A Rn.lCT.IXCE THEOREM 263 



networks, rfi ami th. di aiul rf-, respectively. The total niiinher of pos- 

 sible ways of realizing a given impedance increases \er\' rapidly for. 

 values of w greater than 2; for « = :?, there are, in general, '.i'2 distinct 

 networks giving a specified impetlance. 



l"orinidas (2) ami (H) are to he used for determining the constants 

 of the circuits which have certain specified characteristics, whereas 

 most network fornnilas are for the determination of the character- 

 istics of the circuit from the given constants of the circuit. The ap- 

 plication of these formulas is illustrated !)>• the following numerical 

 problem : 



To design a reactance network which shall be resonant at fre- 

 quencies of HHH). 3000, .5000. and 7000 cycles; anti-resonant at 

 2000, 4000, and (iOOO c\cles, as well as at zero and infinite frequen- 

 cies; and ha\e a reactance of 2.")00 ohms at a frequency of 10,000 

 cycles. 



By formula ( 1 ) the reactance of such a network must be 



V = _ ;. W-^) (Pi-P^) (Pl-P") (P':-P') M^ 



P iP]-p') iPl-f-) iPl-f-) • ^ ^ 



where pi, pi, pi, and p^ are determined by the resonant frequencies 

 to be 1000X2F. 3000X2;r, .50OOX2;r, and 7000X2;r, respectively; 

 p2, pi. and pi are determined by the anti-resonant frequencies to be 

 2000 X27r, 4000 X2jr, and ()000x2;r, respectively; and 11 must be 

 made equal to 0.0.59() in order that the reactance at />=10,000X2t 

 may be 2o00. The variation of the reactance with the frequency 

 is shown by the curv-e of Fig. 2. 



A network having this reactance may be constructed by com- 

 bining « = 4 simple resonant circuits in parallel, or «-|-l =5 simple anti- 

 resonant circuits in series. These two networks are shown by Fig. 2. 

 The numerical values of the elements are determined as follows: 

 Applying formula (2) we have 



J _ 1 _jj (Pl-Pl)(Pl-pi)(pi-f,) no,„ 

 ' Cvpi "{Pi-Pi){pi-M).iPi-p',) ' 



r - 1 _ r AP'^-Pl) (PI- t^iP'.- Pi) _r,ono 

 ' C,pi {pi-pi){pi-PD(Pi-pl) ' 



T - 1 _rT iPi-pi)(P\-p^:)(P',-pl) _r.n.. 



' C,pi '\pi-Pl)iPi-pi.){pi-pD "-''^■ 



'^'~C.pr'\p]-pi)iPl-P':)(Pi-p:)~^-^^^- 



