## 3 Methodology
### Data Evaluation  
To calculate the internal resistance with the current step method, only the Excel file generated from the Neware BTS for the measurement cycle was necessary. After each resting step in the measurement cycle, the voltage difference \(\Delta U\) and the current difference \(\Delta I\) were determined. As mentioned in Section 2.1, the timespan of the voltage and current measurement was 100 ms.  
After loading the values for the voltage/current and time into the Matlab workspace, the Matlab file selected the required data. As with the current step method, only the Excel file of the measurement cycle was needed. The Neware BTS determined the charge/discharge energy after each charging/discharging step. In the corresponding Matlab file, the values of all charging steps and the values for the discharging steps were summed up. The difference between the charging and the discharging energy was the energy loss \(Q_{loss}\) in mWh. The unit is transferred into Ws. This simplification is possible because the value of the current corresponds to a square wave signal and has either the value zero or the value of the selected C-rate. \(I_{m}\) is the magnitude of the current during the experiment. The duration of the current flow is tm. Due to the current profile, it is 900 s for each measurement. The corresponding Matlab file took the temperature and time values from the Excel document generated during the measurement by the LabView software. It also reads the current, voltage, and resulting energy values. These were taken from the Excel document generated by the Neware BTS during calibration. Afterwards, the time segments of the measurement were manually transferred to the Matlab file. The curve-fitting toolbox was then used to extrapolate the rest phases. Subsequently, the starting point, before a temperature rise, and an ending point, after a temperature rise, were determined based on the last intersection of the temperature curve with the extrapolated straight lines. Then, the areas of the triangles above and below the temperature curve were calculated. In a while loop, a variable was generated to change the x-value (time value) of the extrapolated lines. The area of the lower triangle was made up of the temperature curve and the distance of the variable from the starting point. The area of the upper triangle was made up of the temperature curve and the distance of the variable to the endpoint. The areas were determined with the \"trapz\" command. Then, the lower area was divided by the upper area and the result was saved. Thereafter, the value of the variable was increased by one. At the point where both areas were approximately equal, the result was approximately equal to one. The temperature difference was then determined at this point. Based on the temperature difference obtained from the calibration and the measurement, the internal resistance was determined. By using the energy \(Q_{calibration}\) taken from the calibration file and the temperature difference \(\Delta T_{calibration}\) for Equation (7), the heat capacity of the system was calculated. By using the temperature difference \(\Delta T_{measurement}\) in Equation (8), the energy loss \(Q_{loss}\) was calculated. Then, the internal resistance \(R_{i}\) was calculated with Equation (9). For determining \(\Delta T\), the area-compensation method was used. The pre-period (resting phase before a temperature rise) is extrapolated forwards, and the post-period (the resting phase after a temperature rise) is extrapolated backwards until the two triangular areas on either side of the main period (time of temperature rise) are equal (Figure 6). The temperature rise found in this way is the temperature difference \(\Delta T\)[10]. This method was used twice, once for the calibration and once for the measurement. The red lines in Figure 6 are the extrapolated resting phases. Together with the blue line and the black temperature curve, they enclose the mentioned triangles.