THEORY AND DESIGN OE WAVE-FILTERS 15 



(2) Obtain two expressions for the shunt impedance, one derived 

 thru the characteristic impedance of the " constant k " wave-filter 

 and containing the coefficients nti . . . m n ; the other from a con- 

 sideration of its possible most general form corresponding to the 

 series impedance, with coefficients a, b, etc. Equate these expres- 

 sions at all frequencies and thus obtain a set of relations between the 

 coefficients nil . . . m„ and a, b, etc., equal to the latter in number. 



(3) Fix one resonant frequency of the shunt impedance, a fre- 

 quency of infinite attenuation, in each attenuating band, using the 



second expression above which will determine n simple ratios—- etc. 



b 



(4) Solve these simultaneous equations by obtaining an explicit 

 solution for the coefficients m u . . . m n in terms of the critical fre- 

 quencies /o, /i . . . and frequencies of infinite attenuation /i«, . . . /„«,, 

 and a solution for the coefficients a, b, etc., in terms of these fre- 

 quencies and the coefficients mi, . . . m u . 



This method will later be applied to the design of the low-and-band 

 pass wave-filter. 



Mid-Shunt Equivalent Wave- Filter 



The general wave-filter whose mid-shunt characteristic impedance 

 is equivalent to that of the " constant k " wave-filter can be obtained 

 in a manner somewhat similar to the one above. However, it is 

 possible to derive the mid-shunt equivalent directly from the mid- 

 series equivalent wave-filter by a simple process wherein these two 

 are assumed to have equivalent propagation constants. 



Let the series and shunt impedances of this wave-filter be z i2 and 

 Z22, and its mid-series and mid-shunt characteristic impedances 

 K12 and K22, respectively. The fundamental condition here is that 



K 22 = K 2 k- (17) 



Under the assumption that the wave-filter has a propagation constant 

 equivalent to that of the general mid-series wave-filter, where Ku = K\ k , 

 we may write from (1) 



and 



2n_2l2 



221 222 



e 



r = 



2i£i£ — Z\\ 2z22 — K 



2k 



2i£u-f-2n 2z 2 2 + ^2Ar 



These relations and (1) give 



ZnZ 2 2 = Zi2Z2i = K ikK 2 k — z\kZik = R* . (18) 



