Fractional Cole Model with Dual Caputo–Weyl Operators for Complex Frequency Impedance Analysis

Abstract

This paper presents a generalized framework for modeling electrical impedance in the complex frequency domain using fractional derivatives. The approach extends the classical Cole model by incorporating both Caputo and Weyl fractional operators, enabling the representation of systems that exhibit simultaneous transient memory effects and periodic steady-state behavior. The proposed formulation is supported by new theoretical results, including a stability criterion for fractional impedance systems and a convergence property showing reduction to the classical model as the fractional order $\alpha \to 1^{-}$. Applications to electrochemical impedance spectroscopy (EIS) and bioimpedance analysis demonstrate the practical value of the model.