MAXIMALLY-FLA T FILTERS IN WA VEGUIDE 713 



whence 



/ + 2r = ^ (A20) 



which proves the second condition mentioned above, namely, that the sum 

 of the lengths of the transmission Hnes in the cavity and its equivalent cir- 

 cuit is equal to a half wavelength. 



The normaUzed admittance of the cavity terminated in the surge admit- 

 tance of the guide can be written in terms of its loaded Q and a wavelength 

 variable as 



Kr:)-]^ 



YR^ 1 -{- J2Q ^2 \^fj - 1 J . (A21) 



This expression is obtained from equations A8 and A13 by making the as- 

 sumption that the bandwidth is narrow so that the sine of the angle in equa- 

 tion A8 can be replaced by the angle. This admittance is referred to a point 

 slightly inside the cavity, i.e. a distance t' inside. 



The similarity between this expression and the corresponding one for the 

 parallel resonant circuit consisting of lumped elements is evident. (See eq. 

 7 of the text.) 



[M] 



YR = 1+ J2Q ^j. - Jj (A22) 



In the case of the cavity the bracketed term is a wavelength variable; in 

 the case of the tuned circuit it is a frequency variable. 



The loss function for maximally-flat filters in waveguides becomes 



g.,+ 



[*Kr:-')] 



(A23) 



The loaded Q's of the cavities taper sinusoidally from one end of the filter to 

 the other so that 



Qr = Qr sin (^-^^) ^- (A24) 



