We present a general theory of linear continuous circuits (microwave networks, waveguides, transmission lines, etc.) based on Lie theory. It is shown that the fundamental relationship between the low- and high-frequency circuits can be fully understood via the machinery of Lie groups. By identifying classes of distributed-parameter circuits with matrix (Lie) groups, we manage to derive the most general differential equation of the n-port network, in which its low-frequency (infinitesimal) circuit turns out to be the associated Lie algebra. This equation is based on identifying a circuit Hamiltonian derived by heavily exploiting the Lie-group-theoretic structure of continuous circuits. The solution of the equation yields the circuit propagator and is formally expressed in terms of ordered exponential operators similar to the quantum field theory's formula of perturbation theory (Dyson expansion). Moreover, the infinitesimal operators generating the per-unit-length lumped element local circuit approximation appear to correspond to operators (such as observables) in quantum theory. This analogy between quantum theory and circuit theory through a shared Hamiltonian and propagator structure is expected to be beneficial for the two separate disciplines both conceptually and computationally. Several applications are presented in the field of microwave network analysis where we introduce and study the Lie algebras of important generic classes of circuits, such as lossless, reciprocal, and nonreciprocal networks. Applications to the problems of generalized matching and representation theorems in terms of uniform transmission lines are also outlined using topological methods derived from our Lie-theoretic formulation and exact theorems on continuous matching are obtained to illustrate the potential practical use of the theory.
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