652 BELL SYSTEM TECHXICIL JOi'RXAL 



ichere II>{). rti+ofj<0, oioo^O, a3 + a.i<0, ajajSrO, 



/j,+pi3<0, |3o/33>0, (2) 



and b\-(a:i^ — Aa.4)+b-i-\{a-i — d)'- — Aai)ai\ + bj^{ai- — Aa^) 



-•2b^h.\a:i{a«-d)-2a^a^\-2b^b3{alai-2d{a2-d)] 

 - 2h.,b,["iia.2-d) - 2cw,] = 0, (3) 



for (ill values of d>{}, provided 



-aibr+a3bnb3-db3->0, (4) 



-(iJ)x^ + {a.-d)b,bi-atbr>0, (5) 



-dbr + a,bih-2-(h,b-y>0, (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 gi\es the most general form of this type of im- 

 pedance, showing that it is a rational function of the tinu' coefficient/ 

 completely determined, except for a constant factor, by assigning 

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

 subject to certain conditions. The assigned roots and poks arc ihc 

 time coefficients for the free oscillations of the circuit with the dri\ing 

 branch closed and opened, respecti\el\ . That is, the routs and poles 

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



The conditions are as follows; 'I'he real part of each root and pole 

 is negative or zero; the roots and poles occur in pairs of real or con- 

 jugate complex C|uantities; certain additional restrictions must be 

 satisfied, as stated in terms of the symmetric functions of the roots 

 and i)oles by formulas (3) (G). 



!?>• \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\enientl\' illustrated by plotting, in the upper half 

 of the complex plane, the locus of one pole, the other pole being its 

 conjugate. I*"or real poles, a de\ice is used to indicate pairs of |)oiiits 

 on the real axis. Figs. 3-5 show the donuiin of the jxiU-, plotted 

 in this manner, for se\-eral t\pical c.tses. 



Proxided the roots are n<it all rc.il, this domain lonsists of a con- 

 nected region of values, so tli.it it is possible to pass from <ine p.iir 

 of poles to any other pair s,ui>f\ing the same conditions li\ ,i con- 

 tinuous transformation. In the case of four real roots, howe\er, the 

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

 trated in Fig. 5. I'ndcr these circimistances there is a region of real 

 poles which are not continuously transformable into complex poles. 



The networks realizing any specified driving-point impedance are 



^ .Ml rlcctrical oscillalioiis considered in this paper are of the tcinii (■■*■', \\ lure llie 

 lime coeffieienl \ = ip may have any value, real or romplex. 



