Improved DRT Determination Through a Modified Sparse Spike Deconvolution

Abstract

Impedance spectroscopy is a powerful characterization method to evaluate the performance of electrochemical systems. However, overlapping signals in the resulting impedance spectra oftentimes cause misinterpretation of the data. The distribution of relaxation times (DRT) method overcomes this problem by transferring the impedance data from the frequency domain into the time domain, which yields DRT spectra with an increased resolution. Unfortunately, the determination of the DRT is an ill-posed problem, and appropriate mathematical regularizations become inevitable to find suitable solutions. [1]The Tikhonov algorithm is a widespread algorithm for the determination of the DRT g and is given by [2] min g (|| Z - A ⋅ g || + λ ⋅ ||L ⋅ g ||)where the regularization parameter λ scales the penalty term. Unfortunately, this leads to unlikely spectra due to necessary boundaries. The sparse spike deconvolution is using the known positive distribution function for a ZARC-element. Thus this deconvolution leads to more natural spectra for the determination of the DRT. However, the sparse spike deconvolution has a very limited scope by using one single regularization parameter. Consequently, we replaced the scalar regularization parameter with a vector P and added a new regularization parameter k for the simultaneous minimization of the vector and the norm of the residuals. The minimization function of this method is given by [3]min P (|| Z - A ⋅ G( P ) ⋅ c || ⋅ k + ||ln(1 - P )||)Herein the G is a matrix with multiple impulse functions.Literature: [1] Ivers-Tiffée, E., & Weber, A., J.Ceram. Soc. Jpn. 2017, 125, 4. [2] Gavrilyuk, A. L., Osinkin, D. A., & Bronin, D. I., Russ. J. Electrochem., 2017, 53, 6. [3] Bergmann, T. G., & Schlüter, N., ChemPhysChem, 2022