Electrochemical Impedance Spectroscopy (EIS) for Fuel Cells & Electrolyzers

Electrochemical Impedance Spectroscopy (EIS) is a powerful, non-destructive way to “x-ray” an electrochemical device while it is operating. Instead of applying a single DC current or voltage and reading a single point on the V-I curve, EIS perturbs the device with a tiny AC signal (typically 5–10 mV or a few percent of the DC setpoint) and sweeps frequency from high (e.g., 100 kHz) down to low (e.g., 10 mHz). By observing how the device responds at each frequency, we can separate ohmic losses, charge-transfer kinetics, and mass-transport effects—and then map them onto an equivalent electrical circuit that captures the physics.

Two common modes are:

• Potentiostatic EIS: hold a DC voltage (or cell potential) and superimpose a small AC voltage; measure AC current response.
• Galvanostatic EIS: hold a DC current density and superimpose a small AC current; measure AC voltage response.

Either way, the complex ratio of voltage to current defines the complex impedance $Z(\omega)=Z' + jZ''$, where $\omega = 2\pi f$.

EIS can look abstract at first. It’s easier if we begin with simple elements and then assemble them into the models used for fuel cells (FC) and electrolyzers (EL).

Resistor (R).
Impedance: $Z = R$ (purely real).

• Nyquist plot ($-\mathrm{Im}(Z)$ vs $\mathrm{Re}(Z)$): a single point on the real axis.
• Bode magnitude/phase: $|Z|$ is flat at $R$ across frequency; phase = $0^\circ$.

Capacitor (C).
Impedance: $Z = \frac{1}{j\omega C} = -\frac{j}{\omega C}$.

• Nyquist: a point on the negative imaginary axis ($\mathrm{Re}(Z) \approx 0$, $-\mathrm{Im}(Z) = \frac{1}{\omega C}$).
• Bode: $|Z|$ drops linearly with frequency on a log scale (slope −1 decade/decade), phase = $-90^\circ$.

Parallel RC (R ‖ C).
Impedance: $Z = \frac{1}{1/R + j\omega C}$. This is the canonical “relaxation” element:

• Nyquist: a perfect semicircle centered on the real axis, with high-frequency intercept at 0 and low-frequency intercept at R (if there’s no series resistance). The apex occurs at the characteristic frequency $f_0 = \frac{1}{2\pi R C}$.
• Bode: magnitude transitions from capacitive behavior at high f toward R at low f; phase sweeps from ~−90° (high f) through a minimum near $f_0$, returning toward 0° at low f.

In real electrodes, the capacitor is often “distributed” due to roughness, pores, and non-uniform current distribution. We model this as a constant phase element (CPE) with impedance $Z_{CPE} = \frac{1}{Q (j\omega)^n}$, $0 < n \le 1$. When $n < 1$, the Nyquist semicircle is “depressed” (its center drops below the real axis).

On a Nyquist plot we graph $-\mathrm{Im}(Z)$ vs $\mathrm{Re}(Z)$ as frequency is swept. High frequencies typically appear near the origin and the trace moves rightward and upward as frequency decreases. It’s great for seeing relaxation arcs (semicircles), their diameters (resistances), and low-frequency tails (diffusion).

• Real axis $Z'$: in-phase, dissipative losses.
• Imag axis $Z''$: quadrature, energy storage; capacitive $Z''<0$ (plotted up), inductive $Z''>0$.
• Phase $\phi$: recovered from a point via $\phi=\arctan(Z''/Z')$ (sign per your plot convention).
• Magnitude: distance to the origin $|Z|=\sqrt{Z'^2+Z''^2}$.

Each dot is one frequency: its x-position reveals resistive loss at that timescale, its height shows reactive storage, and its angle to the origin reflects the phase relationship. Tracing dots across frequency gives arcs (charge transfer) and tails (diffusion), which we then map to circuit elements.

A practical base model for both PEM fuel cells and PEM water electrolyzers is:

• Ohmic resistance (Rₛ): series resistance from membrane ionic resistance, contact resistances, and leads. In Nyquist, it’s the high-frequency intercept on the real axis.
• Charge-transfer arc (R_ct ‖ C_dl or CPE): represents electrode kinetics and double-layer charging (ORR on the FC cathode; OER on the EL anode is usually dominant). Appears as a mid-frequency semicircle; its diameter is roughly $R_{ct}$ (if arcs are well separated).
• Mass-transport/diffusion element: Warburg or finite-length diffusion (sometimes modeled as a second (R ‖ CPE) at low frequency). This yields a 45° tail (semi-infinite diffusion) or an additional low-frequency arc (finite transport and gas accumulation).

