  Following a consistent geometrical description previously introduced in
Parra-Rodriguez et al. (2023), we present an exact method for obtaining
canonically quantisable Hamiltonian descriptions of nonlinear, nonreciprocal
quasi-lumped electrical networks. Utilising the Faddeev-Jackiw method once
more, we identify and classify all possible singularities arising in the quest
for Hamiltonian descriptions of general quasi-lumped element networks, and we
offer systematic solutions to them--a major challenge in the context of
canonical circuit quantisation. Accordingly, the solution relies on the correct
identification of the reduced classical circuit-state manifold, i.e., a mix of
flux and charge fields and functions. Starting from the geometrical description
of the transmission line, we provide a complete program including lines coupled
to one-port lumped-element networks, as well as multiple lines connected to
nonlinear lumped-element networks. On the way, we naturally extend the
canonical quantisation of transmission lines coupled through
frequency-dependent, nonreciprocal linear systems, such as practical
circulators. Additionally, we demonstrate how our method seamlessly facilitates
the characterisation of general nonreciprocal, dissipative linear environments.
This is achieved by extending the Caldeira-Leggett formalism, utilising
continuous limits of series of immittance matrices. We expect this work to
become a useful tool in the analysis and design of electrical circuits and of
special interest in the context of canonical quantisation of superconducting
networks. For instance, this work will provide a solid ground for a precise
input-output theory in the presence of nonreciprocal devices, e.g., within
waveguide QED platforms.
