Necessary Conditions on the Matrix of an RC Grounded Quadripole

Conjectures of Lucal and Darlington on the synthesis of the matrix of an RC three-terminal network are examined in this paper. Suppose each of the three open-circuit impedances,
z_{11}(s), z_{12}(s)
, and
z_{22}(s)
, is specified as the quotient of two relatively prime polynomials; each is then rewritten as a rational fraction with the polynomial
W(s)
as denominator, where
W(s)
is the least common multiple of the three denominators. Lucal and Boxall conjecture that in this representation every numerator coefficient of each impedance is nonnegative, and in addition, each coefficient in the numerator of
z_{12}(s)
does not exceed the corresponding coefficient in the numerator of
z_{11}(s)
and
z_{22}(s)
. It is observed that these two propositions are equivalent necessary conditions, and a counterexample to the first proposition is exhibited. Darlington conjectures that any realizable RC three-terminal network can be synthesized by the
\pi -T
decomposition method described by Lucal. This counterexample casts some light on Darlington's conjecture. An examination of the above counterexample suggests another condition which may be necessary. It is shown that if the number of nodes of the network does not exceed five, this new condition is indeed necessary; however, it is not known whether it remains valid when the number of nodes exceeds five. In the discussion of these matters a new proof is exhibited of the fact that each root of the determinant of the nodal admittance matrix is a nonpositive real number.