CONSTANT FREQUENCY OSCILLATORS 83 



which may be used to eHmiiuite r,, in (12) and gives 



Xor;-(fxXo + A'o - X,) = X2KX0 - Xy- X^MX, - Xo). (14) 



In order for the frequency to be independent of r„, it is necessary for 

 one of the factors on the left-hand side of the equation to be zero. 

 This, however, necessitates that one of the factors on the right-hand 

 side of (14) also should be zero. Investigation shows that the only 

 pair of factors of (14) that may both be zero, and still be consistent 

 with (13) is the following: 



/X.Y2 + Xo - Xi = 0, (15) 



X2 - Xo = 0. (16) 



Elimination of A'o between these two expressions results in the follow- 

 ing relation: 



(1 + ix)X, = Xr. (17) 



The frequency is then given by the expression : 



Xi -f Xs =-- 0. (18) 



Equation (17j expresses the relation which is required between the 

 reactances of the input and the output network in order to provide for 

 a constant frequency with varying battery voltages. 



In the application of this stabilization to a piezo-electric oscillator 

 such as is shown in Fig. 14-a it sometimes happens that stability im- 

 proves with decrease in the value of the output tuning capacity but 

 oscillations cease before complete stabilization is secured. The expla- 

 nation for this and its remedy may be obtained from (17) and (18) by 

 supposing that the reactance, X2, of the crystal may be represented 

 by an antiresonant circuit, C2 and L2, in series with a capacity, d, 

 while the output reactance, A^i, consists of the antiresonant circuit, Ci 

 and Li. Thus, the value of Ci which satisfies (17) and (18) is 



^ L,2 



^ (1 + m)g ^ 



- C3. (19) 



For discussion, the form which (19) takes in the absence of the stopping 

 condenser, C4, is: 



h 



C, = --[C2+ (1 -f ^)C3] - Cz. 



This shows that a fairly large value of Ci may be required when d is 

 absent, which places the tuning of the plate antiresonant circuit in a 



