288 BELL SYSTEM TECHNICAL JOURNAL 



this is the characteristic impedance of the filter. The width of the 

 pass band is determined by 



1 ^ cosh 0^-1. (31) 



The cut-off frequencies of the filter transformer are given by the 

 formula 



cos — = ± - • (32) 



For relatively narrow bands, the ratio of the band width to the mean 

 frequency is given by the simple formula 



•^^-. (33) 



Jm TTif 



as can readily be shown from equation (32) by using the approximation 

 formula for the cosine in the neighborhood of the angle 7r/2, 



Hence the structure shown in Fig. 5 is equivalent to a perfect trans- 

 former whose ratio is ^^ = 1 + (ZoJZq^) to 1 and a filter whose band 

 width is given by equation (33). Such a filter will have a flat attenua- 

 tion loss over about 80 per cent of its theoretical band when it is 

 terminated on each side by resistances equal to Zoi^p and ZoJ<p on its 

 high- and low-impedance sides respectively. Due to the high Q ob- 

 tainable in the transmission lines, the loss in the band of the filter can 

 be made very low and hence such a transformer will introduce a very 

 small transmission loss. Furthermore, since it is constructed only of 

 transmission lines, it can carry a large amount of power. The com- 

 plete design equations for the transformer are 



Zoi ry, Rl r7 ^I 



<p — ^ ~\~ -^~ ; ■2^01 — — ', Zoo — 



02 ^ ^Iv" — 1; 



/ = 7f; Ro=^, (34) 



where Ri and i?o are respectively the input and the output resistances 

 that the transformer works between. 



Many other types of transforming networks containing only trans- 

 mission lines are also possible. Another simple network which is the 

 inverse of the one considered above is shown in Fig. 7. It consists of 

 a length of line l\ and characteristic impedance Zoj in series with a 

 balanced open-circuited line of length li and characteristic impedance 

 Zog. It is easily shown by employing the equations for a line that 



