Application of the Kramers-Kronig Relations to Multi-Sine Electrochemical Impedance Spectroscopy
Authors: You Chen , Zabara Mohammed Ahmed , Ulgut Burak , Orazem Mark E 
Published: February 2020

Impedance spectra obtained by fast Fourier transformation of the response to a multi-sine potential perturbation are shown to be consistent with the Kramers-Kronig relations, even for systems that are nonlinear and nonstationary. These results, observed for measurements on a Li/SOCl2 battery, were confirmed by numerical simulations. Consistency with the Kramers-Kronig relations was confirmed by use of the measurement model developed by Agrawal et al.1 and by a linear measurement model approach developed by Boukamp2 and implemented by Gamry.
In the experimental work, the impedance response was measured galvanostatically using a Gamry Interface 1000E instrument with both the single-sine and multi-sine method for three systems. For the first system, a Li/SOCl2 battery was discharged under zero direct current (DC), which makes the system nonlinear and nonstationary.3 For the second system, a Li/SOCl2 battery was under constant discharge, which makes the system linear but nonstationary. For the third system, a direct current of 20 μA was applied to a dummy cell with a parallel combination of a diode, a capacitor and a resistor, which makes the system nonlinear but stationary. The single-sine measurements were found to be inconsistent with the Kramers-Kronig relations for nonstationary systems; however, the multi-sine measurements for nonstationary systems were consistent with the Kramers-Kronig relations.
The impedance response was calculated for a system in which a charging current was added to a faradaic current given by a Tafel expression with a time-dependent rate constant. The impedance response was calculated by a Fourier analysis for the single-sine potential perturbation and by a Fast-Fourier-transform (FFT) analysis for the multi-sine measurement. With a linear or an exponential decrease of the rate constant as a function of time, the single-sine measurement was found to be inconsistent with the Kramers-Kronig relations; whereas, the multi-sine measurement was found to be fully consistent with the Kramers-Kronig relations. The apparent charge-transfer resistance for the multi-sine simulation was the average of the time-dependent values. With a large perturbation amplitude and a fixed rate constant, both the single-sine and the multi-sine results were consistent with the Kramers-Kronig relations, yielding a smaller semicircle. The values were noisy at low frequency for the multi-sine measurement, but the noise was reduced by increasing the time used for FFT calculations.
The impedance responses obtained from the multi-sine measurement were found to be consistent with the Kramers-Kronig relations for nonlinear and for nonstationary systems. The present work demonstrates that application of the Kramers-Kronig relations to the results of multi-sine measurements cannot be used to determine whether the experimental system satisfies the conditions of linearity, causality and stability.
References

Agarwal, M. E. Orazem, and L. H. García-Rubio, “Application of Measurement Models to Electrochemical Impedance Spectroscopy: 3. Evaluation of Consistency with the Kramers-Kronig Relations,” Journal of the Electrochemical Society, 142 (1995), 4159-4168.

Bernard Boukamp, “A Linear Kronig-Kramers Transform Test for Immittance Data Validation,” Journal of the Electrochemical Society, 142 (1995) 1885-1894.

Mohammed Ahmed Zabara, Can Berk Uzundal, and Burak Ulgut, Linear and Nonlinear Electrochemical Impedance Spectroscopy Studies of Li/SOCl2 Batteries, Journal of the Electrochemical Society, 166 (2019), A811-A820.