Analysis of time-domain response of series-parallel fractional-order RC alpha and RL beta circuits based on Cole distribution function

Fractional-order circuits, which contain fractance devices (L-beta and C-alpha) described by fractional-order differential equations, are widely used in interdisciplinary engineering research. The application of time-domain response (TDR) measurement increases rapidly in analyzing the characteristics of fractance devices and fractional-order circuits. However, the previous methods are mainly focused on deducing TDR expressions of a single fractance device or series fractional-order circuits in some specific input signals, while overlook the TDR analysis of series-parallel fractional-order circuits (SPFOC) that are commonly applied in modeling electrochemical impedance or electrical bioimpedance. This paper proposes a novel method to derive TDR expressions of SPFOC in an arbitrary periodic signal. First, a discretization method to represent the SPFOC as the sum of first-order circuits is proposed in Cole distribution function, which could approximate the transfer function of fractional-order impedance/admittance. Second, the TDR expression of SPFOC in sinusoidal input signals is deduced in inverse Laplace transform, which can be applied as the components of a general TDR expression of SPFOC in an arbitrary periodic signal based on the Fourier series theory. Finally, both TDR simulations and experiments on Cole impedance model (a basic SPFOC) are conducted to verify the proposed method, which shows excellent agreement compared to theorical analysis. This paper presents a TDR solution to SPFOC that consist of a RC alpha or RL beta element. In future, the theoretical and application extension of the proposed TDR analysis method is still necessary on analysis of more general fractional-order circuits.