EIS & DRT — a simple guide: measure, analyse, plot, and report

EIS (electrochemical impedance spectroscopy) applies a tiny AC perturbation across a range of frequencies and measures the impedance Z(ω) = Z′ − jZ″. Because different processes respond on different timescales, EIS separates them by frequency: series resistance, film/SEI, charge transfer, and diffusion each live in a different part of the spectrum.

DRT (distribution of relaxation times) is a model-free transform of the same EIS data into a distribution γ(τ) of relaxation times. Overlapping arcs that blur into one semicircle on a Nyquist plot become separate peaks on a log τ axis — so you can count and time-resolve the processes without guessing an equivalent circuit first.

One-line warning: EIS and DRT are only as good as the data. Use a small amplitude, a stable (non-drifting) cell, and validate with Kramers–Kronig before you interpret anything. DRT amplifies noise and its peaks depend on a regularisation parameter — always report it.

EIS

What it tells you (by frequency, high → low)

R_s / R_ohmic — high-frequency real-axis intercept: electrolyte + contacts + wiring.
Film / SEI semicircle — high-to-mid frequency (if present).
Charge transfer, R_ct — mid-frequency semicircle; its diameter = R_ct, with a double-layer capacitance C_dl (use a CPE for a depressed arc).
Diffusion / Warburg — low-frequency ~45° line (finite-length diffusion bends to a capacitive tail).

How to run it (design)

Small amplitude so the response is linear: ~5–10 mV (potentiostatic) around a defined DC point, or a current amplitude that keeps the voltage response ≲ 10 mV.
At a defined, equilibrated OCV / SOC — let the cell relax first.
Wide frequency range (e.g. ~100 kHz–10 mHz), enough points per decade (≥ 6–10).
State temperature, and whether it is 2-electrode (whole cell) or 3-electrode / symmetric cell (to isolate one electrode).

Validate the data first — Kramers–Kronig (KK)

KK holds only if the system is linear, causal, and stable (time-invariant). Run a Lin-KK test; if the residuals are small and unstructured, the spectrum is trustworthy. Large or systematic residuals at low frequency usually mean the SOC drifted during the sweep — shorten the sweep, increase amplitude only within the linear limit, or measure faster.

How to analyse / calculate

Read directly: R_s = high-f intercept; R_ct = semicircle diameter; the apex frequency gives the time constant τ = 1/(2πf_apex) and C_dl = 1/(2π f_apex R_ct).
Equivalent-circuit model (ECM) fit: report the circuit you used, the fitted values with uncertainties, and the goodness of fit (χ²). Use CPEs for depressed arcs.
Non-uniqueness: many circuits fit the same data. Keep the circuit as simple as the data justify, and support each element with physical reasoning (or with DRT — next section).

How to plot

Nyquist (−Z″ vs Z′) with equal (1:1) axes — this is essential; unequal axes turn circles into ellipses and mislead the reader. Mark the frequency direction and a few decade points.
Bode (|Z| and phase vs log f) alongside it — Nyquist hides the frequency axis; Bode shows every frequency and reveals features a Nyquist plot buries.
Overlay the fit on the data and show the residuals.
Normalise by electrode area (Ω·cm²) whenever you compare samples.

Cautions

Amplitude too large → non-linear, KK fails. · Cell drifting → low-frequency distortion. · High-frequency inductive tail from cables (below the real axis) — keep leads short, don’t mistake it for a process. · Never publish a squashed (unequal-axis) Nyquist. · One semicircle is not automatically one process.

DRT

What it adds

Z(ω) = R_∞ + R_pol · ∫ γ(τ) / (1 + jωτ) dτ, with ∫ γ dτ = 1. Each peak in γ(τ) is a process; its position is the time constant τ (f = 1/(2πτ)) and the area under the peak is that process’s resistance. DRT resolves overlapping arcs far better than a Nyquist plot and needs no assumed circuit.

How to compute (and the one knob that matters)

Use a tool: DRTtools (MATLAB) or pyDRTtools (Python). The script drt_pyDRTtools.py (in the downloads) is a ready pipeline: load EIS → compute the DRT with pyDRTtools, λ chosen by the L-curve → find the peaks (τ, f = 1/2πτ, and R = peak area) → plot. Install with pip install pyDRTtools cvxopt scikit-learn.
The inversion is ill-posed, so it uses Tikhonov regularisation with a parameter λ:
- λ too small → high resolution but noise-amplified, spurious peaks.
- λ too large → oversmoothed, peaks merge and disappear.
- Choose λ objectively with the L-curve, generalised cross-validation (GCV), or the discrepancy principle — and report the value and the method.
Feed it KK-valid, low-noise data; handle the high-frequency inductance (crop or fit it); keep the same frequency range for every spectrum you compare.

How to interpret & plot

Plot γ(τ) vs log τ (or vs log f). Label each peak with its assigned process and give λ + the tool in the caption.
Assign peaks with evidence, not by eye: watch how a peak moves with temperature (activation energy), SOC, or in a symmetric cell. A peak is a time constant, not automatically a single physical step.
Report each process’s R (peak area) and τ (peak position); these feed a physically grounded ECM.

