Warburg Impedance Calculator

The Warburg impedance models the extra impedance an electrochemical interface shows because reactant ions or molecules take time to diffuse to the electrode surface. For semi-infinite linear diffusion it is Zw = sigma * omega^(-1/2) * (1 - j), where sigma is the Warburg coefficient and omega is the angular frequency.

Because the real and imaginary parts of Zw are exactly equal at every frequency, sigma/sqrt(omega) for both, the ratio between them, and therefore the phase angle, stays fixed at -45 degrees regardless of sigma or frequency. This constant 45-degree line is how a Warburg element is recognized on a Nyquist plot.

Sigma quantifies how strongly diffusion limits the impedance response. It depends on the diffusion coefficients and bulk concentrations of the oxidized and reduced species, the number of electrons transferred, and the electrode area, though in practice it is usually extracted by fitting measured impedance data rather than computed from those quantities directly.

That calculator models the basic Randles circuit, solution resistance plus a parallel charge-transfer resistance and double-layer capacitance, which fits high and mid-frequency data well but ignores mass-transport limitations. This Warburg Impedance Calculator covers exactly the diffusion element that basic model leaves out.

Both the real and imaginary parts scale as 1/sqrt(f), so the Warburg impedance grows without bound as frequency drops toward zero and shrinks toward zero as frequency rises. This is why diffusion effects dominate the low-frequency tail of a Nyquist plot and disappear at high frequency.

Sigma is typically reported in ohms times seconds to the negative one-half power (ohm s^-0.5), sometimes written as ohm Hz^0.5 depending on convention. This calculator expects sigma in ohm s^-0.5 and frequency in hertz.

Yes, a finite-length Warburg element (used for thin films or bounded diffusion) adds a hyperbolic tangent correction term that makes the impedance level off at low frequency instead of growing indefinitely. This calculator implements the simpler and more common semi-infinite case, valid when the diffusion layer never reaches a physical boundary within the timescale of the measurement.

A large Warburg coefficient, and therefore a large Warburg impedance at a given frequency, points to slow mass transport, low reactant concentration, a small diffusion coefficient, or a restricted diffusion path. In batteries this often shows up as poor rate capability, the cell struggles to deliver current quickly because ions cannot reach the electrode surface fast enough.

In a full Randles-with-Warburg model, the Warburg element sits in series with the charge-transfer resistance, and that series combination sits in parallel with the double-layer capacitance, with the solution resistance in series with the whole thing. Fitting software finds Rs, Rct, Cdl, and sigma together from the complete measured Nyquist plot, this calculator lets you evaluate the Warburg piece on its own once you know sigma.

A Warburg element has a fixed 45-degree phase angle at every frequency, reflecting ideal semi-infinite diffusion. A constant phase element generalizes this with an adjustable exponent, letting the phase angle sit anywhere between 0 and 90 degrees, and is used to fit non-ideal, rough, or heterogeneous electrode surfaces that a pure Warburg element cannot capture exactly.