Modeling and Applications of Fractional-Order Mutual Inductance Based on Atangana–baleanu and Caputo–fabrizio Fractional Derivatives

Abstract

Many electrical systems can be characterized more authentically by fractional-order dynamic systems. The Atangana–Baleanu and the Caputo–Fabrizio fractional derivatives have solved the singularity problem in Caputo derivative. This work uses Atangana–Baleanu and Caputo–Fabrizio fractional derivatives to model the fractional-order mutual inductance in the frequency domain. To use the fractional mutual inductance in circuit design, the T-model equivalent circuits are presented with different fractional derivatives. The fractional impedance matching networks based on proposed fractional mutual coupling circuits are simulated as an application. The impedance characteristics of networks with different fractional orders are analyzed. The results indicate that the proposed fractional mutual coupling circuits based on Atangana–Baleanu and Caputo–Fabrizio fractional derivatives can be applied to the complex electrical systems to increase the design degree of freedom, which provides more choices for describing the nonlinear characteristics of the system.

The generalized nonlocal fractal calculus: an efficient tool for fractal circuit analysis

“…the nonsingular kernels (Atangana and Alkahtani, 2016, 2015; Das, 2020; Gómez-Aguilar et al , 2016, 2017; Gómez‐Aguilar, 2018; Liao et al , 2021, 2022a, 2022b; Sheikh et al , 2020). Therefore, the nonlocal fractal calculus is inapplicable to those circuits whose memory effects are described by nonsingular kernels (Atangana and Alkahtani, 2015; Gómez‐Aguilar et al , 2016, 2017; Gómez‐Aguilar, 2018; Liao et al , 2021, 2022b, 2022a; Sheikh et al , 2020). In addition, according to its basis fractional derivatives, the nonlocal fractal calculus-based dynamical equation of any fractal electrical circuit with memory effect can be given in either Riemann–Liouville sense or Caputo sense, where the latter has been chosen (Banchuin, 2022c; Ali et al , 2023; Banchuin, 2023) due to its simpler initial condition handling methodology.…”

“…Unfortunately, albeit such kernel has been often cited, there are many systems, including electrical circuits, whose memory effects have been described by those kernels which are not governed by the power law function, i.e. the nonsingular kernels (Atangana and Alkahtani, 2016, 2015; Das, 2020; Gómez-Aguilar et al , 2016, 2017; Gómez‐Aguilar, 2018; Liao et al , 2021, 2022a, 2022b; Sheikh et al , 2020). Therefore, the nonlocal fractal calculus is inapplicable to those circuits whose memory effects are described by nonsingular kernels (Atangana and Alkahtani, 2015; Gómez‐Aguilar et al , 2016, 2017; Gómez‐Aguilar, 2018; Liao et al , 2021, 2022b, 2022a; Sheikh et al , 2020).…”

A novel generalized fractional‐order memristor model with fully explicit memory description

“…Note also that such Atangana-Baleanu fractional derivative has been applied, cited, and studied in many research areas ranged from electrical engineering to epidemiology. 27,[31][32][33][34][35][36][37][38][39][40][41][42][43][44] In particular, we choose the Atangana-Baleanu fractional derivative in Liouville-Caputo sense rather than the derivative in Riemann-Liouville sense. This is because the former takes the initial condition into its Laplace transformation, 30,33 and we solve the fractional-order memristor's state equation of by means of the Laplace transformation-based methodology.…”

“…Between two well‐known nonsingular kernel fractional derivatives, i.e., the Caputo‐Fabrizio fractional derivative 29 and the Atangana‐Baleanu fractional derivative 30 ; we choose the latter as it is a generalization of the former and also nonlocal. Note also that such Atangana‐Baleanu fractional derivative has been applied, cited, and studied in many research areas ranged from electrical engineering to epidemiology 27,31–44 . In particular, we choose the Atangana‐Baleanu fractional derivative in Liouville‐Caputo sense rather than the derivative in Riemann‐Liouville sense.…”

On the fractional domain analysis of negative group delay circuits

“…This is not the case for singular kernel derivatives that employ singularity at the end of the integrating interval 13 ; thus, their values and so do the resulting circuit element descriptions become inexplicit at such end point of integrating intervals. Since the fractional impedance is caused by the usage of fractional derivative, the advantage of non-singular kernel derivatives is inherited by their resulting impedances that have also been applied in the fractional domain circuit analyses, that is, simple circuits 2,12,14 and mutual inductance circuit, 15 similarly to the well-known singular kernel derivative-based fractional impedances. [16][17][18] However, there exist those circuits that are capable to generate a peculiar negative group delay (NGD) phenomenon.…”