PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 



207 



This is identical to the expression for PI of equation (12.94). PI is therefore 

 the anti-resonant resistance of the crystal and capacitance Ct in parallel. 

 Substitution of the value of m as given by (12.99) into (12.100) gives the 

 anti-resonant resistance at the oscillating frequency. Thus 



Ro = 



CO (Co + Ct) 



1 - 



-/■- 



p2 



(12.102) 



P l-Ct/Co 

 Fig. 12.29 — The error in PI resulting from the use of approximate equation (12.94) 



which is the exact expression for PI. The differential error resulting from 

 the use of approximate equation (12.94) is then 



PI - Ro 

 Ro 



V 



1-1 



p2 



1 + 



^ 



(12.103) 



The per cent error as a function of P is shown in Fig. (12.29). The error 

 diminishes rapidly with increase in P and is negligible for crystals that are of 

 sufficient quality for most oscillator purposes. 



Equation (12.91) for Re is also approximate because of the assumption 

 that the frequency is independent oi Re . A more exact expression will be 

 derived. 



The impedance of the crystal alone is 



^' = .Cod + n'M') '^ - ^'C + '•^'('' - 1»1 <*2.104) 



