414 BELL SYSTEM TECHNICAL JOURNAL 



These attenuation characteristics and their limitations are at once 

 found from a consideration of the impedance frequency curves for each 

 arm shown by Fig. 6C. For a ladder type network it is well known ^ 

 that a pass band will exist when 



OgA=_i, (5) 



where Zi is the impedance of the series arm and Z^ the impedance of 

 the shunt arm. Hence, considering the first filter of Fig. 6, it is ob- 

 vious that the lower cut-off /ci will come at the resonant frequency of 

 the crystal. The upper cut-off /c2 will come some place between the 

 resonant and anti-resonant frequency, the exact position depending 

 on the amount of capacitance in shunt. The anti-resonant frequency 

 will be a point of infinite attenuation since the filter will have an in- 

 finite series impedance at this frequency. 



With the restriction on the ratio of capacitances of the crystal 

 noted in the previous section, it is easily shown that the ratio of the 

 anti-resonant frequency to the resonant frequency is fixed and is about 

 1.004. Hence, we see that the ratio of /oo to /ci can be at most 0.4 

 per cent. The band width must be less than this since /C2 must come 

 between /oo and /ci. A similar limitation occurs for the second filter 

 of this figure, for which case the separation of /<» and/c2 is at most 0.4 

 per cent. For filter number 3, a somewhat larger frequency separa- 

 tion between the points of infinite attenuation results, it being at most 

 0.8 per cent. The addition of any electrical capacitance in series or 

 shunt with any of the crystals results in a narrowing of the band width. 



It is seen then that there are two limitations in the types of filters 

 obtainable with crystals and condensers in ladder sections. One, 

 there is a limitation on the position of the peak frequencies and two, 

 there is a limitation for the band width of the filters. 



By employing the more general lattice type of filter section shown 

 on Fig. 7, the first of these limitations can be removed. By means of 

 this type of section it is possible to locate the attenuation peak fre- 

 quencies at any position with respect to the pass band, but the pass 

 band is limited in width to at most 0.8 per cent. 



For a lattice filter a pass band exists when the impedances of the 

 two arms are related by the expression ^ 



0^1-^^--, (6) 



* See, for example, page 190 in book by K. S. Johnson, "Transmission Circuits for 

 Telephonic Communications." 



9 "Physical Theory of the Electric Wave Filter," G. A. Campbell, B. S. T, J., 

 November, 1922. 



