Günter Rote:

Characterization of the response maps of alternating-current networks

Electronic Journal of Linear Algebra, 36 (2020), 697–703. doi:10.13001/ela.2020.4981, arXiv:1904.02031 [math.CO],  →BibTeX

Abstract

An alternating-current network is an undirected graph in which each edge is weighted by a conductance, a complex number with positive real part. Some nodes are designated as boundary nodes. The response map is the linear map from the vector of voltages at the boundary nodes to the vector of currents flowing into the network through these nodes.

We prove that the known necessary conditions for a linear map to be a response map are sufficient, and we show how to construct an appropriate alternating-current network for a given response map.

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Note on Priority

The result of this paper has been earlier obtained by Graeme W. Milton and Pierre Seppecher:

Realizable response matrices of multi-terminal electrical, acoustic and elastodynamic networks at a given frequency. Proc. Royal Soc. A 464 (2008), 967–986. doi:10.1098/rspa.2007.0345, arXiv:0712.1066.
Theorem 2 of this paper gives this result, and Theorem 3 strengthens it to a realization as a planar network where all terminals are located at the outer boundary.

Last update: September 17, 2024.