Introduction to Formal Realizability 

 Theory — II 



By BROCKWAY McMILLAN 



(M:imi8cri|)t r(>c(>iv'<Ml F('l)ni:iry 15, 1952) 



This pdii of Ihc paper c.rhihil.s d iKhrorlc lu realize a given positive real 

 iuipcdancc matrix. 



1. IXTRODUCTIOX TO PART II 



1.0 111 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 Z,-,(p), 

 1 < '", •'>■ ^ >h where 



(i) Each Z,.,(p) is a rational function 



(ii) ZUP) = Zr.iP) 

 Cm) Zrsip) = Zsrip) 



(i\-) For each set of real constants A'l , ■ • • , k\ , the function 



<Pz{p) = £ Zrsip)lCrks 

 r,s=l 



has a non-negative real part whenever Re(p) > 0. 



Then there exists a finite passive network, a 2/i-pole, 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 

 l)roof we shall recjuire from Part I certain specific properties of positi\'e 

 real operators and matrices. These will be summarized in Section 2 be- 

 low. Further than this, the present part is almost independent of Part I, 



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