Properties of Fractional-Order Magnetic Coupling

Abstract

This paper presents the properties of fractional-order magnetic coupling. The difficulties connected with the analysis of two coils in dynamic states, resulting from the classical approach, provided motivation for studying the properties of fractional-order magnetic coupling. These difficulties arise from failure to comply with the commutation laws, i.e., a sudden power disappearance in the primary winding caused by a switch-mode power supply. Theoretically, under ideal conditions, a sudden power disappearance in the coil is, according to the classical method, manifested by a sudden voltage surge in the form of the Dirac delta function. As is well-known, it is difficult to obtain such ideal conditions in practice; the time of current disappearance does not equal zero due to the circuit breaker’s imperfection (even when electronic circuit breakers are used, the time equals several hundred nanoseconds). Furthermore, it is necessary to take into account phenomena occurring in real inductances, such as the skin effect, the influence of the ferromagnetic core and many other factors. It would be very difficult to model all these phenomena using classical differential calculus. The application of fractional-order differential calculus makes it possible to model them in a simple way by appropriate selection of coefficients and fractional-order derivatives. It should be mentioned that the analysis could be used, for example, in the case of high-voltage generation systems, including spark ignition systems of internal combustion engines. The use of fractional-order differential calculus will allow for more accurate modeling of phenomena occurring in such systems.

Modeling of Internal Combustion Engine Ignition Systems with a Circuit Containing Fractional-Order Elements

“…For many years, there has been a significant increase in the number of research on the practical application of fractional order calculus. A review of the practical applications of this calculus is given, among others, in works on the study of electrotechnical components and devices, such as supercapacitors, resistors with memory referred to as "memristors" [10,11], as well as coils with inductance with a skin effect [12], and fractional mutual inductance coils [13,14]. Fractional order elements C α and L described by fractional-order differentials and integral calculus are introduced as a generalization of classical elements [15][16][17][18][19].…”

Solid-Rotor Induction Motor Modeling Based on Circuit Model Utilizing Fractional-Order Derivatives

“…However, for the noninteger order model, the number of identified parameters is five, which is an advantage of using this model. Physical phenomena occurring in ferromagnetic elements of electrical machines or in the rotor cage bars of induction or synchronous machines can be described using fractional-order differential calculus [19][20][21][22]. The fractional-order operational inductance in the frequency domain represents the impedance whose resistance and inductance are a function of the frequency of the eddy current induced in the solid rotor of the machine.…”