Cautions

Results depend on λ — show a brief sensitivity (2–3 λ values) or state how λ was chosen. · Noise → fake peaks. · Don’t over-interpret tiny peaks. · Same λ and same frequency range across all samples, or the comparison is meaningless.

Worked example

From a spectrum with R_s ≈ 8 Ω (high-frequency intercept) whose DRT resolves two peaks — τ ≈ 2 ms, R ≈ 25 Ω and τ ≈ 50 ms, R ≈ 60 Ω — you get R_ct,total ≈ 85 Ω, and each process’s capacitance from C = τ/R (≈ 80 µF and ≈ 830 µF). The low-frequency 45° tail is diffusion. The script eis.py (in the Resources downloads) draws the correct Nyquist (equal axes) + Bode, reads R_s / R_ct, and shows an illustrative DRT; use pyDRTtools for the real DRT on measured data and report λ.

Equivalent-circuit fitting (ECM) — and choosing the model

Quantitative EIS means fitting a circuit and reporting its elements. The script eis_fit.py (in the downloads) fits Rs–(R∥CPE)–…–Warburg circuits by complex non-linear least squares and returns each value ± uncertainty and a reduced χ², with a data-vs-fit + residual plot.

Elements. R resistance; CPE (constant-phase element) Z = 1/(Q(jω)^a) — a=1 is an ideal capacitor, a<1 a depressed arc; Warburg Z = Aw(1−j)/√ω for semi-infinite diffusion.

Common circuits.

Rs + (R1∥CPE1) — one arc, no diffusion.
Rs + (R1∥CPE1) + W (Randles) — one arc + a diffusion tail.
Rs + (R1∥CPE1) + (R2∥CPE2) [+ W] — film/SEI + charge transfer (+ diffusion).

How to choose it (in order).

Validate the data first (Lin-KK). A bad spectrum cannot be rescued by any circuit.
Let the physics set the number of elements — run the DRT; the number of peaks is the number of R∥CPE arcs you need. Don’t add elements the data can’t resolve.
Use a CPE, not an ideal capacitor, for any depressed/flattened arc.
Add a Warburg only if there is a low-frequency ~45° tail.
Keep it as simple as the data justify (Occam). More elements always lower χ², so watch for the signs of over-fitting: huge parameter uncertainties, parameters that change with the initial guess, or physically impossible values.
Judge by the residuals — they should be small and unstructured vs frequency. A rising or oscillating residual means the circuit is missing something. χ² alone is not enough.
Report the circuit drawn out, every value ± uncertainty, and χ². Tie each element to a physical process — a good fit to the wrong circuit is still the wrong circuit.

The fuller community toolbox (built-in circuits, a string syntax like R0-p(R1,CPE1)-Wo1) is impedance.py (pip install impedance).

Running replicates or several samples — how to report

Fix everything across cells/samples: SOC, temperature, amplitude, frequency range, equilibration time, and (for DRT) λ and range. Otherwise the spectra are not comparable.
Normalise to area (Ω·cm²) so different electrodes can be compared.
Do not just average raw spectra. Instead: fit each replicate, then report the parameters (R_s, R_ct, C_dl, …) as mean ± SD with n. Overlay all spectra as thin lines with a representative (median) one bold.
DRT across replicates/samples: compute each with the same λ and range, overlay the γ(τ) curves, and report peak τ and peak area (R) as mean ± SD.
Account for every cell — how many measured vs included, and why any were excluded.
State n, the frequency range, amplitude, temperature, and SOC in every caption.

References to read (start with 1–2)

A. Lasia, Electrochemical Impedance Spectroscopy and its Applications, Springer. The standard teaching text — start here for EIS.
M. E. Orazem & B. Tribollet, Electrochemical Impedance Spectroscopy, 2nd ed., Wiley. Rigorous treatment, including Kramers–Kronig.
N. Meddings et al., “Application of EIS to commercial Li-ion cells: A review,” J. Power Sources 480, 228742 (2020). Battery-specific practice and pitfalls.
M. Schönleber, D. Klotz, E. Ivers-Tiffée, “A method for improving the robustness of linear Kramers–Kronig validity tests,” Electrochim. Acta 131, 20 (2014). The Lin-KK data-quality test.
T. H. Wan, M. Saccoccio, C. Chen, F. Ciucci, “Influence of the discretization methods on the distribution of relaxation times deconvolution,” Electrochim. Acta 184, 483 (2015) — the theory behind DRTtools / pyDRTtools.
M. Hahn et al., “Computation of DRT by Tikhonov regularization for Li-ion batteries: usage of the L-curve method,” Sci. Rep. 11, 13285 (2021). Choosing λ. https://doi.org/10.1038/s41598-021-91871-3
A recent review: “Distribution of relaxation times: foundations, methods, diagnostics, and prognosis for electrochemical systems,” J. Power Sources / Electrochim. Acta (2025).

Software: DRTtools (MATLAB) and pyDRTtools (Python) — both open source, from the Ciucci group.