Realizability Conditions on n-Port Networks

It is well known that the positive real
(pr)
concept is one of the most important in network theory. Its importance derives from the following two facts: 1) A necessary and sufficient condition for a real rational function to be realizable as the driving-point impedance of a one-port network is that it be a
pr
function. 2) A necessary and sufficient condition for a symmetric nth-order matrix of real rational functions to be realizable as the open-circuit impedance matrix of an
n
-port network is that it be a
pr
matrix. Sets of necessary and sufficient conditions equivalent to the definition of a
pr
function and a pr matrix have been presented in the literature, In this paper new sets of necessary and sufficient conditions are formulated for a rational function and a matrix of rational functions to be
pr
. These conditions give insights that may be useful in research on unsolved synthesis problems; some of these problems are now being studied by the authors. When used for testing purposes none of the new conditions requires root solving, and thus in many cases much of the tedium of previous tests is eliminated.