222 



BELL SYSTEM TECHNICAL JOURNAL 



lattice network with realizable constants, whereas the converse is not 

 necessarily true. 



Let us consider first what types of filter characteristics can be 

 obtained by using a crystal in one arm of a lattice network, and 

 electrical or crystal elements in the other arm. As is well known the 

 equivalent electrical network of a crystal is as shown in Fig. 1. The 



Li C, 



I — "W^^—i 



Ca=Co+c, ; Cb=§- (co + Ci) ' ^i 



C^)' 



fn = 



_ f 



Vttc; 



Co 



Fig. 1 — Equivalent electrical circuit and reactance frequency 

 characteristic of piezo-electric crystal. 



element values, as calculated in a recent paper, for a plated crystal 

 vibrating longitudinally are ^ 



Co 



J\~lw^ t 



X 



1 



A-rrh 9 X 10 



Yj farads; 



^ _ o a 12" Iwly ^, 



(-'2 — — r, -?r, — • -1 — X 



1 



Lx = ^P4r X 9 X 1011 henries, 



Tt'^ 5'22 h '^ 9 X 10" ^^'^"^^^ 



(1) 



where ly, h, U are respectively the length, width, and thickness of the 



crystal expressed in centimeters, K = specific inductive capacity, 



522' = inverse of Young's modulus along the direction of vibration, 



dn is the value of the piezo-electric constant along the direction 



of vibration, and p is the density of the crystal. The resistance 



depends on the clamping resistance, acoustic radiation from the ends 



of the crystal, internal damping losses, etc. In general the ratio of 



the reactance of the inductance Li to the resistance R at the resonant 



frequency Jr is from 20,000 to 300,000, depending on how the crystal 



is mounted, whether it is evacuated, etc. In general this resistance is 



so small that it can be neglected for design purposes, and only the ideal 



reactance characteristic need be considered. 



^"A Dynamic Measurement of the Elastic, Electric and Piezoelectric Constants 

 of Rochelle Salt," W. P. Mason, Phys. Rev., Vol. 55, April 15, 1939, p. 775. 



