Practical Applications of the Kramers–Kronig Relations

Abstract

A technique is presented for using the Kramers–Kronig (KK) relations to evaluate the consistency of impedance data that do not exhibit a DC limit. The approach presented here differs from current algorithms because extrapolation of experimental data, required to evaluate the integral relations, is achieved without the assumption of an equivalent circuit and without extrapolation of polynomial fits of the available data. The impedance values below the lowest measured frequency ωmin are calculated, which would “force” the available data to satisfy the KK relations. Consistent data set have been shown to yield calculated impedance values which are finite and continuous with the experimental data at ωmin. Discontinuities between the caicuiated impedance and the experimental data set can be attributed to violation of one or more conditions of the KK relations. The proposed algorithm is applied to experimental impedance data obtained for a metal hydride electrode and a rotating copper-disk electrode. The technique is shown to be sensitive for testing violations of the stability condition and is used to identify a lower frequency bound for data that satisfy the KK relations. The implications to equivalent circuit fitting procedures and to the estimation of the polarization resistance are discussed.