THEORY AND DESIGN OF WAVE-FILTERS 13 



justifies the assumption regarding z n and shows the latter to give 

 the general case having the specified characteristic impedance. 



The coefficients, m u ■ • ■ m n , may be evaluated by fixing any n 

 physically realizable conditions such as n resonant frequencies of 

 the shunt impedance, which are frequencies of infinite attenuation 

 in the wave-filter. From the foregoing not more than two such 

 frequencies may be included in any internal, and but one in any other, 

 attenuating band. However, since the number of such conditions 

 equals the number of attenuating bands it will be considered most 

 useful to fix one resonant frequency in each attenuating band. If 

 Z\\ has N elements, where N = 2n — 2, 2m — 1, or 2n, the shunt im- 

 pedance will have 2N which may then be found. 



An evaluation process possible here is first to write the expression 

 for Z21 in (15) as the ratio of two polynomials with two variables, in 

 which the assumed relation for z n has been substituted and the 

 variables are an arbitrarily chosen known inductive impedance, z L , 

 and capacitive impedance z c , such, for example, as may occur in 

 z lk . Put each component of the desirable parallel resonant com- 

 ponent form of 02i in terms of these same two variables and two 

 undetermined coefficients, as az L -\-bz c , etc., and write the correspond- 

 ing polynomial ratio expression for s 2 i which will involve the co- 

 efficients. A comparison of the two expressions for s 2 i which must 

 be equivalent gives 2N relations between the coefficients mi, . . . m„ 

 of Zn and the 2N coefficients a, b, etc., of s 2 i- Next fix n resonant 

 frequencies of z 2 i. satisfying the relation 



221 = 0, (16) 



at frequencies fix, . . . f„ x , one arbitrarily chosen in each attenu- 

 ating band. These give n simple ratios — ,etc, which with the other 







relations make a total of 2N-\-n simultaneous equations from which to 

 determine the same number of coefficients. Their solution will give 

 all coefficients explicitly in terms of the independent critical fre- 

 quencies/ ,/i . . . , and frequencies of infinite attenuation fix, . . . /„ M . 

 It is more practical, however, to obtain such explicit solutions for 

 the coefficients m it . . . m n only, and to express the coefficients a, b, 

 etc., as functions of the frequencies and the m's combined. 



That the n additional conditions in (16) are the maximum number 

 which can be imposed may be illustrated in the case of n = 2 by the 

 general low-band-and-high pass wave-filter of Fig. 3 corresponding 

 to the " constant k " wave-filter of Fig. 2. This has a total of twelve 

 elements per section which it will be seen are fully determined by 



