RLC electrical circuit of non-integer order

Abstract

Abstract:In this work a fractional differential equation for the electrical RLC circuit is studied. The order of the derivative being considered is 0 < γ ≤ 1. To keep the dimensionality of the physical quantities R L and C an auxiliary parameter σ is introduced. This parameter characterizes the existence of fractional components in the system. It is shown that there is a relation between γ and σ through the physical parameters RLC of the circuit. Due to this relation, the analytical solution is given in terms of the Mittag-Leffler function depending on the order γ of the fractional differential equation.PACS (2008)

Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel

“…No wonder, then, that the concept of derivative with fractional order is applicable in almost all the branches of sciences and engineering. [11][12][13][14][15] There are few definitions of derivative with fractional order, the old version having a kernel with singularity; this situation does not give a full memory of the description of the physical problem. In order to further enhance the concept of derivative with fractional order, Caputo and Fabrizio 7 have recently proposed a new fractional derivative also designed with the concept of convolution; however, this time the convolute filter is the exponential function, which helps to reduce the risk of singularity.…”

“…8 In their paper, they demonstrated that the derivative possesses very interesting properties, for instance, the possibility to portray physical occurrence with different scales. 7,8 With the old version of fractional calculus, studies towards modelling of electrical circuits, for instance, domino ladders, tree structures, and element coils, have been done; see the work by Losada and Nieto, 8 Razminia and Baleanu, 9 Petra´s 10 and Go´mez et al 11 In the same line of idea, it has been suggested that a fractional differential equation puts together with the simple harmonic oscillations of an LC circuit with the discharging of an RC circuit. With physical observations, it was proven that the wire obtains an inducting behaviour as the current is initiated in it and progressively restores its resisting behaviour.…”

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

“…The first one involves the study of the frequency properties and characteristics of circuits of the class RL β C α [21,25,26,35,36] and circuits of the class RLC α with supercapacitors [9,30,33,34], analysis of transient states in such systems [3,7,9,10,12] and analysis of their stability [11,22].…”

“…There are a few works concerning this issue [3,7,9,10,12]. The paper [9] presents an analysis of an electrical circuit of the class RC α , where the voltage excitation is a unit step function and it analyzes the delay, rise and settling times of the fractionalorder circuit.…”

“…The paper [9] presents an analysis of an electrical circuit of the class RC α , where the voltage excitation is a unit step function and it analyzes the delay, rise and settling times of the fractionalorder circuit. Analysis of free oscillations in the series circuit of the class RL β C α has been performed in [7], and analytical solution for α = β has been obtained. In the paper [3], an analysis of parallel RL β C α circuit has been performed with a unit step function as the voltage excitation.…”