COAXIAL AND BALANCED TRANSMISSION LINES 295 



These equations give the image parameters for a general transforming 

 band-pass filter. The two uses to which such a structure will ordinarily 

 be put are either to obtain a transformer with as wide a pass band as 

 possible for a given impedance transformation or else to obtain a filter 

 without transformation ratio. For the transformer case it can be 

 shown that the widest pass-band occurs when C3 ^ <» , or in other 

 words the condenser Cs is short-circuited. In order to obtain a simple 

 design, it is assumed that each conductor is an eighth of a wave-length 

 at the mid-band frequency or that 



— = 7- (51) 



The mid-band frequency occurs when cosh ^ = 0. Upon substituting 

 the relation Zoy = l/vCo where Co is the total distributed capacity of the 

 input line of the transformer, cosh 6 vanishes when 



Cp^^'_C2_, (52) 



Solving for the frequencies for which cosh = ± 1, it is easily shown 

 that the ratio of the band width to the mean frequency is given by the 

 expression 



_4_ 



fm J I ^' 



(53) 



2(p^Co 



The image impedance Ki at the mid-frequency of the band is from equa- 

 tion (50) 



f. _ TT C2 



Kro = Zo, j 'y^ . (54) 



1+^ ^^ 



4:<p''C 



From the above equations and noting that <p- = 1 + C2/C1, the design 

 equations of the transformer become 



7 - 7? ^ rP + V«p^ - 1 . 7 Zoi 1 ^ 



^ (P — yjif^ — I <P Ojm 



Co = —^ — m ^l^x{ ; Ci = -xl—^ ; 



Zoi TV \ ^'' — 1 



(55) 



C2=^°V(^^- \)>p\ 



