The realizability boundary and the realization of nonsingular fourth-order conductance matrices by 6-terminal networks

A new approach to the realization of
n
th-order conductance matrices by
(n +p)
-terminal,
n
-port networks is described. Results due to Cederbaum and Foster are employed. Cederbaum has given an example of a paramount matrix that is not realizable as either a resistance matrix conductance matrix. Foster has shown that any dominant conductance matrix is realizable. A conductance matrix with any fixed set of off-diagonal elements can be made dominant, by suitably increasing the diagonal elements. Thus we conclude that some sets of diagonal elements will permit a paramount conductance matrix to be realized, while others will not. Therefore, given a conductance matrix with a fixed set of off-diagonal elements, there exists a continuum of
n
-tuples of diagonal elements which are on the border line of realizability. We define this continuum as the realizability boundary. Networks which realize conductance matrices on the realizability boundary are called minimal networks and are characterized by not having shunt conductances at the ports. Next we apply these principles to the solution of the
(n+2)
-terminal 4-port case. Using a procedure derived from Guillemin, we find and catalog all possible minimal networks. For any given situation, a maximum of 22 networks is required to be tested. Synthesis procedures and examples are given. In addition, we show that certain ports of some minimal networks have constant driving-point conductances. We illustrate how to predict this effect, and suggest a practical application of this result.