Constant Phase Element in the Laplace Domain

The constant phase element (CPE) has long stood as an empirical construct bridging the gap between ideal capacitors and the complex reality of electrochemical phenomena. Despite its universal use in the modeling impedance spectra, the time-domain behavior however remains ambiguous yet, particularly under non-stationary preconditions. In this work, we present a rigorous Laplace-domain framework that reveals the true dynamical nature of the CPE as a memory-bearing, fractional-order element. By introducing a general memory function, we derive an equivalent circuit representation in the s -domain that coherently captures both initialization effects and transient responses. This formulation unifies the behavior of the CPE acting both as a dispersive capacitor/inductor, providing clear physical insight into the dual character of this imperfect element. We apply the theory to the key configurations of the series R-CPE circuit and the simplified Randles model, demonstrating how the dispersive coefficient dictates the persistence and magnitude of memory effects observable in the natural response in basic step methods or the step response in cyclic voltammetry. Our analysis reveals that the multiscale memory dynamics governed by the fractional-order derivative dominate far-from-equilibrium conditions, extending beyond the conventional Caputo definition of the fractional-order operator in the transient modeling of electrochemical and electronic systems.