ELECTRICAL WAVE FILTERS EMPLOYING CRYSTALS 245 



tion when tanh P/2 = 1, we have 



w = \/l -/ — ^, (12) 



where cj^ is 27r/x, where f^ is the frequency of infinite attenuation. 

 For a single section since m = VC2/C1, w must be real and lie between 

 and infinity. The possible attenuation characteristics obtainable 

 with the simple section can be calculated from equation (11). It will 

 be noted that when w = 



m — tanh ^ • (13) 



Similar equations for low-pass, high-pass and all-pass filters can be 

 derived from these equations by letting /a go to zero or /b to 00 , or 

 both. These equations are : 



For Low-Pass Filters 



P 



tanh — = lim coa 



^( / I - o^JIc^b' / I - coVa;.4^ \ 



\\\ - O^J/o^A^yi - CoVcObV 



= VI - cobV^oo^/^j :r, — 5 = ^i\h~ ^r^.' (14) 



\ 1 — W jWB" \ i — CO /COjs 



For High-Pass Filters 



tanh y = lim co/i 



/ J l — CJooVcOg- l l — C0''/C0.4- \ 



V\l - a;,>.42\l - co>bV 



= \/l ^1 — s Vl — orl^jT = mnyJl — w''/'*'^-- (15) 



\ 1 — COao'/WA" 



For All-Pass Filters 



^0 / / I - coJ/cob'-^ 

 -> 00 \\ 1 — coJ/oja" 



, P _ lim coyi -^ / /I — cOoo^/cob"- /I — CO" /to .4" 



2 lim COB — ^ ^ \ \ 1 — c<^oo"/wa" \ 1 — a)-/coB" 



= J-^, V^^. (16) 



\ — Wx" 



For this case, since there is no peak in the real frequency range, we 

 must let cooo be imaginary or io)a- Then 



tanh TT- = — V— CO- = nia^ — co-. (17) 



1 co„ 



The band elimination filter cannot be obtained from the band-pass 

 filter by a limiting process. For the simplest band elimination filter 



