Curriculum · Module 3 · Intermediate

Electrochemical Impedance Spectroscopy (EIS)

The most searched electrochemistry topic: the principle of EIS, its representation, modelling and validation of data quality.

What is EIS? Impedance, frequency and Nyquist/Bode plots

Frequently searched:what is EISimpedance spectroscopyhow to read a Nyquist plotBode plot

1 Theory

Linear system theory

EIS applies a small-amplitude sinusoidal excitation (potential or current) to the system and measures the amplitude ratio and phase difference of the response at the same frequency. If the excitation is small enough (typically 5–10 mV) the system is considered linear and the response occurs only at the same frequency. The impedance Z(ω) = E(ω)/I(ω) is a complex number: |Z| and phase φ (polar) or Z′ and Z″ (Cartesian). It is Ohm's law generalised to the frequency domain: Z = R for a resistor, Z = 1/(jωC) for a capacitor, Z = jωL for an inductor.

Nyquist and Bode representations

The Nyquist plot draws −Z″ against Z′; each point is a frequency but the frequency does not appear on the axes. An R‖C circuit gives a semicircle; the apex frequency ω = 1/(RC) sets the characteristic time constant. The Bode plot draws |Z| and φ against log(f); the frequency information is explicit and small high-frequency features are better seen. Admittance (Y = 1/Z), capacitance (C = 1/(jωZ)) and modulus representations make some features more prominent depending on the type of system.

The impedance of an electrochemical system

The simplest interface model is the Randles circuit: electrolyte resistance R_s, charge-transfer resistance R_ct in parallel with the double-layer capacitance C_dl, and a series Warburg diffusion impedance W. In the Nyquist plot the R_s intercept is seen at high frequency, a semicircle of diameter R_ct at mid frequency and the 45° Warburg line at low frequency. In real systems the semicircles are depressed (CPE), several time constants overlap and inductive loops (adsorbed intermediates, cables) may appear.

Frequency range, amplitude and measurement time

A typical sweep is made between 100 kHz and 10 mHz with 6–10 points per decade; the lowest frequency sets the measurement time (a single period at 10 mHz is 100 s). Potentiostatic (PEIS) and galvanostatic (GEIS) modes are equivalent in most systems; GEIS is preferred on low-impedance systems with drifting OCP, such as batteries. Multisine and FFT methods shorten the low-frequency measurement time.

2 Application in measurement

In EIS a small-amplitude sinusoidal excitation is applied to the system and the frequency-dependent impedance Z(ω) is calculated from the amplitude/phase difference of the response. High frequencies reflect the solution/internal resistance and inductive effects, mid frequencies the interfacial processes (double layer, charge transfer), low frequencies diffusion and insertion.

The Nyquist plot (−Im Z vs Re Z) shows processes as semicircles and 45° lines; the Bode plot (|Z| and phase vs log f) preserves the frequency information. Both representations should be evaluated together.

Valid EIS requires three conditions: linearity (small amplitude), stationarity (the system must not change during the measurement) and causality. These conditions are tested with the Kramers-Kronig test and the EC-Lab® quality indicators (THD, NSD, NSR).

Key equations

Z(ω) = Z' + jZ'' = |Z|·e^(jφ); capacitor: Z = 1/(jωC); CPE: Z = 1/(Q(jω)^α)

3 Applicability

Where it applies

Ageing monitoring, coating evaluation, conductivity, separation of internal resistance/interface in fuel cells and batteries, sensor interface characterisation — non-destructive and in situ.

Limits and caveats

To be valid the system must be linear, stable and stationary throughout the measurement; during charge/discharge or on rapidly corroding surfaces the results may be unreliable.
A single spectrum has no unique model; physical interpretation requires independent evidence.

