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A compositional framework for passive linear networks

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John C. Baez and Brendan Fong

Passive linear networks are used in a wide variety of engineering
applications, but the best studied are electrical circuits made of
resistors, inductors and capacitors. We describe a category where a
morphism is a circuit of this sort with marked input and output
terminals. In this category, composition describes the process of
attaching the outputs of one circuit to the inputs of another. We
construct a functor, dubbed the `black box functor', that takes a
circuit, forgets its internal structure, and remembers only its external
behavior. Two circuits have the same external behavior if and only if
they impose same relation between currents and potentials at their
terminals. The space of these currents and potentials naturally has the
structure of a symplectic vector space, and the relation imposed by a
circuit is a Lagrangian linear relation. Thus, the black box functor
goes from our category of circuits to a category with Lagrangian linear
relations as morphisms. We prove that this functor is symmetric
monoidal and indeed a hypergraph functor. We assume the reader is
familiar with category theory, but not with circuit theory or symplectic
linear algebra.

Keywords:
passive linear network, electric circuit, principle of minimum power,
black box, decorated cospan, compact closed category, hypergraph category,
Lagrangian relation

2010 MSC:
18C10, 18D10, 53D12, 94C05

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 38, pp 1158-1222.

Published 2018-11-18.

http://www.tac.mta.ca/tac/volumes/33/38/33-38.pdf

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