Tikhonov regularization for the deconvolution of capacitance from the voltage–charge response of electrochemical capacitors

Abstract

The capacitance of capacitive energy storage devices cannot be directly measured, but can be estimated from the applied input and measured output signals expressed in the time or frequency domains. Here the time-domain voltage–charge relationship of non-ideal electrochemical capacitors is treated as an ill-conditioned convolution integral equation where the unknown capacitance kernel function is to be found. This comes from assuming a priori that in the frequency domain the charge is equal to the product of capacitance by voltage, which is in line with the definition of electrical impedance. The computation of a stable solution to this problem particularly when dealing with experimental data is highly sensitive to noise as it may lead to an oscillating result even in the presence of small errors in the measurements. In this work, the problem is treated using Tikhonov’s regularization method, where a degree of damping is added to each singular value decomposition (SVD) component of the solution, thus effectively filtering out the components corresponding to the small singular values. The regularized time-domain capacitance of a commercial electric double-layer capacitor is found to be in good agreement with the frequency-to-time transformed capacitance obtained from impedance spectroscopy.