4 What did the application notes find?

Battery AN8 — Compares how the same data looks in impedance, admittance, Nyquist, Bode and Black representations; shows which representation reveals which feature (small high-frequency arcs, low-frequency diffusion) better.
Battery AN50-1 — Explains complex-number arithmetic from scratch and shows the hand derivation of the Nyquist/Bode plots of R, C, L and R‖C circuits — the mathematical prerequisite of EIS.
Battery AN50-2 — Shows that the Laplace transform is the bridge between the time domain (step, pulse) and the frequency domain (impedance), and that current interrupt and EIS are two views of the same transfer function.
Electrochemistry AN5 — Lists the set-up measures for reliable EIS — amplitude selection, cabling, reference impedance, Faraday cage — and the typical artefacts that appear when they are neglected.

5 Which instruments perform this measurement?

SP-300 MTZ-35

References

article A. Ch. Lazanas, M. I. Prodromidis. Electrochemical Impedance Spectroscopy — A Tutorial. ACS Meas. Sci. Au 3, 162–193, 2023. doi:10.1021/acsmeasuresciau.2c00070 open access
article S. Wang, J. Zhang, O. Gharbi, V. Vivier, M. Gao, M. E. Orazem. Electrochemical impedance spectroscopy. Nature Reviews Methods Primers 1, 41, 2021. doi:10.1038/s43586-021-00039-w
book M. E. Orazem, B. Tribollet. Electrochemical Impedance Spectroscopy (2nd ed.). Wiley, 2017.

Equivalent-circuit modelling: Randles circuit, CPE, Warburg and Z Fit

Frequently searched:equivalent circuitRandles circuitWarburg impedanceCPE constant phase elementZ Fit

1 Theory

Elements and their physical counterparts

Resistor: electrolyte, charge transfer, film resistance. Capacitor: double layer, coating/dielectric capacitance. Constant phase element (CPE, Q): Z = 1/[Q(jω)^α]; α = 1 ideal capacitor, α ≈ 0.5 diffusion-like, 0.8–0.95 rough/heterogeneous surface. Warburg (W): semi-infinite diffusion, 45° line. Finite-length Warburg (W_δ, 'short' and 'open'): diffusion within a layer of limited thickness — at low frequency it either returns to the real axis or rises capacitively. Gerischer: homogeneous reaction coupled with diffusion. Inductance: cables and adsorbed intermediates (low-frequency inductive loop).

Model selection and the equivalence problem

Different circuits can reproduce exactly the same spectrum (e.g. Voigt–Maxwell equivalence); a good fit alone therefore does not prove the physical model correct. The right route is to build the model from physical prior knowledge (layers, processes, time constants), to fit with the fewest possible elements and to show that the parameters change with potential, temperature or time in the expected way. Weighting (modulus-weighted least squares), initial values and error margins (%) must be reported.

From CPE to effective capacitance

The CPE parameter Q is not a capacitance (its unit is F·s^(α−1)). The effective capacitance is calculated with the Brug (surface distribution) and Hsu–Mansfeld (normal distribution) formulas; coating thickness or double-layer capacitance can be interpreted quantitatively only after this conversion.

Distribution of relaxation times (DRT)

DRT converts the spectrum into a continuous distribution of time constants without assuming a model; it separates overlapping processes as peaks and shows how many time constants the equivalent circuit requires. The regularisation parameter affects the result; DRT does not replace model building, it guides it.

2 Application in measurement

Impedance data is modelled with elements that have physical meaning (R, C, L, CPE, Warburg, transmission line). The Randles circuit (Ru + Cdl ∥ (Rct + W)) is the starting point for single-interface systems. Surface heterogeneity depresses the semicircle; a CPE is then used and the effective capacitance is calculated with the Brug formula.

Model selection must rest on knowledge of the system; a 'good fit' does not prove that the model is correct. Keep the number of parameters small, give initial values according to physical expectation, compare different models by χ² and parameter errors.

EC-Lab® Z Fit offers more than 150 ready circuits and batch fitting; DRT (distribution of relaxation times) is a complementary tool for separating time constants without assuming a model.

