Kramers–Kronig transformation, dc behaviour and steady state response of the Warburg impedance for a disk electrode inlaid in an insulating surface

As the frequency approaches zero, the impedance described by the Warburg function tends to infinity. This means that the resistance of the equivalent circuit representing the electrochemical process has an infinite resistor and then the current cannot flow through it. This asymptotic behaviour also prevents the application of the Kramers–Kronig transformations, a set of integrals which should be fulfilled by any linear system. Using a more general expression of the impedance for a disk electrode inlaid in an insulating surface developed by Fleischmann and Pons (J. Electroanal. Chem. 250 (1988) 277), the Warburg impedance can be deduced and the Kramers–Kronig transformation is possible. An expression is also deduced for the mass-transfer resistance and the frequency at which the Warburg function fails for the representation of the impedance for a disk. A fast algorithm for the calculation of the generalised impedance function is outlined for its implementation in non-linear impedance fitting programs.