The fractal active low-pass filter within the local fractional derivative on the Cantor set

Abstract

Purpose The purpose of this paper is to derive a new fractal active low-pass filter (LPF) within the local fractional derivative (LFD) calculus on the Cantor set (CS). Design/methodology/approach To the best of the author’s knowledge, a new fractal active LPF within the LFD on the CS is proposed for the first time in this work. By defining the nondifferentiable (ND) lumped elements on the fractal set, the author successfully extracted its ND transfer function by applying the local fractional Laplace transform. The properties of the ND transfer function on the CS are elaborated in detail. Findings The comparative results between the fractal active LPF (for γ = ln2/ln3) and the classic one (for γ = 1) on the amplitude–frequency and phase–frequency characteristics show that the proposed method is correct and effective, and is expected to shed light on the theory study of the fractal electrical systems. Originality/value To the best of the author’s knowledge, the fractal active LPF within the LFD calculus on the CS is proposed for the first time in this study. The proposed method can be used to study the other problems in the fractal electrical systems, and is expected to shed a light on the theory study of the fractal electrical systems.

On the zero state-response of the ℑ-order R-C circuit within the local fractional calculus

“…1641 2016) and so on Ghanbari and Abdon, 2020;Wang, 2022b). In recent years, as a new theory of fractional calculus, the local fractional calculus has been successfully used to explain many non-differentiable (ND) scientific problems, for instance, the shallow water surfaces (Yang et al, 2016), rheological (Yang et al, 2017a), physics (Wang et al, , 2023cYang, 2017;Wang, 2023c), circuits (Yang et al, 2017b;Zhao et al, 2017;Wang, 2023b;Banchuin, 2022;Banchuin, 2023;Wang et al, 2020), vibration and others. Inspired by the recent research results on fractal circuits, the purpose of this article is to derive a new I-order R-C zero state-response circuit (ZSRC) within the local fractional derivative (LFD).…”

“…In recent decades, researchers pay more and more attention to the fractional calculus since it can accurately describe some strange phenomena in many scientific research fields, such as the porous media (Xiao et al , 2019; Wang, 2023a; Xiao et al , 2021; Wang and Shi, 2023a), non-smooth boundary (He et al , 2021; Wang et al , 2023a; Wang, 2022a), image analysis (Ghamisi et al , 2012), control (Ladaci and Charef, 2006), diffusion (Ammi et al , 2019; Atangana, 2016) and so on (Wang et al , 2023a; Ghanbari and Abdon, 2020; Wang, 2022b). In recent years, as a new theory of fractional calculus, the local fractional calculus has been successfully used to explain many non-differentiable (ND) scientific problems, for instance, the shallow water surfaces (Yang et al , 2016), rheological (Yang et al , 2017a), physics (Wang et al , 2023b, 2023c; Yang, 2017; Wang and Shi, 2023b; Wang, 2023c), circuits (Yang et al , 2017b; Zhao et al , 2017; Wang, 2023b; Banchuin, 2022; Banchuin, 2023; Wang et al , 2020), vibration (Yang and Srivastava, 2015) and others. Inspired by the recent research results on fractal circuits, the purpose of this article is to derive a new ℑ-order R-C zero state-response circuit (ZSRC) within the local fractional derivative (LFD).…”

The pulse narrowing nonlinear transmission lines model within the local fractional calculus on the Cantor sets

“…Recently, the fractal and fractional calculus have received considerable attention in a number of different fields because they can model many complex phenomena occurring in extreme conditions such as the unsmooth boundary (He et al , 2021a; Wang, 2022; He et al , 2021b; Wang, 2023a), microgravity space (He, 2020; He et al , 2021c), porous media (Xiao et al , 2019; Wang, 2023b; Xiao, 2021; Wang, 2023c) and so on (Atangana, 2016; Wang et al , 2023a). Among the different fractional derivatives, the local fractional derivatives (LFDs) play an increasingly important role in the electrical and electronic engineering involving in the fractal LC-electric circuit (Yang et al , 2017a), fractal RC circuit (Zhao et al , 2017), fractal filter (Wang and Shi, 2023) and so on (Banchuin, 2023; Banchuin, 2022; Sikora and Pawłowski, 2018).…”

Fractional Hamilton’s Canonical Equations and Poisson Theorem of Mechanical Systems with Fractional Factor

“…There are various definitions of fractional derivative and fractional integral, such as Riemann-Liouville (RL), Caputo, Riesz, Grunwald-Letnikov, He's fractional derivative, etc. These definitions are popular among mathematicians and physicists [32][33][34]. However, previous studies are based on fractional derivatives defined in integral form, so it is very difficult to use these definitions.…”