Source-linked AI summary
Electrochemical impedance spectroscopy beyond linearity and stationarity - a critical review
Noël Hallemans, David Howey, Alberto Battistel, Nessa Fereshteh Saniee, Federico Scarpioni, Benny Wouters, Fabio La Mantia, Annick Hubin, Widanalage Dhammika Widanage, John Lataire
TL;DR
Classical EIS assumes electrochemical systems are linear and stationary, although their current-voltage relations can be nonlinear and evolve over time. This review presents methods for estimating impedance beyond those constraints, emphasizing multisine frequency-domain approaches. In battery measurements, operando EIS yields clean time-varying impedance at low frequencies despite drift, and impedance differs between charging and resting.
Problem
Classical EIS assumes linear, stationary behaviour, while electrochemical processes can be nonlinear and evolve over time.
Source: §1 · p. 3, §4.1 · p. 9Method
The review compares methods for estimating time-varying impedance from measured current and voltage, including windowing/filtering and mathematical regression approaches.
Source: §5 · p. 15, §9 · p. 21Results
Operando EIS produced clean time-varying impedance at low frequencies despite drift, with impedance highest at low SOC and different while charging versus resting.
Source: §10.2 · p. 26Takeaways & Limitations
Higher-dimensional impedance data can contain additional information beyond classical impedance and may support applications such as battery health forecasting.
Source: §11 · pp. 27–28Time-frequency methods have difficulty estimating low-frequency impedance because of drift and do not account for nonlinear distortions.
Source: §9.2.3 · p. 23
Abstract
from arXiv · showhide
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.
1. Overview of applications of EIS for batteries
Classical impedance data supports battery health analysis, while nonlinear and time-varying impedance may add information for diagnostics and smart charging. The section connects these possibilities to Li-plating detection, parameter identifiability, and measurement practicality.
1. Overview of applications of EIS for batteries
- Nonlinear and time-varying impedance data may improve battery SOH diagnostics by adding information beyond classical impedance data.Such data have been used to detect Li-plating and have contributed to identifiability of physical SPM parameters.Source: §1 · p. 3
- Time-varying charge-transfer resistance has been tracked while charging and used to propose adapting the charge profile.Impedance-based Li-plating detection has also been proposed for adaptive fast charging.Source: §1 · p. 3
- Although EIS experiments can be expensive and difficult to implement in battery management systems, low-cost apparatus solutions exist.The section identifies multisine excitation and frequency-domain model estimation as promising tools.Source: §1 · pp. 3–4
2. Modelling electrochemical systems
The paper models electrochemical systems through the relation between current, voltage, and potentially changing external parameters. It treats this relation as a dynamical operator with memory and develops the impedance model first under classical constraints, then beyond them.
2. Modelling electrochemical systems
- The current-voltage relation may also depend on external parameters such as temperature, electrode rotation rate, pressure, and concentration distribution.Source: §2 · p. 4
- In galvanostatic experiments, current is applied and voltage is measured, with the impedance represented by an operator acting on current and external parameters.Source: §2 · p. 4
- In potentiostatic experiments, voltage is applied and current is measured; the paper uses current as excitation and voltage as response for its battery-oriented notation.Source: §2 · p. 4
- The system operators are dynamical, or memory-bearing, so they depend on past information as well as the present input.Source: §2 · p. 4
- The paper first models the operator under linearity and stationarity, then introduces models beyond those constraints.Source: §2 · p. 4
3. Classical EIS revisited
Classical EIS represents an electrochemical system with an impedance model that assumes linearity, stationarity, and causality. Measurements therefore use small perturbations around a steady operating point, and the resulting impedance is specific to that point.
3.1. The constraints of classical EIS
- Classical impedance represents the voltage response through an impedance operator under linearity, stationarity, and causality constraints.The model also includes a constant open-circuit voltage.Source: §3.1 · p. 4, Figure 1 · p. 4
- When current is small enough, the linear term of the Butler–Volmer relation dominates higher-order terms, allowing a local linear approximation.A rule of thumb is to keep voltage deviation below 15 mV.Source: §3.1 · p. 5, Figure 2 · p. 5
- Stationarity is maintained by measuring at steady state with zero-mean excitation and constant external parameters.A positive-mean current can charge a battery and cause nonstationary behaviour.Source: §3.1 · p. 5
- For an LTI system, the impulse response is convolved with current in time, and the impedance is its Fourier transform, enabling frequency-domain representation.Source: §3.1 · p. 5
- Kramers–Kronig relations connect real and imaginary impedance and can be used to assess whether linearity and stationarity constraints appear satisfied.Source: Kramers-Kronig · pp. 6–7
- In practice, Kramers–Kronig testing is constrained by discrete, finite-band impedance data and does not account for measurement noise.An alternative is fitting an equivalent-circuit model to one impedance component and checking its agreement with the measured complex data.Source: Kramers-Kronig · pp. 6–7
Models at operating points
Classical impedance depends on the local operating point, while single-sine and multisine excitations offer different ways to measure it. The section recommends multisine excitation for broadband measurements and describes its time, SNR, and frequency-selection trade-offs.
