Inverse problem of reconstructing the capacitance of electric double-layer capacitors

The capacitance is a characteristic function of a capacitive energy storage device that is inaccessible to direct measurements, but can be estimated from input and output signals. Knowing that electric double-layer capacitors (EDLC) exhibit non-ideal capacitive behavior, their capacitance is commonly estimated from the time-domain definition of capacitance times voltage giving charge (using cyclic voltammetry and galvanostatic charge/discharge data), and in the same time from the same multiplicative relationship but in the frequency-domain (using impedance spectroscopy data), as if the capacitance is constant. The purpose of this study is to provide the recommended procedure to compute the capacitance of such types of devices from time-domain data starting from the definition of capacitance being the ratio of charge by voltage both defined in the frequency domain, which is consistent with the definition of conventional impedance. This turns the situation in the time domain to be an ill-posed inverse problem of convolution, wherein the to-be-deconvolved or reconstructed capacitance can be very sensitive to experimental errors. We show results of the procedure using (i) synthetic data generated using a fractional-order capacitor model of impedance Z(jω)=1/[(jω)αCα] and (ii) experimental data of a commercial aluminium electrolytic capacitor and an EDLC for the two cases of linear voltage ramp and constant-current step excitations.