  This paper develops a general approach to nonlinear circuit modelling aimed
at preserving the intrinsic symmetry of electrical circuits when formulating
reduced models. The goal is to provide a framework accommodating such
reductions in a global manner and without any loss of generality in the working
assumptions; that is, we avoid global hypotheses imposing the existence of a
classical circuit variable controlling each device. Classical (voltage/current
but also flux/charge) models are easily obtained as particular cases of a
general homogeneous model. Our approach extends the results introduced for
linear circuits in a previous paper, by means of a systematic use of global
parametrizations of smooth planar curves. This makes it possible to formulate
reduced models in terms of homogeneous variables also in the nonlinear context:
contrary to voltages and currents (and also to fluxes and charges), homogeneous
variables qualify as state variables in reduced models of uncoupled circuits
without any restriction in the characteristics of devices. The inherent
symmetry of this formalism makes it possible to address in broad generality
certain analytical problems in nonlinear circuit theory, such as the
state-space problem and related issues involving impasse phenomena, as well as
index analyses of differential-algebraic models. Our framework applies also to
circuits with memristors, and can be extended to include controlled sources and
coupling effects. Several examples illustrate the results.
