MA XI MA LL Y-FLA T FILTERS IN WA VEGUIDE 691 



where r is the number of the root, n is the total number of branches. Thus 

 the selectivities of the branches follow the relation 



;m I — ) 



\ 2n J 



Qr = C^^sin^^-^^-Jir (14) 



where Qt represents the selectivity of the total filter, and Qr represents the 

 required selectivity of the r*^ branch, e.g., the selectivities of the first, second 

 and third branches are 



ft = Q^ sing (15) 



ft = ft sing. 



This type of filter is particularly practical when a filter is required to give 

 more than a certain amount of insertion loss in an adjacent band, and less 

 than another certain amount of insertion loss at the edges of the pass-band. 

 Putting this information in equation 10 gives two equations containing two 

 unknowns, Qt , the selectivity of the total filter, and n, the number of 

 branches needed to fulfill the stated requirements. The solution for n 

 may be fractional, in which event the next higher integral value of n is 

 chosen, and this value is used to determine the selectivity, Qr , of the filter. 

 From this, the selectivities of all the branches are determined in accordance 

 with equation 14. 



Standing Wave Ratio 



An alternative way of specifying filter performance is to refer to the input 

 impedance mismatch as a function of frequency. The impedance mis- 

 match can be expressed in terms of the direct and the reflected waves and in 

 terms of the standing wave ratio that exists along the transmission line that 

 connects the properly terminated filter with its generator. The standing 

 wave ratio and the insertion loss of a filter bear a definite relationship to 

 each other if the filter is composed of purely reactive elements. This rela- 

 tionship is given by the formula 



Po (S + If 



Pl 45 



(16) 



y 



where 5 is the standing wave ratio, ^rp^, of the maximum voltage to the 



' min 



minimum voltage as measured along the transmission line. 



