Reconstructing the distribution of relaxation times with analytical basis functions

Abstract

The present work proposes the implementation of non-singular basis functions into the algorithm for reconstructing the distribution of relaxation times (DRT) function of impedance data. These functions reflect the dispersed and asymmetrical nature of non-ideal capacitive–resistive processes. Inclusion is achieved by combining the singular Debye distribution basis with distributed relaxation functions, such as those derived from the analytical models of Cole–Cole and Havriliak–Negami. The shapes of the introduced basis functions are described by constant parameters, for which an empirical optimization approach is provided alongside. Using synthetic impedance data of non-ideal capacitive–resistive processes subjected to white noise, it is shown that the demand for regularization can be reduced significantly by using distributed bases. To underline the practical relevance of non-singular basis functions in DRT reconstruction, an experimental study comprising 100 sodium-ion and 80 lithium-ion commercial cells is presented. In this context, it is shown that auxiliary information from the non-ideal nature of real-world electrochemical processes is outsourced into the basis and, hence, easily filtered out of the resulting DRT. This facilitates the separation of single processes without post-DRT curve fitting and thus improves the interpretation and classification of impedance data significantly.