PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 193 



and 



— \ I 8V -\- 8a] -{- — I — -8V + — 8a] \ 



lAdc^\AdV ^ A da J^do:\dV ^ da /J ,,..„, 



8(jO = ^ ^ 7 r-^ ^7 r-s-^ '— (12.60) 



/I dAV /doY 



\A dco) \do:/ 



The variable p in general may be written as the sum of a and iw. With 

 the remembrance that p is the differential operator d/dt and that a set of 

 linear equations expresses the transient condition, it is evident that the 

 current will have the form /e^' which is equivalent to le" ^" . Inspection 

 of this shows that the real part of p, namely a, determines whether the cur- 

 rents in the system are going to build up with time, or die away with time, 

 or remain constant, depending respectively upon whether a is greater than 

 zero, is less than zero, or is actually equal to zero. With this in mind we 

 see that (12.59) and (12.60) state the change in a and co respectively which 

 would result from some change in the circuit condition. Initially the 

 circuit was oscillating in a steady manner so that a was zero and co had some 

 particular value. A change in V then occurred. This produced a change in 

 the amplitude accompanied by a change in the frequency as expressed by 

 (12.60) and a change in the transient term a. Suppose now that the change 

 in V w'ere very small. Then in order for oscillations again to assume a 

 steady value it is necessary for the amplitude "a" to change a sufficient 

 amount to cause a to become zero. Thus in (12.59) we put 8a equal to 

 zero and solve for the required amplitude change. This may then be elimi- 

 nated from (12.60) resulting in the final expression 



1 dA dd _ 1 dA dd 

 80. = -4aFa^ AJ^W ^^ ^2 ^j) 



^dAdd_ 1 ^ cl0 

 A doj da A da dco 



which gives the frequency change Sco in terms of the change of the inde- 

 pendent variable 8V. 



12.62 Frequency Stability of Conventional Oscillator 



In applying this equation to the oscillator circuit, Fig. 12.24, we must 

 first set up the conditions for oscillations. The ijl(3 equation is 



IjlXi X2 Rg 



^^ ^ aXsRpRo - X1X2X3] - [RpX^iXi + X3) + RgXiiX. + Xs)] 



(12.62) 

 The oscillating conditions /x/3 = 1 requires 



XgRpRg = X1A2X3 



