652 BELL SYSTEM TECHSICAL JOi'RSAL 



li'liere //^O, nfi+aii<{), ni«o>0, a3-\-ai<Q, aja^^O, 



/i,+/33< 0,(32^3^0, (2) 



and b\-((ii^ — Aaid)+b-i^[(u-, — d)'-—-^U(,(ii\ + bi-{ai- — Aat4) 



— '2bih-\a3{a« — d) — '2aiai\ — 2b\b3[aia3 — 2d{a« — d)\ 



- 2/)o6,l« i(ao -d)- 2n„fl3] = 0, (3) 

 for all values of d > 0, provided 



-aib-r- + a3b2b3-db3->0, (4) 



-a„b3- + {a-2-d)b3bi-aibi->0, (5) 



- db c + a lb ,/), - «oft-.'- ^ 0, ((>) 



and, conversely, any impedance S of the form (1) satisfying these condi- 

 tions (2)-(6) can be realized as the driving-point impedance of a two- 

 mesh circuit consisting of resistances, capacities, and inductances. 



Theorem I thus gives the most general form of this type of im- 

 pcciaiicc, showing that it is a rational fimction of the time coefficient,'' 

 completely determined, except for a constant facior, In- assigning 

 four niols and two poles, in addition to the poles at zero and infinity, 

 subjiTt lo (HTlaiii cnndilidns. The assigned roots and poles ,irc ilu- 

 time coefficients for the free oscillations of the circuit with tin- dri\ iiiii; 

 branch closed and opened, respecti\el\'. That is, the loots .md ])iiks 

 correspond to the resonant and anti-resonant points of the impedance. 



The conditions are as follows: The real part of each root and pole 

 is negaii\-e or zero; the roots and poles occur in pairs of real or con- 

 jugate complex ciuantities; certain additional restrictions must he 

 satisfied, as stated in terms of the s\nimctric functions of the roots 

 and poles 1)\- formulas (3)-(()). 



\\\ \irtue of these restrictions, the pair of poles, for assigned \alues 

 of the two pairs of roots, is limited to a certain domain of \alues. 

 This domain is con\-eniently illustrated by plfitting, in the upper half 

 of the complex ])laiH-. tin- Idcu^- (jf (uu- |)iik-, tlii' other pok' being its 

 conjugate. I'or real poles, a (k-\ ic e is iist-d to indicate p.iirs ol points 

 on the real axis. I''igs. 3-") show llie doni.iin of the polo, iilotled 

 in this manner, for se\'eral t>i)i(.il i .im>. 



I'rox'ided the roots are imi ,dl ri.d. this domain consists of a con- 

 nected region of values, so that il is possible lo |)ass from one pair 

 of poles to any other pair satisfying the same conditions b\- ,i con- 

 tinuous transformalion. In the case of four real roots, however, the 

 domain consists, in general, of two non-connected regions, as illus- 

 trated in Fig. 5. I'nder these circumstances there is a region of real 

 poles which arc not cf)ntinuously transformable into complex poles. 



The networks realizing any specified dri\ing-point impedance are 



^ .Ml clnlriial oscill.itions i-()iisi(lc'rf<l In this paper are nf the hiiiii < *', uliiir I he 

 limr coefficienl \ = ip may have any value, real or compUx. 



