Abstract
Results of analysis of transient states in a series circuit of the class $${RL}_{\beta }{C}_{\alpha }$$RLβC�, supplied by an ideal voltage source, have been described in the paper. This circuit consists of a coil $${L}_{\beta }$$Lβ and a supercapacitor $${C}_{\alpha }$$C� 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.
Highlights
In recent years, a great interest has been aroused by the analysis of electrical and electronic systems with fractional-order elements, especially with supercapacitors
In contrast to previous works, this paper describes a method of analysis of transient states in a series circuit of the class R LβCα for any kind of the voltage excitation and arbitrary values of parameters α, β ∈ R+, resulting in closed form solutions
The paper is a continuation of work [10] concerning analysis of the transient state in a series circuit of the class RCα with the supercapacitor modeled as a fractional-order capacitor Cα
Summary
A great interest has been aroused by the analysis of electrical and electronic systems with fractional-order elements, especially with supercapacitors. The paper belongs to the first mentioned direction of studies, and it concerns an analysis of transient states in series circuit of the class R Lβ Cα, supplied by a voltage source of any kind. Generalized method of solving transient states in circuits of the class R Lβ Cα, by defining state equations of the concerned circuit, has been described in the work [12]. In contrast to previous works, this paper describes a method of analysis of transient states in a series circuit of the class R LβCα for any kind of the voltage excitation and arbitrary values of parameters α, β ∈ R+, resulting in closed form solutions. The paper is a continuation of work [10] concerning analysis of the transient state in a series circuit of the class RCα with the supercapacitor modeled as a fractional-order capacitor Cα
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