Models at operating points
- Classical impedance depends on the local operating point; in the Samsung 48X battery, low SOC and low temperature show higher impedance.Impedance also differs between cells at the same SOC and temperature but different SOH.Source: Models at operating points · p. 7
3.2.1. Single-sine excitation
- Single-sine EIS applies selected frequencies sequentially, which is possible because stationarity makes the response independent of excitation time.Source: §3.2.1 · p. 7
3.2.2. Multisine excitation
- A multisine applies several sinusoidal frequencies simultaneously, with separate phases, and impedance can still be computed at the selected frequencies.Source: §3.2.2 · pp. 7–8, Figure 4 · p. 7
3.2.3. The choice of excitation signal
- For broadband classical EIS, the paper recommends multisine excitation, while recognizing single-sine experiments are familiar and easy to use with commercial potentiostats.Source: §3.2.3 · p. 8
- Multisine experiments shorten measurement time because frequencies are applied together and transients need only fade once.Source: §3.2.3 · p. 8
- Multisine measurements require frequency-domain analysis to detect nonlinearity and nonstationarity, which time-domain-only extraction may miss along with leakage and nonlinear distortions.Source: §3.2.3 · p. 8
- At equal measurement duration, multisine can provide better SNR because more periods fit into the time used by longer single-sine experiments.Its frequency choices are less flexible at low frequencies because tones must be integer multiples of a fundamental; multiple multisines can address this if required.Source: §3.2.3 · p. 8
4. Nonlinear and nonstationary impedance models
The section extends impedance models beyond classical EIS’s linearity and stationarity constraints, introducing Volterra-series, time-varying, and combined nonlinear time-varying formulations. It discusses how these models relate to NLEIS, BLA, and impedance measurements along operating trajectories.
4.1. Nonlinear models
- Classical EIS excitation amplitude balances signal-to-noise ratio against linearity, while nonlinear models remain valid over larger excitation spans but at fixed local operating points.The section presents the Volterra series, NLEIS, and BLA as ways to model or assess nonlinear behavior.Source: §4.1 · p. 9
4.1.1. The Volterra series
- Volterra series represent dynamical nonlinear behavior as polynomial operators around a local operating point, though systems with subharmonics or chaos may not be captured.Even- and odd-degree terms contribute spectral content at even and odd integer multiples of the excited frequency, respectively.Source: §4.1.1 · pp. 9–10
4.1.2. Nonlinear EIS
- NLEIS measures leading-order nonlinear impedance coefficients at a fixed operating point, choosing excitation large enough to reveal the target harmonic but small enough to limit higher-order contributions.Under these conditions, the first leading-order coefficient equals the regular impedance.Source: §4.1.2 · p. 11
4.1.3. The best linear approximation
- The BLA minimizes nonlinear distortion in a least-squares sense, depends on excitation amplitude, and differs from the first generalized and leading-order nonlinear impedances.Higher-order odd Volterra terms also contribute at the excited frequency, affecting the BLA.Source: §4.1.3 · pp. 11–12
4.2. Nonstationary models
- Stationarity requires a constant operating point, but obtaining impedance at multiple conditions can require repeated steady-state waits, and some processes never reach steady state.Impedance evolution during operation can contain information that classical EIS and stationary NLEIS cannot gather.Source: §4.2 · p. 12
4.2.1. The time-varying impedance
- A time-varying impedance model represents the response to excitation along a changing trajectory, with a drift signal alongside the linear time-varying response.Its two-dimensional impulse response depends explicitly on excitation time.Source: §4.2.1 · pp. 13–14
4.2.2. Models along an operating trajectory
- Varying the slow trajectory or external parameters yields impedance models along an operating path rather than at one point, and simultaneous multisine excitation measures selected frequencies across that path.Sequential single-sine measurements would assign different frequencies to different parts of a changing trajectory.Source: §4.2.2 · p. 14, §4.2.3 · p. 14
4.3. Nonlinear and nonstationary models
- For nonlinear time-varying systems, the time-varying Volterra series provides a general response model, while the BLTVA extends BLA to track impedance during operation.Measuring time-varying leading-order nonlinear impedance functions is described as not yet studied to the authors’ knowledge.Source: §4.3 · pp. 14–15
5. Measuring current and voltage data
Current and voltage are sampled and windowed during a chosen electrochemical reaction, with multisine excitation recommended for extracting time-varying impedance. The frequency-domain workflow uses an integer number of measured periods and can reveal nonlinear distortions, nonstationarity, and noise.
