Analysis of the Transient State in a Series Circuit of the Class RLβCα

Abstract

Results of analysis of transient states in a series circuit of the class (Formula presented.) , supplied by an ideal voltage source, have been described in the paper. This circuit consists of a coil (Formula presented.) and a supercapacitor (Formula presented.) described by fractional-order differential equations. A method for determining the current and voltage waveforms in the analyzed circuit, based on the decomposition of rational functions into partial fractions, has been described. This method allows to determine transient waveform shapes in the system for any kind of voltage excitation. Two cases of the problem solutions have been considered. The first case concerns a situation where poles of rational functions are real, and the second where rational functions have complex poles. Effective relations enabling the determination of transient waveforms in a closed form have been given. Analytical formulae describing transient state waveforms in the system for different types of voltage excitations: constant, monoharmonic, periodic and arbitrary being an element of a Hilbert space, have been determined, too. The obtained results have been illustrated by an example.