Electrochemical impedance spectroscopy (EIS): principle, Nyquist plots and equivalent circuits

In EIS the potentiostat applies a small sinusoidal perturbation, normally a voltage of a few millivolts on top of a DC potential, and measures the current that flows at the same frequency; their ratio and phase shift give the impedance Z(ω) = Z′ + jZ″ at that frequency (Lazanas & Prodromidis 2023). The measurement is repeated across a wide frequency range, typically from some mHz to about 1 MHz (Lazanas & Prodromidis 2023).

The perturbation must be small because real electrochemical cells are not linear: only a small signal keeps the response linear, so that the output is a sinusoid of the same frequency and the impedance does not depend on the amplitude (Lazanas & Prodromidis 2023; Sharma et al. 2026). The cell must also stay unchanged during the sweep (Lazanas & Prodromidis 2023).

The power of EIS is that processes with different time constants respond at different frequencies: high frequencies probe fast processes and low frequencies slow ones (Sharma et al. 2026). In a worked example the double-layer charging, the faradaic reaction and diffusion had characteristic frequencies of 4.1 kHz, 310 Hz and 0.4 Hz, so a sweep from 10 kHz to 10 mHz covered all three (Lazanas & Prodromidis 2023). The cost is time: a single period at 1 mHz lasts 1000 s (calculated from 1/f), and at 10 µHz one measurement takes 27.8 h (Lazanas & Prodromidis 2023).

Equivalent circuits. Spectra are interpreted by fitting circuits of resistors, capacitors and distributed elements (Sharma et al. 2026). The Randles circuit puts the solution resistance Rₛ in series with the double-layer capacitance Cdl, which sits in parallel with the charge-transfer resistance Rct and the Warburg impedance Zw that describes diffusion of the redox species (Lazanas & Prodromidis 2023). Real, rough electrodes give depressed arcs that are fitted with a constant phase element (CPE) in place of the capacitor; its exponent n is 1 for an ideal capacitor, 0.5 for a Warburg element and 0 for a resistor (Lazanas & Prodromidis 2023).

• Nyquist plot. −Z″ is plotted against Z′, one point per frequency, with high frequencies on the left; both axes should have the same range (Lazanas & Prodromidis 2023). A resistor and capacitor in parallel give a semicircle whose diameter equals the resistance (Lazanas & Prodromidis 2023).
• Reading a Randles spectrum. The semicircle meets the real axis at Rₛ on the high-frequency side and at Rₛ + Rct on the low-frequency side, so its width is the charge-transfer resistance; this region is controlled by electron transfer (Lazanas & Prodromidis 2023). At lower frequencies a straight line at 45° (slope 1) is the Warburg diffusion tail, controlled by mass transport (Lazanas & Prodromidis 2023).
• Several arcs. Each process with its own time constant can give its own semicircle; two arcs are well resolved only when their time constants differ by about 100 times or more (Lazanas & Prodromidis 2023).
• Bode plot. log|Z| and phase are plotted against log frequency, which shows directly at which frequency each process occurs and displays the high-frequency region that the Nyquist plot compresses (Lazanas & Prodromidis 2023).
• Finite diffusion. A thin film or porous layer turns the 45° line into an arc (permeable boundary) or a near-vertical capacitive line (blocking boundary) at the lowest frequencies (Lazanas & Prodromidis 2023).
• Inductive points. Points below the real axis at high frequency come from cable self-inductance and are most prominent in low-impedance cells such as batteries and supercapacitors (Lazanas & Prodromidis 2023).

