The Fuel Cell Polarisation Curve

Introduction

The reversible voltage of a fuel cell is measured conceptually at equilibrium, when it delivers no net power. A working cell must draw current, so it pays activation, ohmic and mass-transport voltage losses. Plotting cell voltage against current density gives a polarisation curve . Multiplying voltage by current density gives power density. Together, the curves show why maximum voltage, maximum current and maximum useful power occur at different operating points.

Core explanation

A useful bookkeeping expression is V(i) = E rev − η act,a − η act,c − iR Ω − η mt , with sign and definitions chosen for positive loss magnitudes. Activation terms arise because anodic and cathodic electron-transfer or surface reactions need driving overpotential. For hydrogen–oxygen PEM cells, oxygen reduction at the cathode often causes a substantial activation loss because its multistep kinetics are slower than hydrogen oxidation on suitable platinum catalysts. Butler–Volmer describes the current–overpotential relation near a specified interfacial state; at sufficiently large one-direction overpotential, a Tafel approximation may be useful, but coverage and transport can change its slope.

The ohmic term is often approximated as iR Ω for area-specific resistance R Ω. Proton transport through membrane and ionomer, electron transport through electrodes and contacts, and interconnect resistance contribute. If current density rises while resistance is approximately constant, this voltage drop grows roughly linearly. Water content can alter a PEM membrane's proton conductivity; drying increases resistance, while excess water can flood gas pathways. A measured high-frequency resistance is useful but may not include every distributed resistive process.

At high current, reactants can be consumed at catalyst surfaces faster than gas transport replenishes them. Local oxygen partial pressure falls, causing mass-transport loss. Liquid water can obstruct pores and worsen it. A simple concentration-overpotential form contains a logarithm such as −(RT/nF)ln(1−i/i lim) for an idealised limiting-current model, with sign chosen as a positive loss. As i approaches i lim, the expression rises sharply. Real fuel cells have spatially distributed channels, water and catalyst layers, so one fitted i lim is an effective quantity.

Power density is p = iV per geometric active area. At i=0, voltage may be high but power is zero. At very high i, voltage collapses and efficiency is low. A power curve therefore rises, reaches a maximum, then falls or becomes impractical. Operating exactly at the peak is not always desirable: efficiency, durability, heat rejection and compressor power can favour a lower current density.

Polarisation-curve features are suggestive, not unique diagnoses. A steep initial drop may be activation-related, a nearly linear middle region ohmic, and a high-current bend transport-related. But drying can change ohmic resistance with current, and flooding can begin before the apparent end region. Impedance, gas-pressure variation and humidity tests help separate causes.

Step-by-step reasoning

Begin with E rev under actual gas activities. At the current of interest, subtract positive activation losses, iR Ω and transport loss using consistent area units. Plot V and then calculate p=iV. Examine how changing temperature, pressure or humidity affects each term. If interpreting an experimental curve, test a suspected mechanism independently rather than assigning every segment solely by appearance.

Visual explanation

Draw E rev as a horizontal dashed line. Below it draw a falling voltage curve with labels: activation-dominated near low current, a straighter ohmic region in the middle, and rapid high-current fall near mass-transport limitation. On a second axis draw power density beginning at zero, rising to a peak and declining. Add a small water-balance icon to show how membrane hydration and flooding can affect different regions.

Real-world analogy

A delivery route has a start-up cost, a roughly proportional travel cost per package and a bottleneck when loading docks become saturated. Those three patterns resemble activation, ohmic and transport losses. The analogy is qualitative; electrochemical losses are coupled and their equations depend on surface and gas conditions.

Real-world example

A PEM fuel cell is measured at fixed temperature while oxygen pressure is increased. The high-current voltage improves markedly but the low-current region changes less. This supports oxygen transport as an important high-current limit. A second test increases membrane humidity; if the middle-region slope improves, lower proton resistance may contribute. Neither result alone proves a unique catalyst-layer mechanism.

Why?

Why does voltage fall under load? Reactions and transport need finite driving forces, dissipating part of the available free energy. Why is power zero at open circuit? Current is zero. Why can power decrease at very high current? Voltage falls faster than current rises near severe transport limitation or shutdown.

Common misconception

The point of highest voltage is not the point of highest power. Also, the apparent linear slope of a voltage curve need not be purely membrane resistance; distributed electrode and concentration effects can contribute. A single fitted three-term equation is a useful summary, not a full spatial fuel-cell model.

Worked example

Question: At 0.50 A cm⁻² a cell's E rev is 1.18 V, total activation loss is 0.25 V, area-specific resistance is 0.20 Ω cm² and transport loss is 0.08 V. Find operating voltage and power density.

Reasoning: Ohmic loss is iR=0.50×0.20=0.10 V. Thus V=1.18−0.25−0.10−0.08=0.75 V. Power density is 0.50×0.75=0.375 W cm⁻². The calculation is at one current, so it does not locate the maximum-power point.

Answer: V=0.75 V and p=0.375 W cm⁻².

Quick check

1. Why does a fuel cell produce no electrical power at open circuit even if its voltage is high? Answer: Electrical power is current times voltage, and open-circuit current is zero.

Exam focus

Write losses as positive magnitudes subtracted from reversible voltage. Track area-specific resistance units so iR is volts. Distinguish power density from efficiency and avoid assigning a mechanism from curve shape alone. Explain how a limiting-current model predicts a sharp high-current loss.

Advanced insight

The measured polarisation curve is a cell-level average over many local conditions. Current density can vary along gas channels as oxygen is consumed and water accumulates. Compressor load can rise when higher air flow is used to delay transport losses, reducing net system power. Consequently an operating point chosen from stack power alone may differ from the system-optimal point after parasitic loads and durability are included.

Summary

A fuel-cell polarisation curve links current density to a voltage below the reversible value. Activation, ohmic and mass-transport losses dominate under different conditions but can overlap. Power density equals iV and peaks at an intermediate current. Reliable diagnosis uses supporting tests of kinetics, resistance, gas transport and water management.

Practice questions

1. A cell runs at 0.7 V and 0.4 A cm⁻². What is power density? Answer: 0.7×0.4 = 0.28 W cm⁻².

2. If area-specific resistance is 0.15 Ω cm² at 1.0 A cm⁻², what is the simple ohmic drop? Answer: 0.15 V.

3. Why might increasing oxygen pressure improve the high-current region? Answer: It raises oxygen availability at the catalyst, reducing concentration loss when transport is limiting.

4. Why might the maximum-power point be an undesirable continuous operating point? Answer: It can have lower efficiency, greater heat production or poorer durability than a lower-current operating point.

Sources: US DOE NETL Fuel Cell Handbook, performance chapter; US DOE, fuel-cell system components.