Battery Impedance Spectroscopy: Randles Circuit, Nyquist Plot and Warburg Diffusion (2D Schematic)

Presets

Live readouts

Graphs

Nyquist plot (Re Z against -Im Z)

Bode plot: impedance magnitude

Bode plot: phase

Current and voltage waves at the present frequency

Diffusion length against frequency

Numerical checks

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How to use

Pick a preset and watch the analyser sweep from high to low frequency, building the Nyquist plot point by point. Try: (1) Ideal semicircle and read Rs and Rct off the plot; (2) Aged cell and Cold cell to see Rct grow; (3) Slow diffusion tail and the 45 degree Warburg line; (4) Depressed semicircle by lowering alpha; (5) Single 1 Hz measurement to watch the current and voltage waves, the phase lag and the ion swing; (6) raise the noise and the points scatter. Space pauses, R restarts the sweep, hover a part for its numbers.

The science

Method. In galvanostatic electrochemical impedance spectroscopy a small sinusoidal current I0 sin(wt) is driven through the cell at many frequencies and the voltage response is analysed with a lock-in (correlation with sin and cos). The complex impedance is Z(w) = V(w) / I(w); the amplitude is small enough that the cell behaves linearly.

Equivalent circuit (Randles with bounded diffusion).

Z = j w L + Rs + [ 1/Z_CPE + 1/(Rct + Z_W) ]^-1 Z_CPE = 1 / (Q (j w)^alpha) Z_W = Rd coth( sqrt(j w tau) ) / sqrt(j w tau), tau = L_d^2 / D

Rs collects electrolyte and contact resistance (left intercept of the Nyquist plot), the semicircle of diameter Rct is the charge-transfer process with its double-layer capacitance (apex at w = 1/(Rct Cdl)), and Z_W is solid-state diffusion: a 45 degree line at mid frequency that turns into a steep capacitive line when the particle fills (the cotangent form). The cable inductance bends the high-frequency end upward.

Temperature. Rct follows an Arrhenius law with 50 kJ/mol and the diffusion time with 25 kJ/mol, so a cold cell has a much bigger semicircle. Rs gets a small linear coefficient.

Ions. The diffusion length is sqrt(2 D / w), in units of the layer thickness sqrt(2 / (w tau)). The moving ions in the scene follow this: a small swing near the surface at high frequency, a swing that reaches deep into the electrode at low frequency.

Honest simplifications. Linear small-signal circuit with fixed element values; one lumped charge-transfer and one diffusion process per cell (real cells show separate anode and cathode arcs and SEI contributions), the sweep points are the analytic model plus Gaussian-like multiplicative noise rather than a full instrument simulation, and the electrode geometry is schematic. The check below does integrate the circuit in time.

Checks in the page. (1) a time-domain Runge-Kutta simulation of the circuit (Warburg as a 30-section Foster ladder) analysed with a lock-in agrees with the closed-form Z at four frequencies; (2) the high-frequency intercept equals Rs; (3) the semicircle apex has -Im Z = Rct/2 and Re Z = Rs + Rct/2; (4) the Kramers-Kronig integral over Im Z reproduces Re Z.