Theoretical Foundations of Fractal Electrotechnic. Fractal Elements and its Properties

The article is devoted to the foundations of the theory of fractal electric circuits, which generalizes the traditional theory of circuits to the case of the connection between currents and voltages of elements by derivatives of arbitrary order, including fractional ones. The dependence of the active and reactive components of the complex of the impedance of the fractal element on the degree of the derivative is investigated. The analysis of properties of ideal fractal elements is carried out. Based on this analysis, it was concluded that the traditional inductor, resistor and capacitor are special cases of a fractal element in the basis of powers of derivatives (1, 0, -1). It is shown that, unlike traditional ideal elements, a fractal element can have unusual properties, such as the dependence of active resistance on frequency or the presence of negative active resistance. A family of frequency characteristics of a fractal element is constructed depending on the degree of the derivative. The results obtained in the article can be the basis for studying the properties of fractal electric circuits in various operating modes.