Design of excitation signal
- For time-varying impedance data, use a multisine excitation; single-sine excitation is suited to stationary nonlinear impedance estimates.The single-sine is a special case of the multisine.Source: §5 · p. 15
- ORP EIS uses random-phase multisines with only odd harmonics excited.Odd harmonics left unexcited can help detect nonlinear distortions in the response.Source: §5 · pp. 15–16
Windowing
- Measure an integer number of multisine periods to avoid spectral leakage; this is required for NLEIS and time-varying impedance data.The integer-period requirement applies to frequency-domain estimation, including multisine experiments.Source: §5 · p. 15
Sampling and frequency-domain data
- Use uniform sampling above twice the measurement's highest frequency to avoid spectral aliasing, then compute frequency-domain data from the sampled signals using the DFT.The DFT maps the sampled current and voltage to frequency-domain data on its frequency grid.Source: §5 · pp. 15–16
Frequency-domain data
- With one measured period, intentionally unexcited DFT lines reveal nonlinear distortions, but nonlinear and nonstationary behavior cannot be distinguished when both occur.The illustrated ORP multisine leaves even harmonics and the seventh harmonic unexcited.Source: §5 · p. 16, Figure 10 · p. 16
Measurement noise
- Increasing the number of measured periods improves frequency-domain SNR by distributing noise across more DFT lines and reducing the variance at each line.In practice, the number of measured periods P is increased to improve SNR.Source: §5 · p. 17
6. Classical frequency domain impedance estimation
For periodic excitation measured over an integer number of periods in steady state, impedance is computed as the ratio of voltage and current spectra. Random excitation instead uses cross- and auto-spectra, with averaging across experiments appropriate only under stationary conditions.
Periodic excitation
- For periodic excitation measured over an integer number of periods in steady state, compute impedance as the ratio of voltage and current spectra.At positive frequencies, the voltage's open-circuit-voltage component at DC does not affect the estimate; increasing measured periods reduces uncertainty through averaging.Source: §6 · p. 17
Random excitations
- For random excitation, estimate impedance from the cross-spectrum of voltage and current divided by the current auto-spectrum.Windowing can reduce transients when the DFT does not correspond to the continuous Fourier transform.Source: §6 · pp. 17–18
- Averaging spectra across experiments reduces transients and avoids division by zero, but is appropriate only in stationary conditions where experiments can be repeated.For pseudo-white noise with unit current power in the frequency band of interest, only the cross-spectrum numerator needs to be computed.Source: §6 · p. 18
7. Detection of nonlinearity and nonstationarity
A zero-mean ORP excitation and measured frequency-domain response can reveal nonlinearity, nonstationarity, and noise. The spectral patterns distinguish LTI, NLTI, LTV, and NLTV behavior, while odd-only excitation helps classify nonlinearities and limits their contribution at excited frequencies.
