210 BELL SYSTEM TECHNICAL JOURNAL 



4 



of Ct . For practical considerations we may assume ^ <<C 1 which reduces 



the equation to 



Oil — 01^ Ct , , 



(12.116) 



(12.117) 



Since 



2 2 



Oil Co (Co + Ct) 



o>2 — oj (o)2 — o)) (o;2 + 0)) ^ 2(a;2 — o)) 



then 



0)i wf 0?l 



0) — 0)2 — 1 



2r 



6: - 



(12.118) 



where r is the ratio of the capacitances of the crystal. 



A curve of per cent frequency change multiplied by r as a function of 



— is shown on Fig. 12.28 for comparison with the associated PI change. 

 Co 



12.85 Relation between PI and Oscillator Activity 



The relation between PI and activity obtained in a particular oscillator 

 will now be examined. Let the curves of Fig. 12.30 represent the variations 

 of p with amplitude for two oscillators A and B, or they might be for the 

 same oscillator at widely different frequencies. These are characteristics of 

 the oscillator circuits and may be of any shape. However, for oscillators 

 with grid leak bias, the curves normally have no negative slopes. The rate 

 of change of p depends upon the rate of change of ^t and plate resistance of 

 the vacuum tube as shown by (12.45) for input conductance.* Since p 

 builds up to a value equal to PI we may plot PI for p. The grid current 

 I g is usually taken as a measure of amplitude. Therefore, Fig. 12.30 may 

 be plotted as shown in Fig. 12.31 where PI is the independent variable. 

 These curves are the characteristics of the oscillator circuits A and B with PI 

 defining the quality of the crystal when used with a particular value of Ct . 

 It is characteristic of oscillators to "saturate" as shown by the curves. 



* It is also a function of grid resistance but this does not appear in the approximate 

 equation (12.45) Ijecause of the assumption of no grid current. See Chaffee's'' complete 

 equation for input admittance. 



