Introduction to Formal Realizability Theory - II

This pari of the paper exhibits a network to realize a given positive real impedance matrix. 1.0 Iii this part of the paper we prove the following theorem: 1.1 Theorem: Let Z(p) be an n X n matrix whose elements are Zr ,(p), 1 r, s n, where (i) Each Zra (p) is a rational function (ii) Zr,(p) = Zrs{p) (iii) Zn (p) = Z,r (p) (iv) For each set of real constants ki, · · · , k,, , tlie function n r,s=1 has a non-negative real part whenever Re(p) > 0. Then there exists a finite passive network, a 2/i-polc, which has the impedance matrix Z(p). A dual result holds for admittance matrices Y(p). 1.2 The converse of this theorem was proved in Part I: that if a finite passive 2/i-pole has an impedance matrix Z(p), then this matrix has properties (i) through (iv) of 1.1. 1.3 We recall that in Part I matrices satisfying the conditions of 1.1 were called positive real (PR). 1.4 The proof of 1.1 is a direct generalization to matrices of the Brune process" for realizing a two-pole impedance function f(p). For this proof we shall require from Part I certain specific properties of positive real operators and matrices. These will be summarized in Section 2 below. Further than this, the present part is almost independent of Part I, 541