7.1. An example with ORP multisine excitation
- LTI responses contain spectral content only at DC and excited frequencies; NLTI adds integer harmonics, while LTV and NLTV responses show hyperbolic skirts around excited frequencies and, for NLTV, nonlinear harmonics.The example uses an ORP multisine with excited harmonics {1, 3, 7} over 10 periods.Source: §7.1 · p. 18
- In real measurements, noise is distributed across DFT frequencies, yet the example reports that nonlinearities, nonstationarities, and noise remain distinguishable.In the battery spectrum shown in Fig. 11, the voltage satisfies LTI constraints and noise is at least 1000 times smaller than the linear response.Source: §7.1 · p. 20
7. Detection of nonlinearity and nonstationarity
- An ORP multisine measured over many periods can detect nonlinearity and nonstationarity and estimate noise across a wide frequency range in one experiment.The paper recommends zero-mean ORP excitation because classical impedance data cannot establish whether measured multisine data meets linearity and stationarity assumptions.Source: §7 · p. 18, §7.2 · p. 20
7.2. Advantages of an ORP multisine excitation
- Random phases offer simple implementation, while exciting only odd harmonics distinguishes even from odd nonlinear distortions and keeps even nonlinear distortions off excited frequencies.Even nonlinear distortions therefore introduce no uncertainty when computing the BLA or BLTVA with a multisine.Source: §7.2 · p. 20
8. Nonlinear impedance estimation
The section describes frequency-domain estimation of leading-order nonlinear impedance coefficients with a single sine and estimation of the best linear approximation (BLA) with either single-sine or multisine excitation. For ORP multisines, BLA uncertainty reflects noise and odd nonlinear distortions, with different ways to reduce each contribution.
8.1. Leading order nonlinear impedance estimation
- Leading-order nonlinear impedance coefficients are estimated using a zero-mean single-sine excitation measured over an integer number of periods.Different frequencies are applied sequentially; measuring more periods averages stochastic noise.Source: §8.1 · p. 20
8.2. Best linear approximation
- The BLA can be estimated at different frequencies using sequential single-sine excitation, independently of the excitation amplitude.For multisine excitation measured over an integer number of periods, its estimate is computed as in classical impedance experiments.Source: §8.2 · p. 20
- With an ORP multisine, BLA uncertainty comes from noise and odd nonlinear distortions at excited frequencies; more periods reduce noise, while fewer excited frequencies reduce nonlinear uncertainty.Odd nonlinear distortion uncertainty does not depend on the number of measured periods; Riemann-equivalent multisines can change the number of excited frequencies without changing the BLA.Source: §8.2 · pp. 20–21
9. Time-varying impedance estimation
Time-varying impedance estimation extends classical measurements to evolving systems, with multisine methods ranging from short-window FFT and time-frequency analysis to operando EIS. These approaches trade time, frequency, and noise performance; operando EIS additionally models nonlinear distortions and drift, subject to limits on odd nonlinearities and resolution.
9. Time-varying impedance estimation
- Time-varying impedance cannot be estimated by simply dividing DFT spectra; the review focuses on multisine methods using windowing/filtering or regression.Windowing/filtering yields an average within a selected time frame, while spectral regression extracts time variation.Source: §9 · p. 21, §9.1 · p. 21
9.1. FFT-EIS applied to nonstationary data
- FFT-EIS applies classical impedance measurements to short subrecords along a changing voltage trajectory, assuming impedance is constant within each window.Its short windows restrict measurement to high frequencies; reported trajectory results cover only two frequencies above 300 Hz, though a later implementation reached 50 Hz–50 kHz.Source: §9.1 · p. 21
9.2.1. STFT-EIS
- STFT-EIS estimates impedance from windowed voltage and current spectra at each time, with FFT-EIS as its rectangular-window special case.The estimate uses subrecords centered on the time instant; enough periods and careful window choice are important for resolving excited-frequency peaks.Source: §9.2.1 · p. 22
9.2.2. Dynamic Multi-Frequency Analysis
- Dynamic multi-frequency analysis filters current and voltage spectra around excited multisine frequencies, then inverse-transforms them to obtain time-varying impedance.Enough periods must be measured to provide frequency-domain data around the excited frequencies.Source: §9.2.2 · p. 22
9.2.3. Equivalence between STFT-EIS and DMFA
- For symmetric windows, STFT-EIS and DMFA have equivalent continuous definitions; their practical differences arise from window/filter choice and implementation.The time–frequency trade-off remains: improving time selectivity reduces frequency resolution.Source: §9.2.3 · pp. 22–23
- DMFA uses one DFT of the full record and can yield higher SNR and faster computation, whereas STFT-EIS can estimate impedance in real time.Neither time-frequency approach handles low frequencies well in the presence of drift, and nonlinear distortions are not treated by these methods as described.Source: §9.2.3 · p. 23
9.3. Operando EIS
- Operando EIS models time-varying impedance and nonlinear distortions with basis expansions, then estimates parameters by linear least squares.Legendre polynomials separate time and frequency dependencies; long records are handled in local frequency bands, and the time–frequency resolution trade-off remains.Source: §9.3 · pp. 24–25
- Operando EIS estimates the time-varying impedance or BLTVA, includes uncertainty bounds, and models drift to recover low-frequency information.It also detects, quantifies, and classifies nonlinearities; odd nonlinearities must not be too strong.Source: §9.3 · pp. 23–25
10. A case study on commercial Li-ion batteries
The Samsung 48X case study estimates classical impedance at fixed operating points and time-varying impedance during charging, then fits an equivalent circuit model to both. The results show that impedance and fitted parameters can differ between resting and charging conditions.
