# 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).** 1. **Validate the data first** (Lin-KK). A bad spectrum cannot be rescued by any circuit. 2. **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. 3. **Use a CPE**, not an ideal capacitor, for any depressed/flattened arc. 4. **Add a Warburg only** if there is a low-frequency ~45° tail. 5. **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. 6. **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. 7. **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) 1. **A. Lasia, *Electrochemical Impedance Spectroscopy and its Applications*, Springer.** The standard teaching text — start here for EIS. 2. **M. E. Orazem & B. Tribollet, *Electrochemical Impedance Spectroscopy*, 2nd ed., Wiley.** Rigorous treatment, including Kramers–Kronig. 3. **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. 4. **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. 5. **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**. 6. **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 7. 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.