**1. Introduction**

Increasing the power density of highly utilized traction drives is a frequently discussed research topic. The reduction of losses or the improvement of the heat dissipation capabilities are both potential measures to address this target. A significant influencing factor on the overall efficiency of a traction drive is the selection of the soft magnetic material. The influence of structural material parameters on the efficiency of the electric drive is well studied [1,2]. Eddy losses play a significant role in traction applications, due to their high frequency dependency. In order to reduce this loss share, silicon (Si) and aluminum (Al) can be added as alloy components to the iron matrix. The specific electric resistance *ρ*el is increased, leading to a reduced loss contribution of the eddy losses [1,2]. A direct dependency between the electron contribution of the thermal conductivity *k*e and the specific electric resistance *ρ*el can be found in the Wiedemann–Franz law:

$$k\_{\rm c} = \frac{L\_0 \theta}{\rho\_{\rm cl}},$$

with the Lorenz number *L*<sup>0</sup> and the temperature *ϑ*. As shown within this study, the rule is not fully applicable for alloys, but already indicates a negative impact of increased Si and Al alloy components on the thermal conductivity *k*. Several influencing thermal parameters, such as the heat transition in the air gap, the interfaces between lamination and housing, the impregnation goodness or the end winding correlation are well studied within

**Citation:** Groschup, B.; Rosca, A.; Leuning, N.; Hameyer, K. Study of the Thermal Conductivity of Soft Magnetic Materials in Electric Traction Machines. *Energies* **2021**, *14*, 5310. https://doi.org/10.3390/ en14175310

Academic Editor: Ryszard Palka

Received: 30 July 2021 Accepted: 20 August 2021 Published: 26 August 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the literature [3,4]. A fundamental understanding of the influencing factors of structural soft magnetic parameters on the thermal behavior of electric machines is rare to find. Correlations or validated data for the thermal conductivity of soft magnetic material are not frequently studied. Exact knowledge about the thermal conductivity of soft magnetic materials is crucial for its selection. The selection is especially challenging in the case that the soft magnetic material is placed within the main heat dissipating path. A well suited example for such an application is an Induction Motor (IM) with housing and direct shaft cooling, such as that introduced in [5].

Within this study, an analytical as well as an experimental approach is introduced to obtain data for the thermal conductivity of soft magnetic materials. Eight different soft magnetic materials with different Al and Si content are selected, according to Table 1. The name of the material, an Acronym (Acr.) with the material number from one to eight, the silicon weight content, the aluminum weight content and the nominal thickness are added to the overview. Measurements of the electric resistivity *ρ*el in dependency of the temperature *ϑ* are performed to have sufficient data input for the analytical approach. For the experimental approach, the thermal conductivity *k*m is determined using an indirect measurement technique.

$$k\_{\rm m}(\vartheta) = a(\vartheta) \cdot \rho(\vartheta) \cdot c\_{\rm p}(\vartheta) \tag{2}$$

The thermal diffusivity *a*(*ϑ*) is measured using a Laser Flash Analysis (LFA). The density of the material *ρ* is measured at room temperature, using the Archimedes principle. A simple model is used to adapt the gained data in dependency of the temperature. A modified model of the Kopp–Neumann law is utilized to determine the specific thermal heat capacity *c*p(*ϑ*) of the materials. All measurements and models are developed for a temperature range between room temperature and 225 °C. The results of the thermal conductivity in dependency of the temperature *k*(*ϑ*) are compared between the indirect measurement and the analytical approach. The results are used to investigate the influence of the material choice on the thermal heat dissipating capabilities of a traction drive. An IM with direct shaft cooling and housing cooling, as introduced in [5], is selected as a reference for this simplified case study.


**Table 1.** Alloy weight content and nominal thickness of studied materials.

A study is performed to evaluate the accuracy of the measurement according to [6]. The accuracy is defined as the closeness of agreement between a measured quantity value and a true quantity value of the measurand. The accuracy is not a quantity and cannot be given as a numerical quantity value according to [6]. In order to analyze the accuracy, an estimation of a possible measurement error is performed. This error estimation is aimed to represent the worst possible measurement error. It includes systematic and random measurement errors. The absolute value of the estimated error of a variable *x* is labeled with Δ*x*. The estimated relative measurement error *δx* can be expressed with the following:

$$
\delta \mathbf{x} = \frac{\Delta \mathbf{x}}{\mathbf{x}}.\tag{3}
$$

Please note the difference between accuracy and precision. Precision is the closeness of agreement between indications or measured quantity values obtained by replicate measurements on the same or similar objects under specified conditions [6]. In order to gain good measurement results, both precise and accurate measurement results are necessary. The precision throughout the study shows very good values. The coefficient of variation *cv* is used for investigating the measurement precision. The *cv* is calculated by a division of the standard deviation of a measurement sequence and the gained average value.

### **2. Analytical Formula for the Thermal Conductivity**

The two main mechanisms of heat transfer in metallic alloys, such as soft magnetic material, are the phonon *k*<sup>p</sup> and the electron thermal conductivity *k*e. The total value for the analytical approach *k*calc can be calculated as follows:

$$k\_{\rm calc}(\vartheta) = k\_{\rm p}(\vartheta) + k\_{\rm e}(\vartheta) \tag{4}$$

The mechanism can be subdivided into different scatter processes between phonons, electrons and imperfections. A good overview of the resulting interactions is given in [7,8]. This study mainly uses the correlations as introduced in [8]. The most important assumptions and derivations of the correlations are discussed in the following.
