A recurrent formula for calculating an impedance of the cathode catalyst layer in a polymer electrolyte fuel cell

A recurrent formula for estimating an impedance of the cathode catalyst layer with fast oxygen transport in a polymer electrolyte fuel cell is derived. The catalyst layer is divided into N sub-layers and application of the charge conservation law enables to obtain the nonlinear recurrent relation $Zn = f(Zn-1)$, where $Zn$ is the accumulated impedance of all sub-layers up to the $n$-th one. Numerical solution of this relation gives the total impedance of the catalyst layer with taking into account variation of the static overpotential of the oxygen reduction reaction along the depth of this layer. The model is simple, robust and two orders of magnitude faster than the standard model based on numerical solution of the differential equation.