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

Abstract

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