Impedance Spectroscopy (EIS) Parameter Calculator
Find the real and imaginary impedance, magnitude, phase angle, and characteristic frequency of a Randles equivalent circuit for EIS analysis.
🔬 What is EIS Parameter Calculation?
Electrochemical impedance spectroscopy (EIS) measures how an electrochemical cell responds to a small AC voltage across a range of frequencies, and this calculator computes the impedance components predicted by a Randles equivalent circuit, a solution resistance Rs in series with a parallel charge-transfer resistance Rct and double-layer capacitance Cdl.
Electrochemists use this constantly, to predict the shape of a Nyquist plot before running an experiment, to sanity-check fitted circuit parameters against raw impedance data, to estimate the characteristic frequency where the semicircle peaks, and to teach students how Rs, Rct, and Cdl each shape the impedance response.
A common misconception is that impedance is just resistance measured with AC. It isn't, impedance has both a real part (resistive) and an imaginary part (reactive, from the capacitor), and both parts change with frequency, which is exactly what the Nyquist plot visualizes as a semicircle rather than a single point.
This calculator computes the real part Z', the negative imaginary part -Z'', the magnitude |Z|, the phase angle, and the characteristic frequency fc from your Rs, Rct, Cdl, and frequency inputs, and plots the full impedance locus so you can see where your point falls on the semicircle.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Mid-Frequency Point on a Moderate Semicircle
Rs=10 Ω, Rct=200 Ω, Cdl=20 µF, f=100 Hz
Example 2 — High-Frequency Point, Nearly Resistive
Rs=25 Ω, Rct=500 Ω, Cdl=10 µF, f=1000 Hz
Example 3 — Low-Frequency Point Near the Semicircle's Far End
Rs=5 Ω, Rct=1000 Ω, Cdl=50 µF, f=10 Hz
❓ Frequently Asked Questions
🔗 Related Calculators
What is electrochemical impedance spectroscopy (EIS)?
EIS applies a small AC voltage across an electrochemical cell over a range of frequencies and measures the resulting current to compute impedance at each frequency. Plotting the real part Z' against the negative imaginary part -Z'' produces a Nyquist plot that reveals circuit elements like solution resistance, charge-transfer resistance, and double-layer capacitance.
What is a Randles equivalent circuit?
The Randles circuit is the simplest useful model of an electrochemical interface: a solution resistance Rs in series with a parallel Rct-Cdl combination, where Rct is the charge-transfer resistance and Cdl is the double-layer capacitance. It produces a characteristic semicircle in the Nyquist plot.
How is impedance calculated from Rs, Rct, and Cdl?
Z' = Rs + Rct / (1 + (omega*Rct*Cdl)^2) and Z'' = -omega*Rct^2*Cdl / (1 + (omega*Rct*Cdl)^2), where omega = 2*pi*f is the angular frequency. These come from combining Rct and Cdl in parallel using complex impedance algebra, then adding the series Rs.
Why does the Nyquist plot look like a semicircle?
As frequency sweeps from very high to very low, the Rct-Cdl parallel combination's impedance traces a circular arc in the complex plane, starting near Rs at high frequency and ending near Rs+Rct at low frequency. The peak of the arc occurs at the characteristic frequency fc = 1/(2*pi*Rct*Cdl).
What does the phase angle tell you?
The phase angle is the angle between the current and voltage waveforms. A phase angle near 0 degrees means the interface behaves resistively, while a phase angle approaching -90 degrees means it behaves almost purely capacitively, at the characteristic frequency the phase angle reaches its most negative value for this circuit.
How do I find the characteristic frequency from a measured Nyquist plot?
The characteristic frequency is the frequency at the top of the semicircle, where -Z'' is at its maximum. Algebraically fc = 1/(2*pi*Rct*Cdl), so once you read Rct and Cdl off the semicircle's diameter and peak height, you can compute fc directly.
What are the units for Rs, Rct, and Cdl in this calculator?
Rs and Rct are entered in ohms, Cdl is entered in microfarads (µF), and frequency is entered in hertz (Hz). Internally, Cdl is converted to farads before applying the Randles circuit formulas, so all outputs come out in ohms and hertz.
Does this calculator include the Warburg diffusion element?
No, this calculator uses the basic Randles circuit without a Warburg element, which models pure charge-transfer kinetics. Real cells with significant mass-transport limitations show an additional 45-degree line at low frequency that this simplified model does not reproduce.
Why does the impedance approach Rs alone at high frequency?
At high frequency, the double-layer capacitor's reactance becomes very small, so it acts almost like a short circuit across Rct. Current flows through the capacitor rather than Rct, so the total impedance drops toward just the series resistance Rs.
Can this calculator be used for battery or supercapacitor EIS analysis?
Yes, the same Randles circuit is a standard first-pass model for batteries and supercapacitors near a single frequency region, though real devices often need extra elements (Warburg, constant-phase elements) for a full fit. This calculator gives the basic Rs, Rct, Cdl relationship as a starting point.
What does a small semicircle diameter indicate about a cell?
A small semicircle diameter (small Rct) indicates fast charge-transfer kinetics at that interface, meaning the electrochemical reaction proceeds easily. A large semicircle diameter indicates sluggish kinetics, often from a passivating film, poor catalyst activity, or an inherently slow reaction mechanism.