  This paper presents a general framework for linear circuit analysis based on
elementary aspects of projective geometry. We use a flexible approach in which
no a priori assignment of an electrical nature to the circuit branches is
necessary. Such an assignment is eventually done just by setting certain model
parameters, in a way which avoids the need for a distinction between voltage
and current sources and, additionally, makes it possible to get rid of voltage-
or current-control assumptions on the impedances. This paves the way for a
completely general $m$-dimensional reduction of any circuit defined by $m$
two-terminal, uncoupled linear elements, contrary to most classical methods
which at one step or another impose certain restrictions on the allowed
devices. The reduction has the form $$\begin{pmatrix} AP \\ BQ \end{pmatrix} u
= \begin{pmatrix} AQ \\ -BP \end{pmatrix} \bar{s}.$$ Here, $A$ and $B$ capture
the graph topology, whereas $P$, $Q$, $\bar{s}$ comprise homogeneous
descriptions of all the circuit elements; the unknown $u$ is an $m$-dimensional
vector of (say) ``seed'' variables from which currents and voltages are
obtained as $i=Pu -Q\bar{s}$, $v=Qu + P\bar{s}$. Computational implementations
are straightforward. These models allow for a general characterization of
non-degenerate configurations in terms of the multihomogeneous Kirchhoff
polynomial, and in this direction we present some results of independent
interest involving the matrix-tree theorem. Our approach can be easily combined
with classical methods by using homogeneous descriptions only for certain
branches, yielding partially homogeneous models. We also indicate how to
accommodate controlled sources and coupled devices in the homogeneous
framework. Several examples illustrate the results.