• Cell. Use the same three-electrode cell as for cyclic voltammetry; batteries are two-electrode cells and should be measured in a 2-electrode, 4-terminal connection because the wire resistance is not negligible (Lazanas & Prodromidis 2023).
• Steady state. Let the electrode reach a stable state before measuring: a drifting system gives an unstable Lissajous figure and invalid data (Lazanas & Prodromidis 2023).
• Choose the DC potential. For electrocatalysts, spectra recorded at catalytic-turnover potentials, rather than at open circuit, track OER activity (Sharma et al. 2026); OER studies on MOF-derived catalysts used 1.5 V vs RHE (Beglau et al. 2023; Sondermann et al. 2025).
• Amplitude. 5–10 mV is usual; resistive non-aqueous electrolytes may need more, and the Lissajous ellipse should stay symmetric and steady (Lazanas & Prodromidis 2023).
• Frequency range. Go from at least 10⁴ Hz down to 1 Hz or lower; with a 100 Hz lower limit only the solution resistance and capacitance were seen, not the charge transfer (Sharma et al. 2026). Place the cell in a Faraday cage for high-impedance measurements (Lazanas & Prodromidis 2023).

Batteries. In a lithium-ion cell each interface, from the graphite anode to the metal-oxide cathode, contributes its own circuit elements, and an extra high-frequency semicircle is attributed to the solid electrolyte interphase (SEI) formed by electrolyte decomposition at the anode, which influences cycling stability (Lazanas & Prodromidis 2023). See our pages on battery materials and the graphite anode.

Solid electrolytes. Polycrystalline ion conductors show separate semicircles for conduction inside the grains and across grain boundaries, and the bulk-grain arc can need frequencies above 5 MHz (Lazanas & Prodromidis 2023). See solid electrolytes.

Electrocatalysts. The Nyquist plot gives Rₛ for iR compensation of voltammetry and Rct for comparing catalysts (Sharma et al. 2026); a low Rct means fast charge transfer, as in a NiFe MOF–carbon nanotube OER catalyst with 7 Ω against 39 Ω for RuO₂ (Sondermann et al. 2025). Such MOF-derived catalysts are related to frameworks like HKUST-1 and ZIF-8. Impedance at OER potentials can also measure adsorbate capacitance to estimate the active surface area of Ni-based hydroxides (Sharma et al. 2026).

Sensors and corrosion. In faradaic biosensors a target binding to the electrode blocks the redox probe and raises Rct in a concentration-dependent way, and EIS is also a standard tool in corrosion science and fuel cells (Lazanas & Prodromidis 2023).

EIS sits alongside CV and elemental checks by ICP-MS in our guide to characterisation techniques.

• Fitting before validating. Run the Kramers–Kronig test first; data that fail it come from a non-linear or unstable system (Lazanas & Prodromidis 2023).
• Too large an amplitude. Spectra recorded at different small amplitudes should be identical; if they are not, the system is outside its linear range (Lazanas & Prodromidis 2023).
• A frequency window that misses processes. Stopping at 100 Hz misses the charge-transfer arc (Sharma et al. 2026), and the lowest frequencies cost hours (Lazanas & Prodromidis 2023).
• Unjustified circuits. Many circuits fit the same spectrum; choosing measurement parameters and building a physically meaningful circuit remain the hard part (Moehl et al. 2026), and every element should correspond to a process in the cell (Lazanas & Prodromidis 2023).
• Treating a CPE as a capacitance. The CPE parameter is not a capacitance; an effective capacitance has to be calculated with a formula that depends on the circuit (Lazanas & Prodromidis 2023).
• Squashed axes. Nyquist plots with unequal axis ranges distort arcs, a common error in published figures (Lazanas & Prodromidis 2023).
• Tafel slopes from impedance. Deriving Tafel slopes from Rct is valid only at low overpotential (Sharma et al. 2026).

The LabCanvas figure editor Library has the potentiostat and electrochemical cell as an editable object, including a cutaway view of the cell and the AC signal path, and an editable data plot for Nyquist and Bode spectra. Draw the Randles circuit next to the Nyquist plot, mark Rₛ, Rₛ + Rct and the 45° Warburg line, label the frequency direction, and export the panel for a methods figure, SI or graphical abstract.