PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 



171 



Since this circuit is singly periodic, the differential equation for ii is 

 of the third order and is derived from (12.1) by setting the plate inductance 

 1,2 of the P coefficients equal to zero. The general equation then becomes 



§■ + "'^ + "^t + ^-•' = 



(12.14) 



where 



1 



_ ^ 



+ 



RoCh 



1 



LiCa 



+ 



Ri 



Ri 



P.-. = 



1 



i?2 Ll Ca Cb 



i?2 Ll Cb Rp Ll Cb 

 1 



i?2 Ll Cm R 





(12.15) 



With the substitution of the uncoupled damping factors and frequencies, 

 (12.15) becomes 



Pi = 2aa + 



1 



P2 = /3a + 



P3 = 



RiCb 



lag 

 RiCh 



4- 



+ 



1 



RpCb 

 2aa 



R2 Cb R2 L 

 The frequency as obtained from (12.14) is 



0' = P2 

 with the conditions for oscillation 



RpCb 



(12.16) 



(12.17) 



^-f: 



(12.18) 



obtained by setting the damping factor a equal to zero. The ratio of driven 

 to undriven frequency is obtained by dividing (12.17) and (12,18) by /Sq. 

 That is 



/Q2 p. p. 



(12.19) 



/3a /3' iSaPl 



12.14 Interpretation of the Equations 



It is learned from this analysis that the frequency of oscillation while 

 governed principally by the frequency of the crystal also depends upon all 

 the constants of the circuit. The effect of the plate circuit impedance is 



