Nonlocal fractal calculus based analyses of electrical circuits on fractal set

Abstract

Purpose

The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus.

Design/methodology/approach

The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited.

Findings

The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead.

Originality/value

For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.

Citation

Banchuin, R. (2022), "Nonlocal fractal calculus based analyses of electrical circuits on fractal set", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 41 No. 1, pp. 528-549. https://doi.org/10.1108/COMPEL-06-2021-0210