ELECTRO-MECHANICAL REPRESENTATION OF CRYSTAL 721 



where e is the strain (elongation per unit length) produced in the crystal 

 by an applied potential gradient V. Comparing (5) with (7) and not- 

 ing that Qi -\- Qz = elm, and V'= Ejlc, we have on the insertion of the 

 values for Co and Ce from (1) 



\Ks 



'\4r 



Solving for k we find 



, \ \Ks\ . , /I + 16x^2 1 ^ \^ 



(9) 



when IGird^/Ks is a small quantity as it is for quartz. 



When the crystal is used as an element in an electrical network, and 

 allowed to vibrate freely, the force F of Fig. 2 can be set equal to zero 

 and the network short-circuited. Solving for the impedance on the 

 electrical side we find 



^ -j(l -k') 



^-Plh'+Jlq{'^-k^~) 

 ^-f/P+j/q 



(10) 



where fi^ = Ja^; fi — f.A^i'i- — k"^) (/a being the natural mechanical 

 resonance frequency of the crystal), and q is the ratio of the reactance 

 of the condenser Ce to the resistance Rjl or 



g = IIIRitJaCe. (11) 



It is easily shown that the impedance Zc is also the impedance of the 

 network of Fig. 1 if ^ 



Ca = Co; 



Cb = CokVil -F); 



Lb = ll4TrjA'k'Co; ^ ' 



Rb = l/2Trf ACok'q. 



Hence the representation in Fig. 2 reduces to the well known Fig. 1, 

 when the crystal is free to vibrate. 



A network representing the second case, when one end is supported 

 with the other end used to drive a load, is shown on Fig. 3. The 

 method of deriving the constants is the same and all of the constants 



"• If account is taken of the variation of Ce and Le with frequency, the elements are 

 Ci = Co; Cb = —;rri T^ ! ^b = ■,-,,,. ,^ ; Rb = l/lirfACok-q where q = p^ .. . 



7r-(.l — R-) ilR'jA f^o ttKCeJA 



