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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.
Highlights
The classical approach to the analysis and modeling of electrical systems usually ignores the effects of the non-ideality of elements and allows one to obtain a mathematical model based on integer-order differential equations
Classical differential and integral calculus is a well-studied area of mathematics, and its application can be found in numerous publications on the analysis of differentiation and integration, solving ordinary and partial differential equations, integral equations, etc
It can be stated that the use of fractional-order magnetic coupling between two coils makes it possible to avoid the Dirac impulse that is used in the classical approach
Summary
The classical (standard) approach to the analysis and modeling of electrical systems usually ignores the effects of the non-ideality of elements and allows one to obtain a mathematical model based on integer-order differential equations. The use of this approach does not always lead to precise models. Descriptions employing differential calculus of fractional order have been coils (losstocoils) with the skin occurring effect, especially those with This soft ferromagnetic cores, or coils) containing extended include phenomena in real inductances. The main part of the paper is covered α inverse Laplace transform of fractional-order systems allow the approximation of the s factor by the by Section presenting the fractional‐order coupling coils.byThis section quotient of 4, polynomials withanalysis integerof powers.
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