Analysis of the transient state in a parallel circuit of the class RLβCα

The paper presents the results of the conducted analysis of transient states in a parallel circuit of the class RLβCα, supplied by an ideal current source. The considered circuit consists of a real coil Lβ and a (super)capacitor Cα, modeled as fractional-order elements. The paper proposes a method for determining the current and voltage waveforms, which uses the decomposition of rational functions into partial fractions and inverse Laplace method. By using this method, transient waveforms occurring in the system for any kind of current excitation can be determined. Two cases of the problem solutions have been considered in the paper, for real and complex poles of the rational functions, which result from the analysis of the main fractional-order differential equation, describing the considered circuit. Analytical relations describing transient state waveforms in the system have been determined for different types of current excitations: constant, monoharmonic, polyharmonic and arbitrary being an element of a Hilbert space. The obtained results have been illustrated by an examplary case of a parallel circuit of the class RLβCα. The paper is a continuation of previous studies concerning transient-state analysis in circuits with elements modeled as fractional-order (Jakubowska and Walczak 2015, 2016).