistance of the stator *R*<sup>1</sup> and rotor *R*<sup>2</sup> iron are calculated as follows:

$$R\_{1/2} = \frac{d\_{\rm out,1/2} - d\_{\rm i,1/2}}{k\_{\rm m} \cdot \pi \cdot (d\_{\rm out,1/2} + d\_{\rm i,1/2}) \cdot l\_{\rm i}} \tag{33}$$

with the outer and inner diameters *d*out1 = 282 mm; *d*out2 = 168 mm; *d*i1 = 214 mm and *d*i2 = 100 mm, the active length of the lamination *l*<sup>i</sup> = 285 mm, and the evaluated thermal conductivity of the measurements *k*m. The studied operational point is under steady-state behavior. The two thermal Lumped Parameter Thermal Network (LPTN) circuits are depicted in Figure 10. The maximum allowed temperature of stator winding and rotor bar is *ϑ*max = 180 °C. The maximum power loss *P*loss in the steady-state operation that can be extracted from the rotor or the stator is calculated as follows:

$$P\_{\rm loss} = \frac{\theta\_{\rm max} - \theta\_{\rm fluid}}{R\_{1/2}(\theta\_{\rm avg})},\tag{34}$$

with the average temperature of the stator or rotor iron *ϑ*avg = (*ϑ*max + *ϑ*fluid)/2.

**Figure 9.** Real geometry (**left**) and simplified thermal shell model of the studied machine (**right**).

**Figure 10.** LPTN model of the simplified case study for the stator and the rotor.

## **5. Results**

The thermal conductivity of the measurement approach *k*m(*ϑ*) is calculated utilizing Equation (19). The results of the more accurate Archimedes principle are used for the density values. The results of the measurement procedure are depicted in Figure 11. The results of the analytical calculation based on Equations (9) and (16) are given in Figure 12. Material M1 has the highest thermal conductivity, while M8 has the lowest thermal conductivity in both approaches. In the measurement approach, the difference in the thermal conductivity *k*<sup>m</sup> mainly results from the different values of the thermal diffusivity measurements *a* as shown in Figure 6. The differences in the density *ρ* and the thermal heat capacity *c*p seem to have a minor impact on the difference of the thermal conductivity *k*m.

**Figure 11.** Results of the measurements of the thermal conductivity *k*m.

A comparison between the measurement and the analytical approach is performed. The relative difference between *k*calc(*ϑ*) and *k*m(*ϑ*) is calculated as follows:

$$
\delta k\_{\rm m-calc}(\theta) = \frac{k\_{\rm calc}(\theta) - k\_{\rm m}(\theta)}{k\_{\rm calc}(\theta)} \tag{35}
$$

The value of this difference is plotted in Figure 13. The results of the analytical and the experimental approach show very good agreement for most of the materials. Materials M1 up to M6 show differences smaller than 10% for the entire temperature range. The differences for *ϑ* = *θ*<sup>D</sup> are significantly smaller. Materials M7 and M8 show higher differences, below 18% for material M7 and below 30% for material M8.

**Figure 12.** Results of the calculation of the thermal conductivity *k*calc.

**Figure 13.** Relative difference between thermal conductivites gained from measurement and analytical formula *<sup>δ</sup>k*m−calc(*ϑ*).

A possible reason for the deviation can be evaluated by the analysis of the overall estimated error of the two different procedures. The previously examined errors are multiplied to gain the overall error estimation of the measurement.

$$
\delta k\_{\rm m} = \delta \rho \cdot \delta a \cdot \delta c\_{\rm P} \tag{36}
$$

For the analytical approach, solely the influence of the measurement of the electric resistivity *δρ*el is considered.

$$\delta k\_{\rm calc}(\theta) = \frac{k\_{\rm calc}(\theta\_{\prime}\rho\_{\rm el} \cdot (1 - \delta \rho\_{\rm el}))}{k\_{\rm calc}(\theta\_{\prime}\rho\_{\rm el})} - 1 \tag{37}$$

The value is evaluated at *ϑ* = 398 K. The results of the accuracy studies *δk*<sup>m</sup> and *δk*calc are depicted in Figure 14. The accuracy study shows a clear dependency of the measurement results on the material thickness *d*. In particular, the estimated measurement errors of the thermal diffusivity *δa* has a squared dependency on the material thickness *d*. The influence of the thickness is also visible for the estimated errors of the electric resistivity measurement *δρ*el, where the estimated thickness error has a linear influence. This linear influence shows some impact on the accuracy of the calculated thermal conductivity value *<sup>k</sup>*calc. The deviation between the measurement and calculation results *<sup>δ</sup>k*m−calc(*ϑ*) shows a similar trend as the estimated errors *δk*<sup>m</sup> and *δk*calc. An allocation of the two effects is very likely, but not absolute clearly justifiable in the eyes of the authors. Material M8 is by far the thinnest material *d* ≈ 0.1 mm but also has by far the highest silicon content. It is also possible that the used formula has some inaccuracies in predicting such high silicon contents. Due to the estimated measurement errors for M8, a clear separation is not possible. The second material with higher deviations between measurement and calculation is M7. It is the second thinnest material, i.e., the nominal thickness is *d* ≈ 0.2 mm. M7 has a

significantly lower silicon content than M8, but the highest aluminum content. The alloying contents are close to those of M2 that show the lowest deviation of all materials between the measurement and the analytical approach. This indicates that the formula is accurate for the given alloys, and the differences of M7 occur due to the measurement errors of the material or some other structural influences that are not considered in the given formula.

**Figure 14.** Relative predicted error of the measured values of the thermal conductivity *δk*.

The influence of the alloys on the phonon and electron thermal conductivity can be analyzed based on the introduced formula. The evaluation is performed at *ϑ* = 398 K. The phonon thermal conductivity *k*<sup>p</sup> deviates from 6.8 W/(m K) for M8 up to 15.7 W/(m K) for M1. The electron thermal conductivity *k*<sup>e</sup> deviates from 8.5 W/(m K) for M8 up to 34.0 W/(m K) for M1. Both values show significant deviations, whereas the electron thermal conductivity has the higher impact on the overall value. The influence is limited to the electron–impurity scatter *W*e−<sup>i</sup> according to Equation (13) and the phonon–impurity scatter *W*p−<sup>i</sup> according to Equation (8). Other scattering processes are not influenced according to the used formula.

The results of the simplified case study are depicted in Figure 15. Material M1 shows the most preferable thermal properties. With this material, a maximum of 41 kW stator losses and 22 kW rotor losses could be extracted from the motor in the steady-state operation. With material M8, only 10 kW stator losses and 6 kW rotor losses would be allowed to ensure steady-state operation. It is well visible that all materials with high aluminum and silicon content show significant disadvantages regarding their capability for heat extraction. Please note that this estimation is based on some significant simplifications.

**Figure 15.** LPTN model of the simplified case study for the stator and the rotor.
