The Symmetrical Transfer Characteristics of the Narrow-Bandwidth Four-Crystal Lattice Filter

The conventional four-crystal lattice filter is analyzed, without recourse to image parameter theory, in order to obtain an (approximate) expression for a general symmetrical insertion-loss characteristic. The resulting expression is given as an equivalent low-pass transfer function which is the ratio of two polynomials of the second degree in complex frequency. These in turn have coefficients which are functions of three parameters: 1) shunt capacitance ratio; 2) fractional-bandwidth, crystal-Q products; 3) fractional stagger of the inner critical frequencies of the crystal. Numerous curves illustrate the effect on the insertion-loss characteristic of the three design parameters. Where comparison is possible, the approximate equations reached in this paper are shown to agree with the corresponding approximate image parameter equations of Kosowsky. Also, a variety of responses are shown to result which were not discovered by previous image parameter techniques. The approximations involved are equivalent to ignoring critical frequencies of the impedance elements (and hence of the transfer function) when these are far removed from the filter center frequency. Thus the representation is only valid near the filter center frequency. However, the error involved in the approximation is typically less than 0.1 db to the 40-db attenuation points, with narrow-band filters.