Kron's method and cell complexes for magnetomotive and electromotive forces

TL;DR

This paper develops a topological and geometric framework for analyzing electrical networks, linking currents, fluxes, and forces through Kron's tensor analysis and extending to non-local interactions and network relations.

Contribution

It introduces a novel topological and geometric approach to Kron's method, incorporating cell complexes, Hodge theory, and category theory to analyze network interactions.

Findings

Currents are boundaries of surface fluxes.

Flux creates electromotive forces via a metric and Hodge operator.

The framework extends to non-local interactions and network relations.

Abstract

Starting from topological principles we first recall the elementary ones giving Kirchhoff's laws for current conservation. Using in a second step the properties of spaning tree, we show that currents are under one hypothesis intrinsically boundaries of surfaces flux. Naturally flux appears as the object from which the edge comes from. The current becomes the magnetomotive force (mmf) that creates the flux in the magnetostatic representation. Using a metric and an Hodge's operator, this flux creates an electromotive force (emf). This emf is finally linked with the current to give the fundamental tensor - or "metric" - of the Kron's tensorial analysis of networks. As it results in a link between currents of cycles (surface boundaries) and energy sources in the network, we propose to symbolize this cross talk using chords between cycles in the graph structure on which the topology is based. Starting then from energies relations we show that this metric is the Lagrange's operator of the circuit. But introducing moment space, the previous results can be extended to non local interactions as far field one. And to conclude, we use the same principle to create general relation of information exchange between networks as functors between categories.