Key equations

Randles: Z = Ru + 1/(jωCdl + 1/(Rct + Zw)); Warburg: Zw = σ·ω^(-1/2)·(1 − j)

3 Applicability

Where it applies

Resolving layered systems (coating, SEI, solid-electrolyte interface); tracking resistance/capacitance components with ageing; diffusion-coefficient determination.

Limits and caveats

Fitting non-stationary or non-linear spectra is meaningless — data quality must be verified first.
Circuits with many elements fit almost any data; if the parameter uncertainty is high the model is over-complex.

4 What did the application notes find?

Battery AN14 — Shows the equivalent-circuit fitting process with the Z Fit tool, the physical meaning of the elements and the existence of mutually equivalent circuits on example data; stresses the necessity of physical prior knowledge in model selection.
Corrosion AN20 — Shows fitting of the depressed semicircle with R‖Q and calculation of the 'pseudocapacitance' from the Q–α pair; explains that reporting Q as a capacitance without this conversion is wrong.
Battery AN42 — Introduces the modified inductance element L_a for inductive loops; shows that distorted loops on the real axis can be fitted by applying the CPE logic to inductance.
Battery AN43 — Shows fitting of porous electrodes with the transmission-line model; explains that the Warburg is in fact a transmission line and that the in-pore resistance/capacitance distribution can be represented by a single element.
Corrosion AN45 — Shows batch fitting of many spectra taken as a function of potential or temperature with the same circuit, and automatic plotting of the parameters against the variable.
Battery AN60 — Shows the principle of DRT and how the overlapping SEI, charge-transfer and diffusion processes in a battery spectrum separate as distinct peaks; discusses the effect of regularisation.

5 Which instruments perform this measurement?

EC-Lab®

References

article A. Ch. Lazanas, M. I. Prodromidis. Electrochemical Impedance Spectroscopy — A Tutorial. ACS Meas. Sci. Au 3, 162–193, 2023. doi:10.1021/acsmeasuresciau.2c00070 open access
book M. E. Orazem, B. Tribollet. Electrochemical Impedance Spectroscopy (2nd ed.). Wiley, 2017.
article T. H. Wan, M. Saccoccio, C. Chen, F. Ciucci. Influence of the Discretization Methods on the Distribution of Relaxation Times Deconvolution: Implementing Radial Basis Functions with DRTtools. Electrochimica Acta 184, 483–499, 2015. doi:10.1016/j.electacta.2015.09.097

Can I trust my EIS data? Kramers-Kronig, THD/NSD/NSR and artefacts

Frequently searched:Kramers-Kronig testEIS validationEIS artefactEIS amplitude selection

1 Theory

The four conditions

For a valid impedance measurement the system must be causal (the response follows the excitation), linear (the response is proportional to the excitation and only at the same frequency), stable (it returns to its former state when the excitation is removed) and time-invariant (stationary during the measurement). Electrochemical systems are non-linear by nature (Butler–Volmer is exponential) and usually change slowly (corrosion, ageing, OCP drift); the validity of every spectrum must therefore be tested.

Kramers–Kronig (KK) relations

In a system satisfying the above conditions Z′ and Z″ are not independent; one can be calculated from the other by an integral transform. If the measured spectrum is not KK-compliant at least one of the conditions has been violated. The practical application is to fit with a 'measurement model' (many R‖C Voigt elements) instead of the direct integral; the residuals are plotted against frequency and regions showing systematic deviation are considered unreliable.

Harmonic analysis: THD, NSD, NSR

Violation of linearity produces harmonics in the response; the total harmonic distortion (THD) is the ratio of the harmonics to the fundamental and 5 % is the practical threshold — if larger, the amplitude must be reduced. Violation of stationarity (drift) creates sidebands around the fundamental frequency; the non-stationary distortion (NSD) measures this — if larger, the wait time must be extended. The noise-to-signal ratio (NSR) quantifies measurement noise — if larger, the amplitude must be increased or averaging applied. Since these three indicators can be calculated separately at every frequency point, they show which region of the spectrum is corrupted and why.

