Second-Life EV Battery EIS: Nyquist Fit and Grading

The topology is $R_0 + [\mathrm{CPE}\parallel(R_{ct}+Z_W)]$, with the Warburg element inside the faradaic branch. With $\omega=2\pi f$:

The CPE is represented numerically by its admittance at $\omega_0=1\ \mathrm{rad/s}$, so its displayed unit is S. Conventional Q has the same SI numerical value but units S·sᵅ, which vary with α. The Warburg convention is σ in Ω·s⁻½; it differs by √2 from the coefficient of $1/\sqrt{j\omega}$. The reference input is the real Warburg resistance at ω=1 rad/s, in mΩ; its numeric value equals σ expressed in mΩ·s⁻½. This normalization keeps the complete editable reference value visible.

A three-start bounded Levenberg–Marquardt fit uses analytic complex derivatives, logarithmic positive parameters and 0.2 ≤ α ≤ 1. Each complex residual is divided by the measured |Z|; real and imaginary parts receive equal weight. Up to 65 iterations per start are used. The RMS is $\sqrt{\sum |(Z_{fit}-Z_{meas})/|Z_{meas}||^2/N}$. Local one-sigma uncertainties use the inverse of the unregularized weighted information matrix, with 2N−5 degrees of freedom. Optimizer damping is used only to take fit steps; it contributes no measurement information. The smallest/largest information eigenvalue ratio must exceed 10⁻¹² before covariance is evaluated. Rank-deficient covariance is unavailable rather than regularized into a finite uncertainty. Positive parameters within 0.1 log units of either numerical bound (−24 or +24), and α at its lower bound, are unresolved; α=1 remains the valid ideal-capacitor limit.

The diffusion-free arc frequency is $f_a=(R_{ct}Q)^{-1/\alpha}/(2\pi)$ and its equivalent capacitance is $1/(2\pi f_a R_{ct})$. These describe the isolated CPE arc; the actual full-spectrum peak can shift with diffusion. The diffusion indicator is σ, not a material diffusion constant D.

The largest of R₀/reference, Rct/reference and σ/reference selects the EIS tier. If capacity is supplied, the worse capacity tier governs. A hold takes precedence over any tier. Gamry’s application note (s4, pp. 6–7) supplies the circuit benchmarks; Rastegarpanah et al. (s3, pp. 1–2, 7–9) supports the use and limitations of EIS screening.

Small-signal, linear, stable measurements of one module design at fixed SOC and temperature; reproducible contacts and four-wire compensation. One depressed charge-transfer arc and semi-infinite diffusion describe the selected frequency window. Inductive points are excluded before fitting.

The secondary, ungraded tutorial measurements are the public impedance.py fitting sample, all 57 capacitive points, with unspecified cell provenance and conditions. They are not documented retired-EV measurements. The reference resistances, σ and grading limits are illustrative user policy, not published universal acceptance thresholds. No capacity SOH is inferred from EIS.

The matched-reference assertion defaults to false. Tutorial mode never yields an EV module grade, even if this box is checked. Select entered module measurements, provide a documented matched reference, and confirm matching to enable grading. The cited EV study identifies an open Nissan Leaf dataset (DOI 10.6084/m9.figshare.12227282); its raw files could not be retrieved here, so no matched EV baseline is claimed.

A storage candidate means the entered EIS policy is met; it does not establish fitness or certify a module. Confirm capacity, self-discharge, cell imbalance, insulation, thermal behavior and the intended current/duty cycle before reuse. Installation work belongs to a qualified person.

Use a matched reference with the same chemistry, series/parallel arrangement, SOC, temperature and contact method. No automatic temperature correction is applied. Poor residuals, a rank-deficient Jacobian, a fit at a numerical bound, unavailable or excessive parameter uncertainty, sensitivity inflation ≥10⁶, nonconvergence or missing arc/diffusion coverage withhold grading. Coverage requires at least a decade on either side of the diffusion-free arc frequency, σ/√ωmin ≥ 0.2 Rct, and measured real-impedance span and reactive magnitude each exceeding the entered response floor times max |Z|. The default response floor is 0.0001 (0.01%); set it using the measurement resolution. These are explicit screening heuristics, not universal identifiability guarantees.

Local uncertainty assumes independent relative measurement errors. Model mismatch and correlated errors can make it optimistic; inspect residual patterns even if the fit passes. A low residual does not prove the model is unique or satisfy a Kramers–Kronig validation. EIS data quality must be checked experimentally.

Finite diffusion, multiple arcs, SEI separation and lead inductance require an expanded circuit or a justified measurement window. This tool keeps the full stated Randles–CPE–semi-infinite-Warburg topology. It does not estimate D without electrode geometry and thermodynamic information. User entry supports 8–40 points because of the catalog’s input-field limit; retain measurements across the whole fitted window. All 57 sample points are fitted at defaults. The fitted curves use 360 directly evaluated frequencies without interpolation.

The cited study describes a 2011 Nissan Leaf pack (s3, p. 8), while the dataset’s public DataCite metadata describes a 2012 Leaf. Resolve that provenance discrepancy and verify test conditions before adopting a matched EV baseline.