Calculator inputs
Use a single point or paste a full impedance sweep.
Example data table
Use the following example values for a quick sweep import or to test normalization outputs.
| Frequency (Hz) | Z′ (Ω) | Z″ (Ω) | Area (cm²) | Mass (g) | Volume (cm³) |
|---|---|---|---|---|---|
| 10 | 2.8 | -18.2 | 1.00 | 0.005 | 0.020 |
| 100 | 1.9 | -11.4 | 1.00 | 0.005 | 0.020 |
| 1000 | 1.2 | -6.5 | 1.00 | 0.005 | 0.020 |
| 5000 | 0.9 | -2.3 | 1.00 | 0.005 | 0.020 |
Formula used
Angular frequency: ω = 2πf
Impedance magnitude squared: |Z|² = Z′² + Z″²
Complex capacitance real part: C′ = -Z″ / (ω|Z|²)
Complex capacitance loss part: C″ = Z′ / (ω|Z|²)
Ideal capacitor approximation: C = -1 / (ωZ″)
This approximation is most reliable when the phase angle is close to -90°.
Loss tangent: tanδ = C″ / C′
Relaxation time: τ₀ = 1 / f₀
Areal capacitance: CA = C / A
Gravimetric capacitance: Cg = C / m
Volumetric capacitance: Cv = C / V
How to use this calculator
- Choose single-point mode for one frequency or sweep mode for many frequencies.
- Select the preferred reporting method. C′ is usually the better spectral metric.
- For single-point mode, enter either Z′ and Z″ or |Z| with phase angle.
- Optionally enter area, mass, and volume to obtain normalized capacitance values.
- Optionally enter a characteristic frequency if you already know the relaxation feature.
- Press the calculate button to show the result above the form and the chart below the summary.
- Use the CSV and PDF buttons to export the dataset or report after calculation.
FAQs
1) What does EIS-derived capacitance represent?
EIS-derived capacitance describes how much charge the interface stores at a chosen frequency. It can reflect double-layer storage, pseudocapacitance, porous transport effects, and non-ideal behavior rather than one fixed capacitor value.
2) Why are both C′ and C″ reported?
C′ estimates stored energy, while C″ tracks dissipative losses and relaxation. Reading both together helps separate efficient charge storage from resistive, diffusion-limited, or kinetically delayed behavior.
3) Should Z″ usually be negative?
For capacitive systems, Z″ is usually negative with the common sign convention. A positive value often indicates inductive behavior, wiring artifacts, or a sign convention mismatch in the exported EIS data.
4) Which capacitance should I report in a paper?
Use C′ for frequency-dependent complex capacitance analysis. Use the ideal approximation only when the phase angle is near −90° and the response behaves like a nearly ideal capacitor.
5) Why does capacitance change with frequency?
Porous electrodes, ion transport limits, surface heterogeneity, and Faradaic reactions can all make capacitance frequency dependent. Low frequencies often reveal more accessible storage, while high frequencies emphasize faster processes.
6) What does the C″ peak mean?
The C″ peak frequency is often used to estimate characteristic relaxation time. A lower peak frequency generally means slower charge redistribution inside the electrochemical system.
7) Can I paste a full impedance sweep?
Yes. Enter one line per point in the sweep box using frequency, Z′, and Z″. The calculator computes each row, summarizes key values, and plots the spectral trend.
8) Why are some outputs blank or negative?
Blank results usually come from zero frequency, missing impedance terms, or invalid rows. Negative capacitance can appear when the sign convention is reversed or the data are not capacitively dominated.