A Contribution to the Foundations of the Generalized Theory of Electrical Circuits: Concepts, Methodology and Axiomatics

The aim of this article is to create a conceptual and methodological foundation for a new, generalized theory of electrical circuits. In the generalized theory, it is assumed that the relation between voltage and current of inductors and capacitors can be described by derivatives and integrals of arbitrary order, including fractional ones. Traditional electrical engineering assumes that this relation is defined by first-order operators and represents a special case of the generalized theory. The necessity of a generalized theory is caused by a contradiction. On the one hand, there exists a broad class of elements and circuits whose behaviour is not described by the traditional theory. These include systems with memory effects, distributed and network structures, electrochemical elements, supercapacitors, and composite materials. On the other hand, their behaviour does not have a unified theoretical justification within the framework of classical circuit theory. The article analyses this contradiction and identifies the limitations inherent in the classical approach. A coherent conceptual framework of the generalized circuit theory is introduced. Methodological requirements are formulated, and a minimal axiomatics is constructed to ensure a consistent circuit description of elements with arbitrary temporal dynamics without revising the fundamental laws of electrical circuits. The scientific novelty of the obtained results lies in the formation of a conceptual basis for the further development of a generalized theory of electrical circuits. It is intended to eliminate the theoretical fragmentation and insufficient rigor characteristic of existing fractional and integral models, while preserving the basic concepts and laws of traditional circuit theory.