Comparison of Approaches for Assessing Linearity of Impedance Measurements
Authors: Goh Jie Min , Babauta Jerome , Orazem Mark E. 
Published: August 2024

There are two experimental methods available to assess the linearity of electrochemical impedance measurements. Observation of current as a function of potential, known as Lissajous plots, provides an instantaneous visual representation of the data.1 Deviation from an elliptical shape suggests a perturbation amplitude that is too large, resulting in nonlinear impedance data. A second approach is to measure, along with the fundamental, the higher harmonics of the measurement signal. Under potentiostatic control, the harmonics of current that exceed the noise level suggest nonlinear behavior. These harmonics may be expressed in terms of a frequency-dependent total harmonic distortion, THD, written as the ratio between the sum of the root mean square of the harmonic magnitude and fundamental frequency magnitude for electrochemical applications.
The impedance was measured for a model system consisting of a Pt disk immersed in an electrolyte containing 0.01 M potassium ferricyanide, 0.01 M potassium ferrocyanide, and 1 M KCl. Measurements were performed using a Gamry Reference 3000 with THD measurement enabled and disabled. Low-frequency (10 mHz) Lissajous plots were recorded to facilitate comparison of the two methods for assessing linearity. The impedance measurements with THD enabled or disabled were in good agreement, including the frequency-dependent stochastic error structure, the shapes of Lissajous curves, and the values of the impedance. Both the low-frequency THD and the low-frequency Lissajous plots gave similar sensitivity to nonlinear behavior. Perturbation amplitudes of 1, 5, and 10 mV rms yielded elliptical Lissajous plots with THD values on the order of the background noise. Interestingly, the THD for the 10 mV perturbation was slightly above the background noise with a value of 0.01 (or 1 % of the fundamental). Perturbation amplitudes of 20, 60, and 100 mV rms yielded both non-elliptical Lissajous plots and low-frequency THD values much larger than the background noise.
The regression of the measurement model 2 to the imaginary part of the impedance was applied to assess the consistency of the measured data to the Kramers-Kronig relations.3 Data measured at amplitudes of 1, 5, 10, and 20 mV rms were found to be consistent with the Kramers-Kronig relations, and data collected with amplitudes of 60 or 100 mV rms were found to be inconsistent with the Kramers-Kronig relations. As the measurement with a 20 mV rms amplitude was found to be nonlinear by observation of non-elliptical Lissajous plots and a THD of 4.2% at a frequency of 10 mHz, the post-measurement analysis of consistency with the Kramers-Kronig relations was found to be less sensitive than experimental observation of Lissajous plots and calculation of THD. Comparison of Lissajous plots to THD values suggests that experiments with THD less than 1% of the fundamental may be considered to be linear.
References:

Hirschorn and M. E. Orazem, “On the Sensitivity of the Kramers–Kronig Relations to Nonlinear Effects in Impedance Measurements,” Journal of The Electrochemical Society,156 (2009) C345.

Watson and M. E. Orazem, EIS: Measurement Model Program, Version 1.8, ECSArXiv, 2023.

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.