ELECTRICAL WAVE FILTERS EMPLOYING CRYSTALS 235 

 In either case the impedance of the network can be written in the form 



,.,2 \ / ,.,2 



Z = -jH 



1 - 



OJl' 



1 



C03- 



C02«-l 



1 - 



1 



C02;i-2"' 



(4) 



where /f ^ and = ojo — wi — • • • — W2n-i — W2n = =0. For the 

 series resonant circuits of Fig. 13A, the element values are given by 



Li = 



1 



CjCOi 



lim 



CO — > OOi 



jcoZ 



i = 1, 3, 



2« 



1. (5) 



r-nm^ — ^1 — I 



n-3 n-3 



"2 



nRnn 



L^WJP — 



L, C, 



MM 



"2n-2 



HH 



■2h 



B 



Fig. 13- — Impedances arranged in form for application of Foster's Theorem. 

 For the antiresonant circuits in series the values become 

 1 / iim \ / jco 



Ci = 



lim 



W — > COi 



0, 2,4, ••• 2n. 



(6) 



Z{wi' — CO") 



These values include the limiting values for the series case of Fig. 13B 



Cn = 





Z-o — °0 , Cln — 0; L-in 



H{oi<^Wf • • • C02n-2") 

 (cOi"C03^ • • • C02n-1^) 



(7) 



Hence if the elements of one arm of the lattice are arranged in either 

 of the forms shown on Figs. 13 A or B, the element values can be 

 calculated from equations (5) and (6). 



If they are not in this form, they can be transformed into one of 

 these two forms by well known network transformations. For 

 example, all the filters of Figs. 2, 3 and 4 are of this form or can be 

 put in this form by employing the simple network transformation of 

 Fig. 1. For the two crystal sections shown on Fig. 6, the series 



