On Equivalence of Resistive n-Port Networks

In this paper the problem of realization of a resistive
n
-port network from its paramount admittance matrix
Y
is considered from a geometric point of view. Augmentation of the set of
n
ports to a linear
2n - 1
tree on a complete graph with
2n
vertices leads to a system of
n(n + 1)
equations in
q = n (2n - 1)
unknowns. All the real solutions of this system correspond to equivalent
n
ports. They may be looked on as points of a manifold
L
in the
q
-space
E^{q}
. The realization of Y in the conventional sense being constrained to non-negative resistor values is equivalent to finding a point of intersection of
L
and the non-negative orthant
P
of
E^{q}
. In this paper the theory of equivalence is studied and some properties of L are defined. Similarly to the known method of the decomposition of the admittance matrix Y of a resistive
n
-port network with
n + 1
vertices into a unimodular congruence
Y = CGC'
, this paper proposes a decomposition of a general admittance matrix Y into a subunimodular congruence
Y = CGC'
.