EIS fitter: Nyquist and Bode plots with equivalent circuit fitting (Randles, CPE, Warburg)

What it does

Electrochemical impedance spectroscopy (EIS) applies a small AC signal over a range of frequencies and records the complex impedance Z = Z′ + jZ″. The result is plotted two ways. A Nyquist plot shows −Z″ against Z′, where each process makes a semicircle. A Bode plot shows |Z| and the phase against frequency. To put numbers on the processes, you fit an equivalent circuit.

This tool fits five common circuits:

• R(RC): solution resistance Rs in series with a charge-transfer resistance Rct in parallel with the double-layer capacitance C. It gives one ideal semicircle.
• R(RQ): the same circuit with a constant phase element (CPE) in place of the capacitor. A CPE has Z = 1/(Q(jω)^α) and gives the depressed semicircle seen on real, rough electrodes.
• Randles, R(Q(RW)): adds a Warburg element in series with Rct, for semi-infinite diffusion. The semicircle is followed by a 45° line at low frequency.
• R(RQ)(RQ): two arcs, for example a film or SEI layer plus charge transfer in a battery. The faster arc is listed first.
• RQ: a blocking electrode or an intact coating, with no charge transfer.

The fit is complex non-linear least squares weighted by |Z|, so each frequency counts by its relative error. It starts from several guesses read off the spectrum. You get:

• every value with its standard error
• the rms error of the fit
• the effective capacitance of each CPE (Hsu–Mansfeld)
• each arc’s time constant and peak frequency
• Rct × area and the exchange current density i₀ = RT/(nF·Rct·A)
• a table that fits every circuit to your data, so you can compare them

How to use it

1. Paste three columns or open a CSV, TSV or Excel file: frequency (Hz), Z′ and Z″ in ohms. If your file has |Z| and phase instead, choose that format.
2. The tool detects whether the third column holds Z″ or −Z″. If the arc comes out upside down, set the sign yourself.
3. Choose a circuit. Check the comparison table, and prefer the simplest circuit that fits within the noise with every value well defined.
4. Enter the electrode area to get capacitance per cm², Rct × area and i₀.

Worked example

The example is a simulated Randles cell: Rs 20 Ω, Rct 100 Ω, Q 20 µF·s^(α−1) with α = 0.9, σ = 30 Ω·s^−½, and 0.4 % noise, measured from 100 kHz down to 30 mHz. The Randles fit returns Rs 19.98 Ω, Rct 100.1 Ω, Q 20.04 µF·s^(α−1), α 0.900 and σ 29.99, with a 0.08 % rms error. For a 3 mm disc (0.0707 cm²):

• the effective capacitance is 10.02 µF (142 µF/cm²)
• the time constant is 1.0 ms, so the arc peaks near 159 Hz
• i₀ is 3.6 mA/cm²

The one-arc circuits miss by about 10 %, because they cannot follow the diffusion tail. The two-arc circuit runs one resistance off to infinity, so it is flagged as not pinned down.

Questions

Why is my α below 0.8? Rough or porous electrodes, uneven current distribution or a distribution of time constants all lower α. Below about 0.5, the element behaves more like diffusion than a capacitor.

Some high-frequency points sit below the axis. At high frequency, cable and cell inductance makes Z″ positive, so −Z″ dips below zero. Leave those points out, because none of these circuits contains an inductor.

Is anything sent anywhere? No. Your data stays in your browser, and the page works offline once loaded.

Last updated 2026-11-08.