A Compositional Framework for Passive Linear Networks

TL;DR

This paper develops a categorical framework for passive linear networks, representing circuits as morphisms and using a functor to capture their external behavior through symplectic geometry, unifying circuit composition and behavior analysis.

Contribution

It introduces a categorical model for passive linear networks and constructs a functor mapping circuits to symplectic vector spaces, capturing their external behavior in a mathematically rigorous way.

Findings

The black box functor is symmetric monoidal.

Circuits correspond to Lagrangian linear relations.

The framework unifies circuit composition with symplectic geometry.

Abstract

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 the category of symplectic vector spaces and Lagrangian linear relations. We prove that this functor is 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.