176 BELL SYSTEM TECHNICAL JOURNAL 



where Reg is the effective resistance and Xcg the effective reactance of the 

 crystal, the grid resistance Rg and the circuit capacitance Cg in parallel at 

 the oscillating frequency. Upon substitution of these in (12.21) 



i_ ^ [RccRp - X.jXcg + Xs)] +j(XrRcg + RpXs) ^ J 

 M/3 IJiXlXcg — jllXlRcg 



(12.22) 



where 



1 

 Thus — is of the form 



M(3 



Ns — X]_ -\- Xcg -\- Xz 



which means that P — 1 and () = 0. 



This results in the following two equations obtained from the real and 

 imaginary parts of (12.22) both of which must be satisfied for oscillations 

 to be maintained. 

 The real part of (12.22) gives 



Xi(m + 1) (Reg + Xcg) + XcgXz ,.^ ^ s 



and from the imaginary part is obtained 



Rp<f>. 



Xs = ^^^ ^^-^ (12 24) 



where ^c^ = — ^ (This ratio of reactance to resistance of the crystal circuit 



Reg 



will appear in various equations later.) 



Equation (12.24) may be said to define the oscillating frequency and is in 

 a convenient form to examine the effect of the various circuit variables upon 

 the frequency. The impedances -Yi , Reg , X,g and A'3 may be thought of as 

 forming an oscillating loop (See Fig. 12.11). For oxcillations to be main- 

 tained in such a loop the sum of the reactances must equal zero and the 

 sum of the resistances must equal zero. But the sum of the resistances 

 cannot equal zero since Reg is the only resistance in the loop and it is posi- 

 tive. It is therefore necessary for the driving voltage fxVg to act upon the 

 circuit and supply the energy dissipated by the resistance Reg (and also R^ 

 through which the energy is supplied). This alters the frequency some- 

 what and it is no longer determined by setting the three reactances equal 

 to zero as may be seen by equation (12.24). Nevertheless, the right side 

 of this equation is small and approaches zero when Reg approaches zero. 

 It also becomes very small when the reactance Xi becomes small and Reg 

 is not too great. This is the same condition as found by the differential 



