Electrical circuits RC and RL involving fractional operators with bi-order

Jf Gómez-Aguilar,Vh Olivares-Peregrino,Rf Escobar-Jiménez,Gv Guerrero-Ramírez,Ma Taneco-Hernández

doi:10.1177/1687814017707132

Abstract

This article describes electrical series circuits RC and RL using the concept of derivative with two fractional orders α and β in Liouville–Caputo sense. The fractional equations consider derivativ...

Highlights

We describe electrical series circuits RC and RL using the concept of derivative with two fractional orders a and b in Liouville–Caputo sense
The fractional calculus is applied to various electrical circuit problems as a useful mathematical tool
The experimental results do not comply with standard theoretical calculations; this is due to the effects of ohmic friction and temperature, and these losses are not taken into account in the standard approach

Summary

Introduction

Fractional calculus (FC) allows to research the nonlocal response of multiple phenomena,[1,2,3,4,5,6] in electrical circuits, several authors studied the behavior of capacitors, coils, memristors, and meminductors.[7,8,9,10,11,12,13,14,15,16,17,18,19,20] These elements involve irreversible dissipative effects (ohmic friction) and nonlinear effects due to the electric and magnetic fields.[17,21,22] Rousan et al.[23] suggested a fractional order differential equation to study an LC and RC circuit. Gomez-Aguilar et al.[27] presented some electrical circuits in terms of Caputo– Fabrizio fractional operator; they obtained numerical simulations of these circuits by applying the numerical Laplace transform algorithm, and more works related to this fractional operator are given by Atangana and Alkahtani,[28] Atangana and Nieto,[29] Atangana and Alkahtani,[30] Gomez-Aguilar et al.,[31] and Alsaedi et al.[32,33] Recently, Atangana and Baleanu proposed a new definition with non-local and nonsingular kernel based on the Mittag-Leffler function, this definition The aim of this contribution is to present an alternative representation of the electrical series circuits RC and RL using the concept of derivative with two fractional orders a and b.

Results

Conclusion