Optional but common additions:

• Inductor (L) in series: wiring/fixtures can cause a slight inductive loop at very high frequencies (far left of Nyquist, small positive Im(Z)).
• Multiple arcs: distinct processes (e.g., anode vs cathode kinetics, or separate gas-transport regimes) can show up as multiple depressed semicircles at different frequencies.

How these map to the real device

• Rₛ (ohmic):
  Fuel cell: membrane hydration, temperature, and compression dominate. Better hydration and higher temperature reduce Rₛ.
  Electrolyzer: membrane thickness and water purity/temperature are key; higher current densities increase Ohmic heating but Rₛ is set by materials and stack compression.
• R_ct (kinetics):
  Fuel cell: ORR kinetics at the cathode dominate. Higher temperature and better catalysts (e.g., higher ECSA, optimized ionomer) reduce R_ct.
  Electrolyzer: OER kinetics at the anode dominate. Ir-based catalyst layers, proper ionomer content, and improved interfacial transport are crucial.
• C_dl/CPE (interfacial capacitance):
  Sensitive to roughness, porosity, and wetting. Changes in humidity (FC) or water delivery (EL) shift the apparent capacitance and CPE exponent n.
• Mass transport (Warburg/low-f arc):
  Fuel cell: oxygen transport in the cathode (flooding or dry-out) and water management show up at low frequencies as tails or additional arcs.
  Electrolyzer: oxygen bubble dynamics and transport through PTL/flow fields create low-frequency features; improved PTL architecture and flow can reduce these.

Fuel cell at moderate current density

• Nyquist:High-f intercept at Rₛ (x-axis). One depressed semicircle (mid-f) for ORR charge transfer (diameter ≈ R_ct). At lower frequencies, a tail (sometimes a second arc) reflecting mass transport.
• Bode:|Z| shows a plateau at Rₛ at high f, then rises through the R_ct band. Phase dips negative (toward −90°) across the charge-transfer band, sometimes with a second phase feature at low f for transport.

Electrolyzer at a given DC bias/current density

• Nyquist:Similar structure: Rₛ at the high-f intercept, a mid-f OER arc (often larger than FC’s ORR arc for comparable conditions), and a low-f tail/arc tied to bubble dynamics and diffusion in the anode PTL and channels.
• Bode:A clear mid-frequency loss peak from kinetics; low-f deviations from transport and gas evolution.

Rule-of-thumb frequencies (very device-dependent): High-frequency ohmic region: ~10 kHz to 100 kHz. Charge-transfer arc: ~1 Hz to 1 kHz. Mass-transport phenomena: below ~1 Hz (down to tens of mHz). These bands shift with temperature, loading, gas flow/water feed, and materials.

Reading the plots like a diagnostician

• High-f intercept increases? Ohmic problem—check membrane hydration (FC), water purity/temperature/stack compression (EL), and contact resistances.
• Mid-f semicircle gets larger? Slower kinetics—catalyst degradation, poisoning, or poor ionic/electronic pathways in the catalyst layer.
• Low-f tail steepens or new arc appears? Transport/bubbles—flooding, poor water removal (FC), or gas-evolution management issues (EL).
• Semicircle becomes more “depressed”? Increased heterogeneity—CPE exponent $n$ dropping indicates broader distribution of time constants (e.g., aging, uneven wetting).
• Small inductive loop at very high f? Likely cabling/fixtures; keep leads short and use proper 4-wire connections.

Practical measurement tips

1. Small-signal: keep the AC amplitude small (e.g., 5–10 mV) so the system behaves linearly around the DC operating point.
2. Steady state: allow the device to stabilize at the chosen current/voltage before the sweep.
3. Frequency density: more points per decade where you expect arcs (e.g., 8–10/decade around the charge-transfer band).
4. Four-terminal wiring and short leads reduce parasitic inductance and contact artifacts.
5. Temperature and humidity control (FC) or water temperature/flow (EL) are critical—EIS is very sensitive to these.
6. Model wisely: start with the simplest circuit that captures the features (Rₛ + (R_ct ‖ CPE) + transport element). Add complexity only when justified by the data.

• EIS separates losses by timescale: ohmic (fast), kinetics (mid), and transport (slow).
• Nyquist emphasizes resistive contributions and relaxation arcs; Bode highlights where processes live in frequency.
• For FC/EL, a compact but expressive model is Rₛ + (R_ct ‖ CPE) + transport (plus optional L).
• Small changes in the plots map directly to physical changes in materials, interfaces, or operating conditions—making EIS a go-to tool for both R&D and field diagnostics.