200 BELL SYSTEM TECHNICAL JOURNAL 



and 



OJo — CO^ 



n = -i 2 (12.80) 



0)2 ~ COl 



and with the assumption that over the narrow frequency range that the 

 crystal will operate 



2 2 2 2 



C<J2 — «l _, C02 — COl 



COj COiW 



equation (12.78) is reduced to 



(12.81) 



_ 1 +nM\n - 1) 

 <Pc = j^ (12.82) 



In this equation it will be observed are only two variables, namely, n 

 which varies widely as the frequency is varied between /i and f^ and M 

 which is substantially constant over the same range. 



A set of curves is plotted in Fig. 12.26 for a hypothetical set of crystals 

 having values of M of 1, 2, 5, and 10 with n varied over a range that falls 

 between measured frequencies /i and fo . Studies will show that whenever 

 M increases (pc will increase. M is readily calculated from measured 

 constants as seen from the following: From (12.79) 



Rl CO- Rl COl CO 



With the assumption in (12.81) 



Ri Co COl Co Ri 



which is a simple expression containing three of the four measured quanti- 

 ties mentioned above, and which bears a direct relation to activity for a 

 given value of the frequency variable n. M is the new figure of merit of 

 the crystal. 



Figure 12.26 contains a further indication which is useful on occasions. 

 Here (pc is not positive at any frequency unless M is greater than 2. But (pc 

 must be positive for the crystal to oscillate in the two general types of cir- 

 cuits considered here.* It provides a measurable index to separate com- 

 pletely non-useful crystals from those that can oscillate in a given circuit. 



Equation (12.84) may be written 



M = ^ (12.85) 



r 



* For a description of oscillator circuits which do not require the crystal to exhibit a 

 positive reactance see: "A New Direct Crystal-Controlled Oscillator for Ultra-Short- 

 Wave Frequencies" by W. P. Mason and I. E. Fair, Proc. LR.E., Vol. 30, p. 464, Oct. 

 1943. 



I 



