PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 199 



the crystal from the oscillator circuit. Equation (12.44) which gives the 

 conditions necessary for oscillations to exist, may be written in the form 



coGp ^ "^ (12.76) 



In order for oscillations to start, the right side of this equation must be 

 equal to or greater than the left side. If it is greater, oscillations build up 

 causing p to increase until the equality is satisfied. The difference between 

 these two terms before oscillations start is therefore a relative measure of 

 the final amplitude for a particular oscillator. The absolute value of am- 

 plitude cannot be obtained from equation (12.76) since we do not know the 

 relation between p and amplitude. However the greater the magnitude of 



—-^ the greater will be the amplitude of oscillations for a given set of oscil- 



Rc 



lator conditions. This term may therefore be considered a measure of 

 crystal quality. It is the effective Q of the crystal unit as measured at its 

 two terminals and at the operating frequency. To distinguish this from 

 the Q of the crystal as usually spoken of, it will be called (fc . 



In the same respect the left side of equation (12.76) may be thought of as a 



1 



measure of quality of the oscillator circuit, that is, pc>}Ct = — , then (12.76) 



becomes 



^c<P, ^ 1 (12.77) 



12.72 Figure of Merit of Crystals —M 



It is very inconvenient to use ipc as a figure of merit of the crystal because 

 it is a complex function of the constants of the crystal circuits, Figure 12.3, 

 and also the frequency. The computation of (pc from such measurable 

 characteristics as frequency of resonance f\ , frequency of anti-resonance 

 /o , resonant resistance Ri , and static capacity Co , requires considerable 

 time and effort. 



It is highly desirable that a simple, easily determined expression for a 

 figure of merit be found. The steps to indicate a suitable one are as follows: 



The equation for <pc in terms of measurable quantities for computing it is 

 derived from equation (12.27) and given by the formula 



CoZ-i (cO — COl)(co" — CO2) 



^'- r: ^^ 



By letting 



(12.78) 



M = ""-^ ""' , ^ (12.79) 



Ri CO- 



