A NEW STUDY OF THE PROBLEM OF COMPATIBLE IMPEDANCES

It is a classical result, known as Darlington's theorem, that every ra tional positive-real function z(p) is realizable as the input impedanc e of a lumped reciprocal reactance two-port tuner N-t closed at the fa r end on 1 Omega. The theorem is evidently false if the 1 Omega termin ation is replaced by some prescribed non-constant positive-real impeda nce z(1)(p). Any z(p) synthesizable in this more restrictive manner is said to be compatible with z(1)(p) and we write z similar to z(1) to indicate the correspondence. The determination of necessary and suffic ient conditions for the validity of z similar to z(1) is the problem o f compatible impedances. Of the four better-known network treatments, only that of Schoeffler (IRE Trans. Circuit Theory, CT-8, 131-137 (196 1) is completely correct, although severely restricted in scope. In pa rticular, the remaining three contain a common error which appears to have propagated because a constraint on the ratio of even parts z(e)(p )/z(le)(p) derived by Schoeffler is unnecessary if z(p) is not minimum -reactance. The main theorems of Wohlers (IEEE Trans. Circuit Theory, CT-12, 528-535 (1965)) and Satyanaryana and Chen (J. Franklin Inst., 3 09, 267-280 (1980)) are very similar in structure to our Theorem 3 but considerably more complex and do not provide a sufficiently explicit description of the associated tuner Nt. In fact (Theorem 1), with the exception of a physically irrelevant degeneracy (which is easily detec ted), Nt, when it exists, must possess an impedance matrix Z(p). Moreo ver, the latter can be effectively parametrized in terms of z(p), z(1) (p) and a regular-allpass b(p) found as the solution of a standard int erpolation problem of the Nevalinna-Pick type. Three fully worked exam ples clarify the theory and also illustrate many of the numerical step s. (C) John Wiley & Sons, Ltd.