PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 195 



By substitution of these values in (12.61) and disregard of Xg in comparison 

 with all other A''s the equation for frequency stability is obtained as 



1 ^M V V V 



7 - TT> A1A2A3 



^" - ^^^ (12.65) 



dV 





From this we learn that the values of the reactances Xi , A'2 , and X3 

 should be small and the values of Rp and Rg large to give small changes in 

 CO when V is varied. These variables are more or less limited, however, by 

 the conditions necessary for sustained oscillations according to equation 

 (12.63). It is important to notice that the denominator of (12.65) contains 

 functions which do not appear in equation (12.63) and hence may be of any 

 value. These factors are the rates of change of the various reactances with 

 frequency. For given values of circuit constants, the equation shows that 

 the frequency stability increases as these rates of change increase. 



12.63 Frequency Stability Coefficient of Crystals 



The rate of change of the reactance of an element is referred to as the 

 "frequency stability coefficient"* of the element. Expressed in per cent, 

 we have for the frequency stability coefficient of a reactance 



F{X) =f.^ (12.66) 



dco X 



Let us now examine the frequency stability coefficient of a crystal which 

 is used as the reactance .Y2 when connected between grid and cathode of the 

 tube and as .Y3 when connected between grid and plate (See Fig. 12.24). 

 The resistance of the crystal will be assumed to equal zero due to the negli- 

 gible effect of the resistance variations upon the reactance for crystals with 

 average Q and operated at a frequency not too near the anti-resonant 

 frequency. (This may be observed in Fig. 12.21.) 



The reactance of the crystal then is 



X,^-i,%^, (12.67) 



where 



/.r' 2 2 



WL-o CO — a;2 



CO = 27r X frequency 



coi = 27r X resonant frequency 



coo = 27r X anti-resonant frequency 



First suggested by N. E. Sowers. 



