382 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



or zero, when the real part of p is positive, or zero. In view of conditions 

 (1) and (2) above, only one test^^ need be applied to each pair of terms 

 of the series (1): the sum of a pair of terms will be a positive real func- 

 tion if, and only if, the real part of their sum is greater than, or equal to, 

 zero for all purely imaginary values of p. 



The general term pair for which a network representation is sought is 



F,Xp) = -^^ + -^^ + ^^ + i^ = P^Xp) - PniO) (2) 



P — Pn P — Vn Vn Vn 



Evidently two cases can be distinguished at the outset, depending upon 

 whether P„(0) is positive or negative. If P„(0) is positive, the network 

 branch, in order to be realizable, should be designed to represent Pniv)- 

 The left-over negative term, — P„(0), then can be absorbed in the posi- 

 tive first term, F(0), of the series (1); more will be said of this later. If, 

 on the other hand, P„(0) is negative, the network branch should repre- 

 sent the whole term, P„(p) — P«(0). This procedure insures that the 

 real part of the branch impedance will be positive, or zero, at zero and 

 infinite frequencies. To guarantee that the resistance is positive at all 

 other frequencies requires further tests now to be specified. 



Let the real and imaginary coefficients of the poles and residues of 

 the n^^ term be 



(With this notation, «„ and /3„ are always positive ; a„ and 6„ can be either 

 positive or negative.) Then (dropping the subscripts) 



p. . _ 2{aa — hl3) + 2ap 

 ^^ ~ a^ -f ^2 _^ 2ap + p2 



„.p,. ., 2{aa - h^)(a' -f /3') + 2o:\aa + 6/3) ,^. 



R[P{tco)\ = ., , ., — (3) 



[a- + («-)- + 2(,}-{a- - I3-) -fco* 



P(0) = 



R[P{io^) - P(0)] = 



2{aa - 5/3) 

 «- + ^- 



-2{aa - Sab(3 - 3aa;g^ + 5/j^)co' - 2{aa - hjSW 

 (a2 + ^'')[(a'' + /32)2 + 2(a2 - /32)a;2 + co*] 



The necessary and sufficient conditions for the real part of a rational 

 function of p to be positive, or zero, for purely imaginary values of p are 

 that the function be positive for p — > ±t=o and have no imaginary roots 

 of odd multiphcity. When this test is applied to the functions P(p) and 



