PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 



209 



The angular frequency cos is eliminated by multiplying this by (12.108). 

 Thus 



2 2 



0)2 — COl 



-Co 



Co + Ct 



1 + 



/- 



or 



-Co 



COo — COl 



Co + C, 



1 + 



/' 



+ 1 



(12.111) 



(12.112) 



This is the value for ;; at the oscillating frequency and may be reduced to 

 the form, 



n = 



V 



-h'* 



Vr- 



p2 



(g-)J 



(12.113) 



When this value of n is substituted in (12.105), the value of Re is found to 

 be 



Re = 



Ri 



M' 



+ 



1 - 



4/1 -I 1 + 4/1- 



&:-) J 



(12.114) 



For crystals of usable quality "2 < < 1 and by this assumption the equation 

 reduces to ' 



Re = 



Ri 





(12.115) 



This again reduces to (12.91) when if^ > > (^" + 1 j . 



12.84 Frequency Change Resulting from Paralleling Capacitance 



It is often desirable to know how much the frequency of an oscillator may 

 be changed by varying the capacitance Ct across the crystal. This is de- 

 termined from (12.112) which gives the oscillating frequency as a function 



