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BELL SYSTEM TECHNICAL JOURNAL 



to know the constants of an electrical circuit which has the same im- 

 pedance characteristic as the crystal. If attention is confined to the 

 single prominent resonance, the electrical circuit representing the 

 crystal is as shown by Fig. 2. Some theoretical consideration has been 



r^Hh 



Co 



r-nnnp-n 



Ca 



--Hh 



cb 



Ca-Co + C| 



fR = 



06=1; (co+c,; 



Lb = 



c-i^r 



rv i-iC| 



C| 

 C|Co 



2Tr;L|C| " 2Tr^ 



Fig. 2 — Equivalent electrical circuit of piezo-electric crystal. 



given to the electrical network representing perpendicularly cut crys- 

 tals by Cady,^ Van Dyke,^ Dye,^ Vigoureux ^ and others. Assuming a 

 quartz plate to have only plane wave motion, Vigoureux has investi- 

 gated the motion in such a plate, and has shown that there will be 

 resonances at odd harmonics of a particular frequency determined by 

 the length, and mechanical constants of the plate. In the neighbor- 

 hood of the fundamental resonance of the crystal, with the electrical 

 plates placed on the crystal surfaces, he finds the equivalent circuit 

 shown by Fig. 2A, the elements of which in practical units have the 

 following values: 



Co = :; — ; — . . ^ ^ , . ^f i = capacitance m larads, 



Ci = 



Li = 



Airle X 9 X IQii 

 loLSEdn'' 



TvHe X 9 X 10 



lelmP X 9 X 10" 



SloEHn'' 



Yl = capacitance in farads, 

 inductance in henries, 



(1) 



where /o, Im, h are respectively the lengths of the optical, mechanical, 

 and electrical axes in centimeters, 



K = specific inductive capacitance = 4.55 for quartz, 

 E = Young's modulus = 7.85 X 10" for quartz, 



dn = piezo-electric constant = 6.4 X 10~^ for quartz, 



p = density = 2.654 for quartz. 



2 W. G. Cady: Phys. Rev. XIX, p. 1 (1922); Proc. I. R. E. X, p. 83, (1922). 



3 K. S. Van Dyke: Abstract 52, Phys. Rev., June, 1925; Proc. I. R. E., June, 1928. 

 ' D. W. Dye: Proc. Phys. Soc, XXXVIII (5), pp. 399-453. 



^ P. Vigoureux: Phil. Mag., Dec, 1928, pp. 1140-53. 



