246 BELL SYSTEM TECHNICAL JOURNAL 



with a single peak as shown on Fig. 2, filter 2, the equations are 



P_ - co^LiGd - ccy^s') 1 / - co^(l - COVCOB^) . ..^- 



Hence when the position of the peak of infinite attenuation and the 

 characteristic impedance Zq at zero frequency are specified Li and Cz 

 can be determined. 



We next consider the case of a filter with a total of three resonances 

 rather than two. For a band-pass filter this will be represented by 

 the impedance arms shown on Fig. 23B. The impedance of the 

 series and lattice arms will be 



z.,-i[ a-w)a--'W) i ,^^=JU-^V (19) 



wCl L 1 — CO7CO2 J C0C2 \ 0)2^/ 



Combining these to form the propagation constant and characteristic 

 impedance we find 



Zo = j^rT^ (1 - C0VC0^2)(1 _ ^2/^^2)^ 

 \ CO UlL-2 



t^nUf._ '^2 yi - cu /c>;^ A^ - a; /cog ; 



^^"^ 2 - V G (1 - coVa;2^)^ ^^^^ 



^ ^ / (I - coVcoA^)(l - coVcog^) 



V (1 - C0Vc02-)2 



We wish to show now that this type of section has an attenuation 

 characteristic equal to that obtained by two sections of the kind shown 

 in Fig. 23A. To show this we write 



tanh -^ + tanh ~ 



tanh 2 = Y, T2 ' ^ ^ 



1 -f tanh — tanh — 



Substituting the value of tanh P/2 given by equation (14) in (21) 

 and letting the two cutoff's co^ and cojs coincide for the two sections, 

 we have 



2 1 + W1W2 \ (1 — (ji^jia^y 



coa^cob2(1 + rmmi) , . 



W2 = 5—] r, (23) 



Wa' + C0£"WiW2 



where 



