PRinXG-POINT IMPEDASCR OP TirOMIiSH CIKCIIIS 659 



fithfr .1 pair of ronjujsate coinplex iju.iniitirs or a pair of real quan- 

 tities, with the added provision thai the rt'.il pan of each root and 

 pole is negati\e or zero. 



Stated in terms of (lb), these general reslrirtions (2) reipiire all the 

 coetticients to Iw real and to have the s;inie sign. Throughout this 

 paper these signs will always be taken positive: thus all the a's and 

 /»'s are positive or zero. In order to provide that the real part of 

 each root lie negative or zero, the coethiients of the numerator must 

 satisfy the additional requirement 



— (J4rti--f aiOoOa — aoaa=>0, (22) 



ami also n." — -taoa4^(). (2:?) 



The second condition (23) is satisfied aulumaiically by \irtuc of the 

 tirst condition (22), unless both ai and a,i are zero; in that case (23) 

 is re(iuired. These are precisely the necessary and suftirient condi- 

 tions that the numerator of (lb) be factorable into two real quadratic 

 factors with positive coefficients. 



In addition to the general restrictions (2) upon the individual roots 

 and poles, there are certain additional conditions which must be 

 satisfied by all the roots and poles together. These conditions are 

 more conveniently stated in terms of the coefficients by prescribing 

 a certain domain of \'alues of the eight coefficients (ao. Oi, as, fla, ««, 

 ^i. (>2. bi) such that the coefficients of any driving-point impedance 

 of a two-mesh circuit lie in this domain, and, converseK', any set of 

 values in this domain can be realized as the coefficients of a driving- 

 point impedance of a two-mesh circuit. 



By a realizable circuit is unilerstood a circuit consisting of resist- 

 ances, capacities, and self-inductances, with positive or zero values, 

 together with mutual inductances with values such that every prin- 

 cipal minor of the determinant of the inductances is positi\e or zero. 

 In the case of two self-inductances with mutual inductance between 

 them, this reduces to the well known condition LxL^— M-^0. 



The domain is defined analytically by formulas (3)-0)), in terms 

 of a parameter d. This parameter is intimately related to the resist- 

 ances in the circuit, as will lie shown later. In order that this domain 

 ma>- contain real \alues, the following relation must In? satisfied: 



-(/' + 2<j//'--(ai(i.i-f-(ir-4a,K/t)'/+( -« 411 |-+(i|(ij(ri -«,/!. r) >(), (24) 



or in equivalent form, 



— [d — ao{ai+at)iat+ai)][d — ao(ai+a3)(ai+aO] 



ld-ao{ai+at){ai + a3)]^0. (25) 