10. A case study on commercial Li-ion batteries
- The case study uses a pristine 4.8 Ah Samsung 48X cell at 5°C or 25°C; classical EIS is performed after two hours of relaxation at selected SOCs.Source: §10 · p. 25, §10.1 · p. 25
10.1. Estimating classical impedance data
- At 10% state of charge and 25°C, the chosen excitation produces nearly linear behavior, with no detected nonlinear distortions or nonstationarities and an SNR of at least 1000.A small drift signal and measurement noise are present.Source: §10.1 · p. 25
10.2. Estimating time-varying impedance data
- During charging at 5°C, the voltage reaches the 4.2 V safety bound before the expected two hours, allowing 29 and 31 excitation periods at 5°C and 25°C, respectively.The multisine is added to the 2.4 A DC charging current; the state of charge rises from 0% to 72.5% while surface temperature increases slightly.Source: §10.2 · p. 26
- Impedance while charging differs from resting impedance, and it is highest at low state of charge.Despite drift hiding low-frequency content, operando EIS yields clean impedance data at those frequencies.Source: §10.2 · p. 26
10.3. Estimating equivalent circuit model parameters
- The selected equivalent circuit maps its branches to electrolyte resistance, diffusion, and electrochemical kinetics, with classical and time-varying fits having mean relative errors below 0.2% and 0.5%, respectively.Fits cover 16.7 mHz to 50 Hz; the parameter cost function is nonlinear and is minimized using particle swarm optimization with MATLAB lsqnonlin.Source: §10.3 · pp. 26–27
- Fitted equivalent-circuit parameters from operating trajectories are not necessarily equal to parameters from classical EIS at fixed operating points.The experiments assume approximately constant temperature; prior work found charge-transfer resistance at a given state of charge is smaller while charging than while resting.Source: §10.3 · p. 27
11. Conclusions & outlook
The conclusion argues that classical impedance is limited to small excitations at fixed operating points, while extended models yield higher-dimensional data across larger excitations and operating trajectories. It presents multisine excitation as a means to detect nonlinear and nonstationary behavior and extract impedance beyond those constraints.
11. Conclusions & outlook
- Classical EIS yields data only for small-amplitude excitations at fixed operating points, while remaining a tool for monitoring electrochemical systems.Source: §11 · p. 27
- Extended impedance models provide higher-dimensional data valid for larger excitation amplitudes and along operating trajectories, with potential applications including battery health forecasting, corrosion detection, and coating studies.The additional information may improve Li-ion battery health-forecasting accuracy.Source: §11 · p. 27
- Multisine excitation can reveal nonlinear and nonstationary behavior in measured current and voltage, from which higher-dimensional impedance data can still be extracted when linearity or stationarity constraints are not met.The figure illustrates equivalent-circuit branches and estimated parameters for the Samsung 48X cell.Source: §11 · pp. 27–28, Figure 17 · p. 28
Appendix B. Volterra series coefficients
The appendices describe linearizing a nonlinear time-invariant system around a slow operating trajectory with a fast perturbation, and identify rescaled Legendre polynomials as basis functions over a finite interval.
Appendix C. Linearising an NLTI system around an operating trajectory
- A slow excitation sets the operating trajectory, while a small fast perturbation is treated using only the first-order Volterra term; the resulting fast response uses a time-dependent two-dimensional impulse response.The slow response is the drift signal and depends only on the slow excitation.Source: Appendix C · pp. 28–29
Appendix E. Equivalence between (60) and (62)
- The text invokes Fourier-transform properties and states X(ω) = F{x(t)} for x = x, y, w, i, v (E.1a), specifying that F acts on t′.Fourier and inverse Fourier transforms act on t′ and ω′, respectively.Source: Appendix E · p. 29
Appendix F. Legendre polynomials
- The basis functions are rescaled Legendre polynomials defined over the interval [0, T].Source: Appendix F · p. 29