ELECTRICAL WAVE FILTERS EMPLOYING CRYSTALS 247 

 Comparing (22) with (20) we see that 



P = P, + P^ and g ="!+"" =tanh(^;^-gi-). (24) 



1 + W1W2 \ 2 / 



We see then that a section with three resonant frequencies can be 

 made to have the same attenuation characteristic as the sum of two 

 simple sections. It is, however, more general since in equations (23) 

 and (24) real values of co2^ and B can be obtained by taking 



OTi — nii^ + imi^\ m-i = Wi^ — imi.; ■ (25) 



that is, the parameter nii can be made complex if the second parameter 

 W2 is made its conjugate. Such complex sections can be made to 

 have attenuation peaks which are finite even in the absence of 

 dissipation.'^ 



By letting coa — > or coy? — ^ oo , the equivalent relations for low-pass, 

 high-pass and all-pass filters can be obtained. These are 



(26) 



(27) 



p I ^2 J 



tanh y = (w-i + W2) xlTi 21 2N2 ! ^^2^ "" (28) 



2 \(1 — coVco2 ) W1W2 



Band Elimination Filter 



For a two-peak band elimination filter such as shown in Fig. 3, 

 filter 2, the equations are: 



^ _ /Li (1 - coVa>A^)(l - coVa;^^) 

 ^^ \C2 (l-a;Va;2^)^ 



, P 1 / - a;2 , , 



tanh - = — ylj- — -^ ; (29) 



Z COa \(1 — CO-/COA )(1 — CO2-/COB) 



LlC'2 \(1 



" VZTC; \(1 - CO^iVcOa2)(1 - W.lVcOB^) ' 



W00IW002 = CUACO/i. 



