Investigation of impedance spectra of systems with distinctive resistive-inductive behavior using the generalized distribution of relaxation times analysis

The distribution of relaxation (DRT) times is a well-established method for the characterization and process identification of electrochemical systems in the frequency domain. In measured impedance spectra, it is in general a very challenging task to reliably extract system characteristics, to identify processes, and to evaluate the impact of operational variables. In many cases, changes in temperature, pressure, or aging conditions lead to only minor visible mutations in the impedance spectra.

However, with the DRT it is possible to transform the impedance spectra into a more distinguishable form in time-domain using a data-based determination of lumped elements , , and &nbsp”;”and a series of &nbsp”;”parallel circuits. This combination of elements represents a generic approach to describe resistive, resistive-capacitive, capacitive, and inductive effects of an arbitrary system. However, since many systems additionally show resistive-inductive behavior, Danzer [1] introduced the generalized DRT (gDRT) including a second series of &nbsp”;”parallel &nbsp”;”circuits. The time constants are predefined, and the distribution of polarization contributions of the resistors are obtained for both the resistive-capacitive and the resistive-inductive behavior by optimization. The results are used afterwards for the analysis of the underlying processes. This method has already been used for the characterization of a variety of electrochemical systems in different fields, like batteries [2, 3] and fuel cells [1]. Since the resulting optimization problem is linear, ill-posed and ill-conditioned, a Tikhonov regularization is used to solve it. The required regularization parameter is usually quantified via the L-curve method, proposed by Paul et al. [4].

In this work, we introduce an efficient and elegant method to describe resistive-inductive effects without having to add the second series of &nbsp”;”circuits to the optimization problem. This can be achieved by allowing negative polarization contributions in the transfer function of the &nbsp”;”elements and by correcting the resulting lumped &nbsp”;”element by the absolute value of the sum of the negative polarization. With this, one can represent both resistive-inductive and resistive-capacitive behavior with a single series of generalized parallel &nbsp”;”circuits, reducing the size of the equation system and thus leading to a more efficient solution of the optimization problem.

The modification of the transfer function of the &nbsp”;”elements is analytically derived and further validated with simulated and measured impedance spectra of electrochemical and photo-electrochemical systems by comparing the results of the gDRT to the newly proposed method. The solution for the &nbsp”;”distribution can be extracted by selecting the contributions with negative sign of the &nbsp”;”distribution. With this, it can be shown, that the obtained solutions are practically identical, and that the new method can be used without limitations instead of the gDRT.

References

[1] Danzer, M.A. Generalized Distribution of Relaxation Times Analysis for the Characterization of Impedance Spectra. Batteries 2019, 5, 53. https://doi.org/10.3390/batter…

[2] Katzer, Felix and Rüther, Tom and Plank, Christian and Roth, Felix and Danzer, Michael A., Investigation of Polarisation Effects Arising from Lithium Deposition on Experimental Half-Cells and Commercial Full-Cells. http://dx.doi.org/10.2139/ssrn…

[3] Markus Hahn, Dominic Rosenbach, Alexander Krimalowski, Tobias Nazarenus, Ralf Moos, Mukundan Thelakkat, Michael A. Danzer, Investigating solid polymer and ceramic electrolytes for lithium-ion batteries by means of an extended Distribution of Relaxation Times analysis, Electrochimica Acta, 2020, https://doi.org/10.1016/j.elec…

[4] Paul, T., Chi, P.W., Wu, P.M. et al. Computation of distribution of relaxation times by Tikhonov regularization for Li ion batteries: usage of L-curve method. Sci Rep 11, 12624 (2021). https://doi.org/10.1038/s41598…