Analysis of the Transient State in a Series Circuit of the Class $$RL_{\beta }C_{\alpha }$$ R L β C α

Abstract

Results of analysis of transient states in a series circuit of the class RL β C α , supplied by an ideal voltage source, have been described in the paper. This circuit consists of a coil L β and a supercapacitor C α 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.

A Harmonic Balance Methodology for Circuits with Fractional and Nonlinear Elements

“…-for selected transient problems, the solutions can be compared with results obtained through the evaluation of analytical solutions based on the Mittag-Leffler function [18,22,23,34]; -steady-state linear AC problems solved with the newly designed numerical methods can be compared with solutions obtained through the application of complex numbers.…”

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

“…The analyzed system can be described by a differential equation with derivatives of fractional order in the Caputo sense. The paper is a continuation of previous studies concerning the analysis of transient states in circuits with fractional-order elements of the class RC α [19] and of the class RL β C α [20] . To solve the derived equation, the Laplace transform method has been applied and the idea of rational function decomposition into partial fractions, originally proposed in [27] for fractional-order differential equations.…”

“…(20) . The inverse transform of function F ( s ) has been determined using Borel convolution theorem and the following relation [25] :…”

“…The relations have been obtained in closed form solutions. The paper is a continuation of works [19,20] concerning analysis of the transient state in a series circuit of RC α and RL β C α class with the supercapacitor and the coil modeled as fractional-order elements.…”

Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions

“…Thus, (1) can be regarded as the fractional analogue of equation of motion. In case we fix p = α, q = β, f (t, x(t)) = (R Z /L)x(t), f (t, x(t)) = (1/LC)[−x(t) + e(t)], (1) takes the form of a fractional-order differential equation of the voltage function x(t), see Equation (4) in [2]. The nonlocal conditions involved in the problem (1) appear in several applications of diffusion processes, computational fluid dynamics (CFD) studies of blood flow problems, bacterial self-regularization models, for instance, see [3][4][5].…”