## 2 Materials and Methods  
### Test Definitions  
A standard test procedure is introduced to cover the relevant thermal cell behaviour. Using the procedure from [20; 23], the plan utilises seven measuring points at logarithmically distributed frequencies from 3 mHz as the first measurement point to 160 \(\upmu\)Hz as the last tested frequency. Also, the number of repetitions per point is changed by modifying the first measuring frequency to 50 repetitions, allowing the cell temperature to reach a quasi-steady state in the first 40 oscillations. A carrier signal utilising a frequency of 5 Hz is overlaid to each corresponding measuring frequency to maintain a stationary state. Here, mainly the testing system defines the carrier frequency. The frequency band in Table 1 defines the limits of the upcoming spectra. The upper boundary results from the minor measurable temperature response at a particular excitation current and the corresponding generated heat. In contrast, the lower boundary corresponds to the compromise of the overall test duration and the benefit of using lower frequencies to the resulting spectra and fitting. Also, the influence of heating of ambient materials, for instance, the climate chamber and the air around the tested setup, limits the lower end of the test [22]. The first quarter of the resulting data points per frequency for further investigation gets cut out. This results in reducing the number of observed oscillations to seven or three. As described in [20], this adjustment helps to minimise inaccuracies due to a change in the occurring \(\Delta T\) per measurement point. The paper investigates the cell's thermal behaviour according to the SOC and ambient temperature. Therefore, a test matrix considering the SOC at 20%, 35%, 50%, 65% and 80% and \(T_{amb}\) of 10 \({}^{\circ}\)C, 15 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, 35 \({}^{\circ}\)C and 45 \({}^{\circ}\)C is spread out. The SOC starts at 50% in 15% steps in both directions. The limits at 20% and 80% rely on the safety limits of the cell to maintain the cell's voltage range. The temperature values cover a broad range in which a battery is typically used. Here, the datasheet defines the lower limit at 10 \({}^{\circ}\)C. \begin{table}
\begin{tabular}{l c c c c c c c} \hline Frequency [mHz] & 3 & 1.8 & 1.1 & 0.7 & 0.43 & 0.26 & 0.16 \\ \hline Repetitions [-] & 40 + 10 & 10 & 10 & 4 & 4 & 4 & 4 \\ \hline \end{tabular}
\end{table} 
Table 1: Signal configuration per test cycle to acquire the significant spectra. It is the minimum ambient temperature at which a high charging current, as utilised in the test, lies within the safety limits. The upper temperature at 45 \({}^{\circ}\)C, again, is a value which lies within the boundaries of the cell datasheet and represents a temperature in a real-world application. A consistent current amplitude is specified at \(I=92\) A, representing a maximum current of a 2C-rate. The modulated current signal is shown in Figure 1.  
The thermal response of the battery cell and the minimum sensor excitation define the current rate since the influence of the noise on the temperature signal at the measurement points on the cell's surface gets more significant at the higher frequencies caused by less overall generated heat. A direct current offset using a third of the amplitude (\(I_{\mathit{off}}=30.66\) A) is overlaid to minimise extensive cooling of the cell during phases near the zero crossing of the signal [22]. The influence can be seen in Figure 2a, as the minimum heat generation is 0.6 W.  
The following Equation (1) usually defines the generated heat \(\dot{Q}\). It covers the main heating aspects [24; 25]. The first term on the right side describes the irreversible or joule heating using the current \(I\) and the ohmic resistance \(R_{i}\). The second term represents the reversible heat by entropic changes during the change of the SOC.