On the test of novel constitutive relation of capacitor for electrical circuit analysis: a fractal calculus-based approach

Abstract

Purpose The purpose of this paper is to test the capability to properly analyze the electrical circuits of a novel constitutive relation of capacitor. Design/methodology/approach For ceteris paribus, the constitutive relations of the resistor and inductor have been reformulated by following the novel constitutive relation of capacitor. The responses of RL, RC, LC and RLC circuits defined on the fractal set described by these definitions have been derived by means of the fractal calculus and fractal Laplace transformation. A comparative Hamiltonian formalism-based analysis has been performed where the circuits described by the conventional and the formerly proposed revisited constitutive relations have also been considered. Findings This study has found that the novel constitutive relations give unreasonable results unlike the conventional ones. Like such previous revisited constitutive relations, an odd Hamiltonian has been obtained. On the other hand, the conventional constitutive relations give a reasonable Hamiltonian. Originality/value To the best of the author’s knowledge, for the first time, the analysis of fractal set defined electrical circuits by means of unconventional constitutive relations has been performed where the deficiency of the tested capacitive constitutive relation has been pointed out.

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

“…Recently, the local fractional derivative (LFD) (Yang, 2011; Yang et al , 2015a, 2015b) is a new definition of the fractional derivative and has been successfully applied to describe many complex phenomena involving in the wave (Yang et al , 2019), diffusion (Yang et al , 2017a, 2017b), physics (Zhang and Yang, 2016; Wang and Si, 2023; Yang, 2017; Wang and Shi, 2023a, 2023b; Wang, 2023a, 2023b) and so on (Yang et al , 2014; Yang et al , 2016). As an important field, the fractional calculus has been used widely to model the fractal electrical systems such as the fractal LC electric circuit (Yang et al , 2017a, 2017b), fractal RC circuit (Zhao et al , 2017), fractal RL high-pass filter (Wang and Li, 2020), fractal high-pass filter (Wang, 2020) and so on (Banchuin, 2023; Banchuin, 2022; Wang et al , 2020). Inspired by recent research results on the fractal electrical systems, in this paper, we will derive a fractal active low-pass filter (LPF) based on the LFD, and investigate the dynamic characteristics via the local fractional Laplace transform (LFLT) and inverse local fractional Laplace transform (ILFLT).…”

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). Inspired and encouraged by the research results in electrical and electronic engineering, here in this paper, we aim to develop a new fractional pulse narrowing nonlinear transmission lines model based on the LFD for the first time and present a novel method to seek for the nondifferentiable (ND) exact solutions.…”