IDEAL FILTERS 



237 



Image Impedance Characteristics 

 In virtue of the relationship previously developed between the 

 image impedance of a given filter and the transfer constant of its 

 complement, the curves just given might also be used to determine 

 the impedance characteristic. However, the precision required In the 

 approximation of Z/ to i? in practical filter design Is much less than 

 that required In the approximation of tanh 0/2 to unity. A satis- 

 factory characteristic can therefore be obtained with a much smaller 

 number of critical frequencies. In a low-pass filter, for example, one 

 or two impedance controlling frequencies Is usually sufficient. With 

 such a small number of critical frequencies the analytical machinery 

 we have set up Is unnecessarily cumbersome. The problem can be 

 solved more effectively by simple cut and try methods, or by the 

 methods advanced by Cauer '' and Zobel.^^ For the sake of complete- 

 ness, however, several Illustrative characteristics are given in Fig. 13. 

 They correspond to the choice of Impedance controlling frequencies 



Zl 



1.04 



1.02 



0.94 



0.92 



0.90 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.( 



FREQUENCY 



0.9 1.0 



Fig. 13 — Typical low pass filter image impedance characteristics. 



' "Siebschaltungen," loc. cit. 



22 This Journal, Apr. 1931, p. 284. Zobel's work is not stated in terms of the 

 lattice pararneters. A simple w-type termination (of low-pass or high-pass type) 

 can be identified with a lattice image impedance having one impedance controlling 

 frequency while an mm'-type termination can be identified with a lattice impedance 

 having two such frequencies. The numerical values he gives can therefore readily be 

 adapted to the lattice design problem. 



