## 3 Case Study
### Results and Discussion  
Prior to NFR data analysis and the development of the degradation model, appropriate frequency ranges for the relevant processes had to be identified in the NFR spectrum of the ageing training data sets [24; 26]. In Figure 6, NFR is shown over the frequency at specific cycle steps from 0 to 400 cycles, at each 50 th cycle, respectively. Processes in the low-frequency range I from 0.02 Hz to 1 Hz can be attributed to diffusion processes in active material particles and processes in the mid-frequency range II from 1 Hz to approximately 300 Hz to electrochemical reactions. Processes in frequency range III, which show significantly lower NFR than the processes in ranges I and II, can most probably be attributed to ionic transport processes between and in the SEI and electrolyte. According to the literature, capacitive processes show almost minor or constant NFR in comparison to Faradaic processes with a Butler-Volmer kinetic. Furthermore, the separation of ranges I and II at approximately 1 Hz is distinct in the training data sets and, furthermore, is not affected by cell ageing. The identified process ranges with the corresponding time constants \(\tau\) and frequencies \(\omega\) are listed in Table 2. The correlation of processes to the frequency range in the NFR spectrum is supported by the corresponding impedance spectrum of the cell, which is shown in the inset of Figure 6 and discussed in detail in [24]. The NFR spectra in Figure 6 further illustrate that nonlinearities in the training data sets increase continuously with cycle ageing, particularly in frequency ranges I and II. In frequency range III, NFR spectra seem to be ageing-independent. However, without using data analysis methods, it is not possible to state with sufficient certainty an explicit ageing-to-NFR-correlation. Figure 6: NFR spectra during cycle ageing at each 50 th cycle, measured with \(I_{AC}\) = 1.6 C; the impedance spectrum initial to cycle ageing in the inset, is measured with C/15 C.  
Therefore, in the next step, a correlation analysis of the NFR and cycle number as a function of the analysed frequencies is calculated using the Spearman rank correlation measure. In Figure 7, the correlation coefficients for Spearman \(\rho_{S}\) correlation is shown over frequency to identify the frequencies with relevant ageing information in the training data sets. Three characteristic frequency ranges with different ageing correlations are identified and can be distinctly separated at approximately 300 Hz. For frequencies higher than 300 Hz, range C, \(\rho_{S}\) varies strongly and therefore NFR data extracted at those frequencies have no valid information about the ageing of the analysed LIB. For frequencies between 0.2 Hz and 300 Hz, range B, \(\rho_{S}\) is exactly 1, which indicates a perfect positive degree of association between NFR and cycle number. In the low frequency range A, \(\rho_{S}\) differs from 1; \(\rho_{S}\) extends from 0.93 to 0.98, thereby indicating that NFR and the cycle number have a weaker positive relationship at these frequencies. In the next step, the identified correlation ranges are interpreted by comparing correlation ranges A to C with the identified processes in frequency ranges I to III, as shown in Table 3. The most sensitive processes in the LIB are in the mid-frequency range from 0.2 to 300 Hz with a correlation coefficient of 1 for both types of correlation analysis methods. Thus, NFR data measured at these frequencies are highly suitable for LIB ageing quantification and the degradation model development. NFR data measured at frequencies lower than 0.2 Hz show a high correlation to ageing, as both correlation coefficients have values of \(>\)0.95. However, the data are not perfectly suitable in comparison with the higher frequency NFR data from 0.2 Hz to 150 Hz. For frequencies higher than 150 Hz, the NFR data have lower and strongly varying correlation coefficients for both correlation methods.