Analytical Solutions for the Impedance of Cauchy‐ and Gauss‐Type Relaxation Processes

This work presents analytical solutions for the impedance of a relaxation process whose time‐constant distribution follows 1) a Cauchy‐distribution or 2) a Gauss‐distribution. Both functions are commonly used as basis for the numerical reconstruction of the time‐constant distribution from experimental impedance data (i.e., the opposite way). Herein it is, however, demonstrated that neither a single Cauchy, nor a single Gauss function, can resemble common experimental features of impedance—especially the nonperpendicular traversal in the Nyquist representation, which is oftentimes seen for experimental data. While both functions will (depending on their width) qualitatively introduce a ‘depressed semi‐circle’ in the Nyquist plot, a Gaussian relaxation function leads to a perpendicular intercept with the abscissa and a Cauchy‐type relaxation function to an angle of zero degree at high and low frequencies.