Analytic Solution for theRLElectric Circuit Model in Fractional Order

Abstract

This paper provides an analytic solution ofRLelectrical circuit described by a fractional differential equation of the order0<α≤1. We use the Laplace transform of the fractional derivative in the Caputo sense. Some special cases for the different source terms have also been discussed.

On the Dimensional Consistency Aware Fractional Domain Generalization of Simplest Chaotic Circuits

“…Note also that x(t) � X(ξ) where ξ � (t α /Γ(1 + α)) [32]. At this point, it can be seen that (11) has been transformed to (14).…”

“…erefore, the solution of (11) can be conveniently obtained by solving (14) and keeping the above relationships between x(t) and X(ξ) in mind. Here, we let A � 0.2A, B � 1.7 Ω, L � 3.3H, and C � 1F similar to [20].…”

“…Here, we let A � 0.2A, B � 1.7 Ω, L � 3.3H, and C � 1F similar to [20]. We also assume that α � 0.9, σ � 1 sec, X(0) � 0.1, U(0) � 0, and V(0) � 0.1 Ω and numerically solve (14) with MATHEMATICA with the above relationship between ξ and t in mind. As a result, the phase portraits of x(t), i L (t), and v C (t) can be simulated by also keeping in mind that i L (t) � Au(t) and v C (t) � Av(t) as depicted in Figures 3-5, where the strange attractors, which refers to the chaotic behaviors, can be observed.…”

“…e fractional calculus and its related differential equation, i.e., FDE, which are the extensions of the conventional integer calculus and the ordinary differential equation (ODE), have been extensively utilized in various research areas, e.g., signal processing, biomedical engineering, electronics, robotics, and control theory [1][2][3][4][5][6][7][8][9][10][11]. e FDE has been used in the fractional domain analysis of electrical circuits in which their orders are fractional instead of being strictly integer, as proposed in the previous research studies [12][13][14][15][16]. Such fractional domain analysis has been found to be necessary because the electrical circuit components in practice have irreversible dissipative effects, e.g., Ohmic friction, thermal memory, and electromagnetic field-induced nonlinearities, which cannot be precisely analyzed by using the conventional integer domain methods.…”

Analytic solution of fractional order differential equation arising in RLC electrical circuit

“…Mathematical models' involving fractional order derivatives has become a powerful and widely used tool for better modeling. Fractional model for electrical circuits such as RL, RC, RLC have already been proposed by many researchers, for details, see [1][2][3]. In order to stimulate more interest in subject and to show its utility, this paper is devoted to new and recent application of fractional calculus.…”