678 BF.LI. 5K5T£.U TECHXICAI. JOVRNAl. 



Thf U\cl\t; nclwurks of Theuit'in III arc iiuliRlcd in Llu' sctond 

 column of Tal)lf III and shown in detail by Fig. 2. Formulas for the 

 romjjutation of their element;* are gi\en by Table II. The \ahies of the 

 elements ran be computed b\- the same rule as that given abo\i- for 

 Table I. 



Kach of these tweKe networks realizes those impedances which 

 have jx)les lying in a certain restricted area or region of the entire 

 domain of possibilities, as indicated for each network in the table by 

 a specilication of the boundary curves of the area. For each par- 

 ticular impedance in the domain various sets of these twehe networks 

 are mutually equivalent. Some points in the domain cannot be 

 realized by networks without mutual inductance. Of the remaining 

 points, each is realizable, in general, by at least three, and by not 

 more than five, of these tweKe networks. This region of the domain 

 which is realizable without mutual inductance is covered, with no 

 overlapping. In- each of tlie four following sets of networks: 13, 17, 

 and 21; i:5, 18, and 22; 14, 17, and 2:5; 15, 19, and 21; the nimibers 

 refer to the networks of Fig. 2. 



That portion of the domain wiiicii cannot i)e realized b\ networks 

 without mutual inductance comprises the three regions boimded by 

 I'l and •'), i'j ,in(i 7, and V-, and (i, respectively, as illustrated by F'igs. 

 4 and 5. 



The third and fourth colunms of Table 111 show a total of 23 net- 

 works, each with six elements, realizing lines in the domain. Of 

 these, eleven are derived as special cases of the networks of both 

 Figs. 1 and 2, six as special cases of Fig. 1 but not of F"ig. 2, and six 

 as special cases of F'ig. 2 alone. The fifth column of the table shows 

 the eleven networks, eacli with fixe elements, realizing pninis in the 

 domain. 



(). F'oKMl l.AS FOR C.M.C fL.XTION Of ( ".ICNICRAI. Nl.IWOKK 



i-"ormulas for the calculation of the xalues of the elements ot the 

 general network of Fig. 7 are given in Theorem 1\'. 'These are gi\cn 

 in the form of nine etiualions (7) (1.")), in<-lusi\T, invoking the twelve 

 elements of the network and two paratiieters, d and k. The para- 

 meter d, however, is fixed by the impedance, since the left-hand mem- 

 l)ers of equations (13) (lo) satisf\- the identity (20). Upon sub- 

 stituting the right-hand members in the identity and rationalizing, 

 equation (21) is obtained, this being a quadratic etiualion in d with 

 coeflicients which are functions of the known coefiicients of the im- 

 l)e<lance. Since d is fixed in this way, there are essentially eight 

 e(|u.ilions in thiili'en \ariablcs, the twcKc elements and the arbi- 



