278 



BELL SYSTEM TECHNICAL JOURNAL 



mission lines in ladder filter structures. The results obtained are 

 briefly reviewed here. 



One of the simplest filters considered is shown on Fig. 1. The filter 

 consists of a length 2/i of transmission line shunted at its center by a 

 short-circuited transmission line of length U. To determine the trans- 

 mission bands of a filter, it is necessary to neglect the dissipation oc- 

 curring in the elements and hence we assume that R and G are zero. 

 In any case these values are small for transmission lines since they have 

 a high Q. Neglecting R and G, equations (1) become 



C2 = ei cos — — jtiZo sm — 



V V 



t2 



ol 



ei 



Oil 



tx COS 7 -7^ sm 



V Zo V 



(8) 



jjjjjjjjjjj?jjjj/jj^jjjf?jj7jjjj?jjj?? 



V/.ZJ./Z/IZ/.l/ 



-^- 



ii 



Fig. 1 — -Band-pass filter constructed from coaxial conductors. 



where v, the velocity of propagation , and Zo, the characteristic impedance, 

 have the values 



V = 



LC 



=^/; 



(9) 



Using these equations the characteristics of the filter illustrated by 

 Fig. 1 are easily calculated. With reference to Fig. 1 and equations (8) 

 we can write the equations of the network as 



coil • • rr • ^''1 



62 — ei cos niZo, sm — , 



V V 



toll . ei . co/i 



H = 1 1 COS 1 -77- sm - — , 



V -^ Zoi V 



Co/l • • rr • <Jill 



eo = «2 COS 7^4^01 sm — , 



V V 



to = ti COS 



coll 



. 62 ■ coll 



j-yT-sm — 



?0i V 



ii = is + ii, 



62 



ts = 



7Z02 tan 



C0^2 

 V 



(10) 



Combining these equations and eliminating all terms except the input 



