Abstract
In this work we obtain analytical solutions for the electrical RLC circuit model defined with Liouville–Caputo, Caputo–Fabrizio and the new fractional derivative based in the Mittag-Leffler function. Numerical simulations of alternative models are presented for evaluating the effectiveness of these representations. Different source terms are considered in the fractional differential equations. The classical behaviors are recovered when the fractional order α is equal to 1.
Highlights
In several works, fractional order operators are used to represent the behavior of electrical circuits; for example, fractional differential models serve to design analog and digital filters of fractional-order, and some works concern the fractional-order description of magnetically-coupled coils or the behavior of circuits and systems with memristors, meminductors or memcapacitors [1–16]
The solutions obtained preserve the dimensionality of the studied system for any value of the exponent of the fractional derivative
We can conclude that the decreasing value of α provides an attenuation of the amplitudes of the oscillations, the system increases its “damping capacity” and the current changes due to the order derivative, the response of the system evolves from an under-damped behavior into an over-damped behavior
Summary
Fractional order operators are used to represent the behavior of electrical circuits; for example, fractional differential models serve to design analog and digital filters of fractional-order, and some works concern the fractional-order description of magnetically-coupled coils or the behavior of circuits and systems with memristors, meminductors or memcapacitors [1–16]. These research works address the study of the described electrical systems. These models have been extended to the scope of fractional derivatives using Riemann–Liouville and Liouville–Caputo derivatives with fractional order; these two derivatives have a kernel with singularity [17]. Atangana and Baleanu suggested two news derivatives with Mittag-Leffler kernel, these operators in Liouville–Caputo and Riemann–Liouville have non-singular and non-local kernel and preserve the benefits of the Riemann–Liouville, Liouville–Caputo and Caputo–Fabrizio fractional operators [28–33]. This work aims to represent the fractional electrical RLC circuit with the Liouville–Caputo, Caputo–Fabrizio and the new representation with Mittag-Leffler kernel in the Liouville–Caputo sense, Entropy 2016, 18, 402; doi:10.3390/e18080402 www.mdpi.com/journal/entropy. Entropy 2016, 18, 402 considering different sources terms in order to assess and compare their efficacy to describe a real world problem
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