*2.3. Measurements of the Temperature Dependent Electric Resistivity*

Measurements of the electric resistivity are performed using the measurement setup as depicted in Figure 1. The setup is designed following the recommendations of the standard DIN EN IEC 60404-13 [14] on a smaller scale because the specimens are not available in the recommended size. Probes of 120 mm × 20 mm are used. In order to gain sufficient accuracy, an analysis of the measurement uncertainties is performed. The specimen is inserted into a measurement fixture and placed inside an oven. The electric resistivity *ρ*el(*ϑ*) is calculated with the following equation:

$$
\rho\_{\rm el}(\vartheta) = \frac{\mathcal{U}(\vartheta) \cdot d \cdot w}{I \cdot l}. \tag{17}
$$

**Figure 1.** Measurement setup for the electric resistivity.

A DC-current *I* between 0.2 A and 2 A is introduced into the sample using a FLUKE 5500A Multi-Product Calibrator [15]. The maximum measurement error of this current is given by Δ*I* = 0.64 mA in the manufacturer data sheet [15]. The voltage is measured with a separate device to improve the accuracy of the measurement. A HP 3458A Multimeter is used for this purpose. The maximum measured voltage during the procedure is around 20 mV. With an maximum measurement error of 16.4 ppm of the reading and 22 ppm of the measurement range, a maximum measurement error for the voltage of Δ*U* = 2.5 μV is specified in the data sheet of the device [16]. Two measurement tips are placed on the probe for the voltage measurement. The distance between the measurement tips is equal to the measuring length *l*. The measuring length *l* as well as the width of the specimen *w* are measured using a digital caliper from Mitutoyo. The maximum measurement error of the measurement device is given with 0.02 mm. In order to include measurement errors that result from geometrical deviations, a total error estimation of Δ*l* = Δ*w* = 0.05 mm is included in the accuracy calculation. The measurement of the two values is repeated at least three times and the average value is calculated. The thickness of the sheet *d* is measured after removing the insulation, using an outside micrometer from Mitutoyo. The measurement is repeated six times and the average value is used. The coefficient of variation *cv* of the iterative geometrical measurement procedure is given in Table 2. *cv* is well below 1% with one outlier of material M8 for the thickness measurement. The measurements show good precision.


**Table 2.** Coefficient of variation *cv* for measurements of geometrical values in %.

The maximum measurement error of the micrometer is given with 0.001 mm. In order to account for geometric errors, a total measurement error estimation of Δ*d* = 0.01 mm for the thickness *d* is used for the accuracy evaluation. The measurement error estimation Δ*ρ*el is calculated at room temperature *ϑ* = 293 K using the following formula:

$$
\Delta\rho\_{\rm el}(\theta) = \frac{(\ell l(\theta) + \Delta l l) \cdot (d + \Delta d) \cdot (w + \Delta w)}{(l - \Delta l) \cdot (l - \Delta l)} - \rho\_{\rm el}(\theta). \tag{18}
$$

The results of the accuracy study (*x* = *ρ*el) are depicted in table Table 3. A trend of increasing measurement errors with decreasing specimen thickness can be observed. One exception of this trend is the decreased accuracy of M1 in comparison to M2. This exception is caused by an increased influence of Δ*U* for this material. Due to the high thickness of this material, the resulting measured voltage *U* is relatively low at a comparable current *I*. For decreasing thickness, the error caused by the thickness *d* dominates the overall influence. For material M8 for instance, the measurement error of the thickness measurement Δ*d* causes about 10.6%, while other influences only contribute by 1.1%. Please note that this influence also could not have been changed utilizing the recommended setup in DIN EN IEC 60404-13 [14], as the thickness of the specimen would have a similar influence. The results of the measurement of the electric resistivity in dependency on the temperature *ρ*el(*ϑ*) are plotted in Figure 2. The results show a significant difference for the electric resistivity *ρ*el between the different materials. At first glance, the material *d* seems to influence the *ρ*el. This impression is not correct because the alloy components Si and Al are the primary influencing factors. As an example, M1 is the thickest selected material and has a very low silicon and aluminum content. Material M8 is the thinnest material with the highest silicon content. A plausibility check can be performed utilizing the ternary plot of the electric resistivity *ρ*el as a function of the silicon *c*w,Si and aluminum *c*w,Si and weight content published in [17]. The eight different materials are added to the plot in Figure 3 based on their silicon and aluminum content. The gained experimental results show very good agreement with this plot gained from the literature at room temperature.

**Table 3.** Estimated measurement error *δρ*el(*ϑ* = 293 K) in %.


**Figure 2.** Results of the electric resistivity measurements *ρ*el(*ϑ*).

**Figure 3.** Theoretical values for the electric resistivity *ρ*el(*ϑ* = 298 K) in Ω mm2/m based on the silicon and aluminum content (Source: Data from [17]).

#### **3. Experimental Evaluation of the Thermal Conductivity**

An indirect measurement technique is used for the evaluation of the thermal conductivity *k*m(*ϑ*), i.e., the thermal diffusivity *a*(*ϑ*) is measured. The thermal conductivity of the measurement *k*m is calculated using the following formula:

$$k\_{\rm m}(\vartheta) = a(\vartheta) \cdot \rho(\vartheta) \cdot c\_{\rm P}(\vartheta) \tag{19}$$

The procedure of evaluating the thermal diffusivity *a*(*ϑ*), the mass density *ρ*(*ϑ*), and the specific heat capacity *c*p(*ϑ*) is introduced in the following.
