PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 203 



curve for (pc are represented by the dotted curves. Note that the oscillating 

 frequency /o for the new curve is closer to/2 than it is for/i , therefore (pc„ is 

 less than <pco and the amplitude of oscillations will be less. Thus two 

 crystals may have the same value of M but will not give the same output 

 unless operated at the same relative frequency with respect to their resonant 

 and anti-resonant frequencies. It would not have been possible to increase 

 the oscillator output by increasing Ct so as to lower the frequency /» because 

 by doing so <pg is decreased more rapidly than <pc is increased, and the result 

 would be a further decrease of activity. It is therefore necessary in deriving 

 an expression for activity to include the variable of relative frequency or n 

 which we have shown to be a function of the reactance of the circuit and 

 the crystal constants. 



12.81 Derivation of Performance Index of Crystals — PI 



It will be assumed in the first derivation that the negative resistance p of 

 the circuit is much greater than the effective resistance of the crystal Re 

 under stable oscillating conditions. Equation (12.43) which expresses the 

 frequency then becomes 



- TLeQ 



(12.86) 



This leads to a very simple solution from which a more exact expression is 

 later obtained. 



Equation (12.44), which expresses conditions necessary for oscillations, 

 may be written 



coLc 



P\ ^ 



oiCtR 



(12.87) 



As before, the numerical difference between p and the right side of the 

 equation is a measure of activity. In fact, the right side of the equation 

 may be considered to be an expression of the activity performance provided 

 the terms are themselves independent of p. This is not quite true, since 

 previous sections show that the capacitance Ct is not entirely independent of 

 the activity. (See equations (12.30) and (12.46).) However, this effect 

 may be considered negligible for most practical purposes and the value of 

 the right side of (12.87) called the Performance Index (PI) of the 

 crystal. From equations (12.86) and (12.87) the performance index is found 

 to be 



P^ = ^^7^2 (12.88) 



KcW Ct 



This equation may be greatly in error under operating condition which makes 

 Re large compared to p. Also Re varies rapidly with frequency and is 



