The Bode plot and the Randles circuit — the rest of an EIS analysis
The Nyquist plot is where EIS starts, but it hides frequency and can blur overlapping processes. The Bode plot puts frequency back on the axis, and the Randles circuit turns the whole spectrum into numbers.
A Bode plot shows impedance magnitude |Z| and phase angle φ against log frequency — the same EIS data as a Nyquist plot, but with frequency made explicit, so the phase bump locates the RC time constant the Nyquist semicircle hides. The Randles circuit is the equivalent-circuit model behind both: a series resistance Rs in series with a double-layer capacitance in parallel with (charge-transfer resistance Rct plus a Warburg diffusion element). Fitting it maps each spectral feature to a physical quantity.
Why a Bode plot, when you have Nyquist
The Nyquist plot is great for seeing Rs, Rct, and the Warburg tail at a glance, but frequency is buried — you can't tell where on the curve a given frequency sits, and two processes with similar resistances overlap into one arc. The Bode plot fixes both: |Z| and phase are plotted against frequency directly, so each process shows up at its own frequency and the time constants separate.
How to read the Bode plot
1 · Magnitude plateaus → resistances
At very high frequency |Z| flattens to Rs; at very low frequency it flattens to Rs+Rct (plus diffusion). The step between the plateaus is the resistance of the process in that frequency band.
2 · Phase peaks → time constants
Each capacitive process produces a peak in the phase angle at its characteristic frequency. One peak means one dominant RC; two peaks means two processes you'd struggle to separate on a Nyquist plot.
3 · Slopes → element types
A −45° phase region is Warburg diffusion; a region approaching −90° is near-ideal capacitance. The phase tells you what kind of element dominates each band.
The Randles circuit — turning the curve into numbers
Both plots are the response of the same model. The classic Randles circuit is Rs in series with a parallel combination of the double-layer capacitance Cdl and the faradaic branch (Rct in series with the Warburg element W). Each element owns one feature of the spectrum:
- Rs — the high-frequency real-axis intercept (Nyquist) / high-f magnitude plateau (Bode).
- Rct — the semicircle diameter / the magnitude step between plateaus.
- Cdl — sets the semicircle's apex frequency / the phase-peak frequency.
- W (Warburg) — the 45° tail / the low-frequency −45° phase region.
Fitting the circuit to the measured spectrum extracts all four as physical numbers — which is what lets you track, say, Rct rising as a cell ages.
Common pitfalls
- Over-fitting. Adding circuit elements always improves the fit numerically; only add an element if there's a physical process to justify it.
- Ideal capacitor assumption. Real interfaces need a constant-phase element, not a perfect Cdl — forcing ideal capacitance distorts the other parameters.
- Non-unique fits. Different circuits can fit the same data; the model has to be chosen from the physics, not just the residual.
- Reading Nyquist alone. Overlapping time constants hide in a single arc — the Bode phase reveals whether it's one process or two.
Where this gets slow by hand
Equivalent-circuit fitting is finicky and easy to do inconsistently — model choice, initial guesses, and constant-phase handling all shift the answer. Across a study of many spectra, fitting each one by hand and keeping the model consistent so the extracted parameters are comparable is the slow, error-prone part.
Consistent circuit fits across an entire study
Niobia fits the Randles (or constant-phase) equivalent circuit to every spectrum in a study with a consistent model and initial conditions, so the extracted Rs, Rct, Cdl, and Warburg parameters are directly comparable across state of charge, cycle, and cell. It reads the Bode phase to decide when one time constant has become two and the model needs another element — and tracks each parameter so a rising Rct surfaces as an aging trend. Finicky, inconsistent hand-fitting becomes one comparable parameter set per spectrum.
Frequently asked
What is the difference between a Nyquist plot and a Bode plot?
Both display the same EIS data. A Nyquist plot graphs negative imaginary impedance against real impedance with frequency implicit, making Rs, Rct, and the Warburg tail easy to see. A Bode plot graphs impedance magnitude and phase against frequency explicitly, which separates overlapping processes and locates each RC time constant.
What is the Randles circuit?
The Randles circuit is the standard equivalent-circuit model for a simple electrochemical interface: a series resistance (Rs) in series with the double-layer capacitance (Cdl) in parallel with the faradaic branch — the charge-transfer resistance (Rct) plus a Warburg diffusion element. Each element corresponds to one feature of the impedance spectrum.
What does the phase peak in a Bode plot tell you?
The frequency at which the phase angle peaks identifies the characteristic time constant of a capacitive process. A single peak indicates one dominant RC process; two peaks indicate two processes that would overlap into one arc on a Nyquist plot.