Abstract
Fractional circuits have attracted extensive attention of scholars and researchers for their superior performance and potential applications. Recently, the fundamentals of the conventional circuit theory were extended to include the new generalized elements and fractional-order elements. As is known to all, circuit theory is a limiting special case of electromagnetic field theory and the characterization of classical circuit elements can be given an elegant electromagnetic interpretation. In this paper, considering fractional-order time derivatives, an electromagnetic field interpretation of fractional-order elements: fractional-order inductor, fractional-order capacitor and fractional-order mutual inductor is presented, in terms of a quasi-static expansion of the fractional Maxwell’s equations. It shows that fractional-order elements can also be interpreted as a fractional electromagnetic system. As the element order equals to 1, the interpretation of fractional-order elements matches that of the classical circuit elements: L, C, and mutual inductor, respectively.
Highlights
The realization and generalization of fractional calculus has become of great significance in many fields [1] such as determining voltage-current relationship in a non-ideal capacitor, fractal behavior of a metal insulator solution interface, electromagnetic waves, and recently in electrical circuits such as filters [2]-[7], oscillators [8] [9] [10], passive realization [11] [12] and energy-related issues in supercapacitors [13]
In this paper, considering fractional-order time derivatives, an electromagnetic field interpretation of fractional-order elements: fractional-order inductor, fractional-order capacitor and fractional-order mutual inductor is presented, in terms of a quasi-static expansion of the fractional Maxwell’s equations. It shows that fractional-order elements can be interpreted as a fractional electromagnetic system
As the element order equals to 1, the interpretation of fractional-order elements matches that of the classical circuit elements: L, C, and mutual inductor, respectively
Summary
The realization and generalization of fractional calculus has become of great significance in many fields [1] such as determining voltage-current relationship in a non-ideal capacitor, fractal behavior of a metal insulator solution interface, electromagnetic waves, and recently in electrical circuits such as filters [2]-[7], oscillators [8] [9] [10], passive realization [11] [12] and energy-related issues in supercapacitors [13]. Tarasov [26] formulated the fractional Green’s, Stokes’ and Gauss’s theorems and realized the proofs of these theorems for simple regions. He considered fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations. The characterization of the four classical circuit elements can be given an elegant electromagnetic interpretation by the quasi-static expansion of Max-well’s equations [31] [32] [33]. This paper is organized as follows: in Section 2, the notions of fractional calculus and fractional-order elements and fractional Maxwell’s equations are presented.
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