[Paper Review] Electrochemical impedance spectroscopy beyond linearity and stationarity - a critical review

Noël Hallemans, David A. Howey+8 more|arXiv (Cornell University)|Apr 17, 2023

Advanced Battery Technologies Research121 references4 citations

TL;DR

This paper reviews advanced electrochemical impedance spectroscopy (EIS) methods that extend beyond classical linear time-invariant assumptions, enabling accurate impedance estimation in nonlinear and nonstationary systems such as Li-ion batteries during charging. By using multisine excitation and frequency-domain analysis within Volterra series frameworks, the approach captures dynamic impedance changes in real time, offering a more realistic characterization than traditional EIS.

ABSTRACT

Electrochemical impedance spectroscopy (EIS) is a widely used experimental technique for characterising materials and electrode reactions by observing their frequency-dependent impedance. Classical EIS measurements require the electrochemical process to behave as a linear time-invariant system. However, electrochemical processes do not naturally satisfy this assumption: the relation between voltage and current is inherently nonlinear and evolves over time. Examples include the corrosion of metal substrates and the cycling of Li-ion batteries. As such, classical EIS only offers models linearised at specific operating points. During the last decade, solutions were developed for estimating nonlinear and time-varying impedances, contributing to more general models. In this paper, we review the concept of impedance beyond linearity and stationarity, and detail different methods to estimate this from measured current and voltage data, with emphasis on frequency domain approaches using multisine excitation. In addition to a mathematical discussion, we measure and provide examples demonstrating impedance estimation for a Li-ion battery, beyond linearity and stationarity, both while resting and while charging.

Motivation & Objective

• To address the limitations of classical electrochemical impedance spectroscopy (EIS), which assumes linearity and time-invariance in electrochemical systems.
• To develop and review methods for estimating impedance in nonlinear and time-varying electrochemical systems, such as Li-ion batteries during charge/discharge cycles.
• To enable dynamic impedance characterization under realistic operating conditions where traditional EIS fails due to inherent nonlinearity and time-variation.
• To provide a comprehensive review of frequency-domain techniques using multisine signals for nonlinearity and nonstationarity compensation in EIS.
• To demonstrate practical applicability through experimental examples on Li-ion batteries under resting and charging conditions.

Proposed method

• Utilizes multisine excitation signals to probe electrochemical systems across multiple frequencies simultaneously, enabling efficient data collection for nonlinear and time-varying systems.
• Applies Volterra series expansion to model nonlinear time-invariant (NLTI) systems, decomposing the system response into polynomial kernels representing linear, second-order, and higher-order nonlinearities.
• Derives frequency-domain expressions for harmonic responses (e.g., V2,1, V3,3) based on generalized impedance functions Zn(ω1,…,ωn), enabling spectral decomposition of nonlinear contributions.
• Introduces a two-timescale decomposition: a slow trajectory (e.g., battery charging current) and a fast multisine perturbation, allowing time-varying impedance estimation via convolution with a time-dependent impulse response.
• Employs discrete Fourier transforms (DFT) to compute complex impedance from measured voltage and current signals, enabling frequency-domain analysis of nonlinear responses.
• Uses rescaled Legendre polynomials as basis functions for modeling time-varying system behavior, particularly in the context of nonstationary impedance estimation.

Figure 1 : Equivalent electrical schematic of an electrochemical system under LTI constraints.

Experimental results

Research questions

• RQ1 How can electrochemical impedance spectroscopy be extended to accurately characterize systems that are inherently nonlinear and nonstationary?
• RQ2 What are the mathematical and experimental foundations for estimating time-varying and nonlinear impedances using multisine excitation?
• RQ3 How does the Volterra series framework enable the decomposition of nonlinear and nonstationary responses in electrochemical systems?
• RQ4 What are the practical limitations and advantages of applying dynamic EIS to real Li-ion batteries during charging?
• RQ5 Can frequency-domain methods based on multisine signals provide reliable impedance estimates beyond the linear and stationary assumptions of classical EIS?

Key findings

• The paper demonstrates that classical EIS fails to capture the true impedance behavior of Li-ion batteries during charging due to inherent nonlinearity and time-variation.
• Using multisine excitation and Volterra series modeling, the study successfully estimates nonlinear and time-varying impedance components, such as second- and third-order harmonic responses.
• Experimental results show measurable differences in impedance magnitude and phase between resting and charging states, confirming time-variant behavior even within short measurement windows.
• The derived expressions for harmonic voltages (e.g., V3,3 = 1/8 Z3(ω,ω,ω)I³) provide a quantitative framework for identifying and quantifying nonlinear contributions in EIS data.
• The time-dependent impulse response z(τ,t) explicitly depends on the operating trajectory i₀(t), proving that stationarity is violated during dynamic processes like battery charging.
• The method enables real-time impedance tracking during operation, offering a path toward in-situ diagnostics and state-of-charge estimation in electrochemical energy storage systems.

Figure 2 : Illustration of the linearisation of the Butler-Volmer equation ( 7 ) for a small amplitude sinusoidal excitation.