Characteristic frequency of the diffusion response of the blocked-diffusion Warburg impedance with frequency dispersion

The blocked-diffusion Warburg impedance with frequency dispersion (BDWf) can be considered in equivalent electrical circuits to characterise the low-frequency impedance response of batteries. In this study, the characteristic frequency of the diffusion response $f_d$ of the BDWf impedance is calculated using the analytical transfer function reported in a previous study and the Newton-Raphson iteration method. The results show that the characteristic frequency of the diffusion response $f_d$ is a numerical value x (calculated from the Newton-Raphson iteration) greater than the frequency calculated from the time constant ${\tau }_{\mathrm{BW}}$ of the BDWf impedance as follows: $f_d=\frac{x}{2\pi {\tau }_{\mathrm{BW}}}$. In this study, the characteristic frequency of the diffusion response $f_d$ is also represented in the impedance response of the BDWf impedance. The characteristic diffusion time constant ${\tau }_d$ calculated from the characteristic frequency of the diffusion response $f_d$ of the BDWf impedance and estimated from battery impedance measurements, could provide a new estimation of the effective anomalous diffusion coefficient of charge carriers in battery electrodes with heterogeneous particle structure. Additionally, the output voltage of a battery measured at the characteristic diffusion time constant ${\tau }_d$ during current pulse tests could be considered a valuable ageing signature for assessing the state of health of the battery.