296 



BELL SYSTEM TECHNICAL JOURNAL 



where Ri is the input impedance from which the transformer must 

 work. 



When the structure of Fig. 9 is used as a filter without transforma- 

 tion, we have Ci = C3; Zoi = Z02 = Zq. For this case the image 

 parameters become 



2 a)/ 



2Ci + C2 



sm 



, . Ci + C2 2co/ V 



cosh Q = 7; cos 1 ^r— 



L\ V 2 



OiZ^Cx' 



— wL'oZn 



Ki = K2 = K 



1 - 



(2Ci + C2)cot^ 



V 



2coZoCi(Ci + C2) 



, Z0C0C1C2 , '^n 

 "^2(Ci + C2)^^"V 



II + 



(2Ci + C2)tan^ 

 2coZoCi(Ci + C2) 



(56) 



'^2(c7Tg) V 



For narrow-band filters it is easily shown that 



4 



/2 — /l ^ ^ TT ^ (2Cl -f C2)Co _ TT^ C2 



/. 



1 +^+_^ 



Ci 2Co 



Ci^ 



16 Co 



(57) 



At the mid-band of the filter, since the constants were worked out on 

 the assumption that each conductor was an eighth of a wave-length at 

 the mid-band frequency, the mid-band filter impedance can be obtained 

 from the last part of (56) by setting ccl/v = 7r/4, giving 



Kn = Zn 



'1 I ^2 TT C2 



"^G 4 Co 



1 I C2 ■ TT C2 



"^c;^4Co 



(58) 



Hence solving for the constants of the filter on the assumption that A 

 is a small quantity, we find 



Ci = 



4Co 



C2 = 



I6C0 



SR 



7r(2 + 7r)A ' 



/ = 



(TT-f 2)A ' 



8/.' 



33 31 

 Co = — ^in /jL/xf, (59) 