Instrument and set-up artefacts

At high frequency: cable inductance, the reference-impedance–input-capacitance filter, discontinuities at current-range changes. At low frequency: drift, temperature change, measurement time. Remedies: four-terminal connection, short cables, low-impedance reference, 'dual reference', suitable current range, checking that the accuracy contour plot covers the measurement region.

2 Application in measurement

The Kramers-Kronig relations test whether the real and imaginary parts can be derived from each other; a violation shows that one of the linearity/stationarity/causality conditions is broken. Bio-Logic's THD (linearity), NSD (stationarity) and NSR (noise) indicators quantify this check at every frequency: high THD → reduce the amplitude; high NSD → extend the wait time or use drift correction; high NSR → increase the amplitude.

Artefacts: high-frequency arcs from cable inductance and reference-electrode impedance, stray capacitance, measurement in regions outside the instrument's accuracy contour plot. On low-impedance batteries 4-terminal connection and galvano-EIS, on high-impedance coatings ULC and shielding are essential.

When comparing instruments ask for the accuracy contour plot: the statement 'EIS up to 7 MHz' does not say with which error band and in which impedance range the measurement is made.

3 Applicability

Where it applies

Every EIS measurement — documenting spectrum validity for publication and quality control.
Selection of measurement parameters (amplitude, wait time, frequency limit) on time-varying systems such as batteries and corrosion.

Limits and caveats

The KK test does not say the cause of the violation; THD/NSD/NSR are more informative in distinguishing the cause.
At very low frequencies (< 1 mHz) no method can completely exclude drift; the measurement time must be shortened or time-varying EIS methods used.

4 What did the application notes find?

Battery AN15 — Discusses when KK transforms can be applied and the limitation imposed by the finite frequency range; shows how measurement compliance is tested with the KK tool in the software.
Corrosion AN64 — Defines the THD, NSD and NSR indicators; shows with examples which violation each corresponds to and how the measurement parameters are corrected according to these indicators.
Corrosion AN65 — Examines the dependence of THD on amplitude, frequency, harmonic count and system type; shows by comparison with Lissajous and KK that the 5 % threshold is only a guideline and must be evaluated per system.
Battery AN9 — Compares the EIS response of linear and non-linear systems; shows experimentally how harmonics and 'wrong' impedance appear as the amplitude increases.
Battery AN44 — Generates the high-frequency artefacts of reference impedance and instrument parasitics by simulation; shows which set-up change removes which artefact.
Battery AN54 — Explains reading the accuracy contour plot and determining into which error band of the plot the measurement falls.
Corrosion AN69-1 — Shows how EIS is tested and corrected on non-stationary systems (e.g. a surface undergoing corrosion); extraction of drift with the NSD indicator and successive measurements.
Battery AN49 — Shows that potentiostatic and galvanostatic EIS are equivalent in most cases, but that GEIS is the right choice on low-impedance batteries with drifting OCP and PEIS on high-impedance systems.

References

article B. A. Boukamp. A Linear Kronig-Kramers Transform Test for Immittance Data Validation. J. Electrochem. Soc. 142, 1885, 1995. doi:10.1149/1.2044210
article A. Ch. Lazanas, M. I. Prodromidis. Electrochemical Impedance Spectroscopy — A Tutorial. ACS Meas. Sci. Au 3, 162–193, 2023. doi:10.1021/acsmeasuresciau.2c00070 open access
article S. Wang, J. Zhang, O. Gharbi, V. Vivier, M. Gao, M. E. Orazem. Electrochemical impedance spectroscopy. Nature Reviews Methods Primers 1, 41, 2021. doi:10.1038/s43586-021-00039-w

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