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Methods / 2026.07.24

Why EIS frequency and DRT relaxation time cannot be compared point by point

A DRT axis is not created by converting every frequency in an EIS dataset. Understanding it requires separating physical characteristic time from the numerical grid used by the solver.

Oparic Team3 min read

Summary

If the relaxation-time range in a DRT plot differs from the range you estimated from EIS frequencies, that difference is not automatically an error. The ideal-RC relation estimates the frequency associated with a peak; the DRT axis is a numerical grid used to solve for an entire distribution. Their endpoints serve different purposes.

1. How are frequency and characteristic time related?

For an ideal, single RC relaxation process, the response near the peak satisfies:

ωτ=1,ω=2πf\omega\tau=1,\qquad \omega=2\pi f

The characteristic time of the peak is therefore approximately:

τ≈12πf\tau\approx\frac{1}{2\pi f}

This relation identifies the frequency at which that process responds most strongly. A higher frequency represents a faster process and a smaller τ\tau; a lower frequency represents a slower process and a larger τ\tau.

It does not convert each measured frequency into one independent τ\tau sample. DRT uses the entire impedance spectrum to infer a continuous distribution. Real processes overlap, broaden, and depart from ideal single-RC behavior, so the equation is useful for estimating peak position—not for defining the DRT-axis endpoints.

2. How does Oparic choose the τ\tau range?

Oparic first establishes one numerical solver coordinate for each measured frequency:

τi=1fi\tau_i=\frac{1}{f_i}

Here, 1/f1/f is a numerical coordinate for the solver. It does not redefine the physical characteristic-frequency relation. The impedance calculation still uses ω=2πf\omega=2\pi f, and physical interpretation of a peak still uses f≈1/(2πτ)f\approx1/(2\pi\tau).

For measured bounds fmin⁡f_{\min} and fmax⁡f_{\max}, Oparic uses:

τgrid,min=10−1/2fmax⁡,τgrid,max=101/2fmin⁡\tau_{\mathrm{grid,min}}=\frac{10^{-1/2}}{f_{\max}},\qquad \tau_{\mathrm{grid,max}}=\frac{10^{1/2}}{f_{\min}}

The reciprocal-frequency range is extended by half a decade at each end, then evaluated on a denser grid. This displays the distribution edges but does not create new experimental information.

A frequent source of confusion

In the current drt_data.csv, freq_hz records the reciprocal of the time-grid coordinate, 1/τ1/\tau. It is not the physical characteristic frequency of an ideal RC peak, which is 1/(2πτ)1/(2\pi\tau).

3. How should peaks near a boundary be interpreted?

DRT infers a continuous distribution from finite, noisy impedance data. Extending the numerical grid does not add constraints, so peaks near either boundary require more validation than well-constrained interior peaks.

Very short τ\tau

Check cable and fixture inductance, phase error, and high-frequency instrument noise. Modeling a series inductance does not guarantee that every high-frequency artifact has been removed.

Very long τ\tau

Check whether the low-frequency response was measured fully and whether temperature, state of charge, or the system itself drifted during the long test. Non-stationary data may look like a slow process.

For every peak

Return to the Nyquist and Bode data, reconstructed impedance, and residuals; compare replicate measurements. If possible, extend the frequency range. A real, stable peak should not simply move with a new axis boundary.

Key point: the axis is the solver's working range, not a confidence guarantee. Interpretation depends on whether the peak is constrained by the data, reproducible, and consistent with the original EIS response and experimental mechanism.

4. Suggested review process

  1. Identify what you are comparing

    Use τ≈1/(2πf)\tau\approx1/(2\pi f) to estimate the characteristic time of an ideal RC peak, but do not substitute it for Oparic's output-grid bounds.

  2. Review raw data and reconstruction

    Inspect Nyquist/Bode curves for jumps, drift, or abnormal curvature; then examine reconstruction quality and structured residuals.

  3. Treat boundary peaks as provisional

    Before naming a peak at either extreme, check high-frequency parasitics and low-frequency non-stationarity.

  4. Check stability

    Compare replicates and, where possible, extend the frequency range. A peak that follows the numerical boundary is not yet well constrained.

  5. Assign a physical process last

    Make a physical assignment only after the peak is reproducible, the fit and residuals are reasonable, and the explanation matches the material and test conditions.