THEORY AND DESIGN OF WAVE-FILTERS 



23 



To fix one resonant frequency of z 21 in each of the two attenuating 

 bands, at/i* and/ 2 oo, we may put 



and 



which give finally 



and 



(az L + bz c )f t =0, 



J loo 



(cz L +dz c )f 2aa =0, 



a = f of if 2 



b (/o-/i+/ 2 )/?< 



c hfih 



(35) 



(fo-/i+W! 



r . 



These eight simultaneous equations in (34) and (35) are sufficient 

 to determine all the coefficients m u m 2 , a, b, c, d, e, and / in terms 

 of the critical frequencies f Q , f u and / 2 , and frequencies of infinite at- 



o nm^ 



C,k 



m 





Fig. 8 — General Mid-Series Equivalent Low-and-Band Pass Wave-Filter 



tenuation, /i M and fi*,. The method of solution here used will be 

 indicated only and the final results given in Appendix II. The 

 combination of (35) and the first three equations of (34), makes it 

 possible to eliminate all coefficients but m,\ and mi and to obtain 

 formulae for the latter explicitly in terms of the frequencies. From 

 (35) and the first equation and last three equations of (34), b, d, and/ 

 are calculable in terms of m u m 2 , and the frequencies. These com- 

 bined with (35) and the first equation of (34) furnish the values of 

 a, c, and e. The formula for the dependent frequency of infinite 

 attenuation, f' ix , results from putting {ez L -\-fz c )f> =0. 



J loo 



The general mid-shunt equivalent wave-filter, having impedances 

 2i2 and 222, will be derived from the general mid-series equivalent 

 wave-filter above through the inverse network relations of (18); 

 namely, 2nZ22 = Zi2Z2i = ^ 2 - For the series impedance we have upon 

 the substitution of z 2 i 



R 2 R 2 R 2 



_R*_ 



12 2 2 i az L + bz c czL+dzc ez L + fz c ' 



(36) 



