212 BELL SYSTEM TECHNICAL JOURNAL 



Some oscillators saturate very rapidly and completely according to curve B 

 and no further output is obtained regardless of the improvement in the 

 crystal quality. For this reason it has not been possible in the past to 

 separate the performance of the oscillator and the crystal since both were 

 based upon the grid current as a measure of quality. By defining crystal 

 activities and oscillator sensitivity in the manner outlined, the crystal and 

 circuit can be studied separately. The per cent of crystals obtainable with 

 PI above a certain value will be known and the design and improvement of 

 oscillator circuits will be facilitated. 



12.86 Use of PI in Crystal Design 

 The expression of PI in terms of the crystal constants and Ct as given by 

 equations (12.89) or (12.92) assists in the design of crystals. As an ex- 

 ample, the effect of changing the area of the crystal electrodes will be com- 

 puted. The () of a crj^stal is defined as 



e = ^. (12.119) 



By introduction of the ratio of capacitances of the crystal ^ = ^ equation 

 (12.119) becomes 



Q = — ^ (12.120) 



or 



- = -4r^ = M (12.121) 



Assuming Q and r do not vary, that is, disregarding effects such as secondary 

 modes, change in damping produced by the mounting etc., and substituting 

 (12.121) in (12.94) we obtain 



PI = ^- fr^ir^' (12.122) 



r co(Co + Ct) 



where — is considered constant. Differentiating (12.122) with respect to 

 r 



Co we find that PI is a maximum when Co = C< . Since Co is proportional 

 to the area of the electrodes this establishes the optimum area for a par- 

 ticular value of circuit capacitance. 



The capacitance of BT-cut plates is 1.68 mmf. per square centimeter per 

 megacycle.* Substitution of this for Co in (12.122) gives 



^ j 68 X lO^M^ 



^^ (1.68 Af -f QV ^ • ^^ 



* All frequencies are referred to the time interval of one second throughout this paper; 

 i.e. megacycles per second is called simply megacycles as is customary in the radio field. 



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