234 BELL SYSTEM TECHNICAL JOURNAL 



coils on the ends. By making the coupling in the secondary high 

 these coils can be made very small and can usually be neglected. 



Another method for reducing balanced lattice filters to unbalanced 

 circuits is to employ crystals with two sets of plates as described in 

 section IV. 



III. Method for Calculating the Element Values 

 OF the Filter 



The curves in Figs. 2 to 6 give a qualitative picture of what type of 

 characteristics can be obtained by the use of crystals in filter networks. 

 In order to determine what band widths and dispositions of attenuation 

 peaks are realizable with crystals it is necessary to calculate the 

 element values, since a crystal cannot be made with a ratio of capaci- 

 tances under 125. 



The actual process of calculation can be divided into two parts. 

 The first part consists in a determination of the critical frequencies of 

 the arms of the network in terms of the desired attenuation character- 

 istic. The second part consists in calculating the element values from 

 the critical frequencies by means of Foster's theorem. 



The attenuation characteristics obtainable with filters are discussed 

 in Appendix I, and it is there shown that the attenuation characteristic 

 of a complicated filter structure can be regarded as the sum of the 

 attenuation characteristics of a number of elementary filters. The 

 critical resonant frequencies of the filter are evaluated in terms of the 

 cutoff frequencies and the position of the attenuation peaks with 

 respect to the cutoff frequencies. The ratios of the impedances of the 

 two arms at zero or infinite frequencies are evaluated in terms of the 

 network parameters. With the aid of these equations, and Foster's 

 theorem discussed below, the element values can be evaluated for any 

 desired attenuation characteristic. Whether the characteristic is 

 realizable or not depends on whether the element values of the equiva- 

 lent circuit of the crystal calculated have a low enough ratio of capaci- 

 tances to be realized in practice. The actual value of the series 

 capacitance Ci of the equivalent circuit of the crystal shown in Fig. 1 

 may also be too large to be physically realizable. 



Having obtained the critical frequencies by the calculations given 

 in the appendix, the element values can be calculated by using Foster's 

 theorem. Foster's theorem ^ deals with impedances in the form of a 

 number of series resonant circuits in parallel as shown on Fig. 13A or 

 a number of antiresonant circuits in series as shown on Fig. 13B. 



3 See "A Reactance Theorem," B. S. T. J., April 1924, page 259. 



