On the Compactness of an RC Quadripole

Consider an RC quadripole with open-circuit impedances
z_{11}(s)
,
z_{12}(s)
, and
z_{22}(s)
, and let
s_v
be a pole of any of these functions. If
k_{11}(v), k_{12}(v)
, and
k_{22}(v)
are the respective residues of these three functions at
s_v
, then it is well-known that
k_{11}(v)k_{22}(v) - [k_{12}(v)]^2 \geq 0
. If the inequality is an equality, then
s_v
is called a compact pole; if every such pole is compact, the network is called compact. In this paper two new properties of compactness are exhibited and discussed. It is shown that if the network is grounded, noncompactness implies certain degeneracies in the determinant of the nodal admittance matrix and its cofactors. If the network is terminated with a resistance
R
, it is shown that for all but a finite number of values of
R
, the overall terminated network is compact. Thus, a noncompact resistance terminated RC quadripole can be approximated to any degree of accuracy by a compact network of this type, which implies that noncompactness is not detectable by terminal measurements of the open-circuit impedances.