72 BELL SYSTEM TECHNICAL JOURNAL 



In order for (5) to be true, both the real and the imaginary portions 

 must separately be equal to zero. If (5) (which comes naturally from 

 (3) with the given substitutions) is separated into its real and imag- 

 inary parts, the resulting two equations must be simultaneous and 

 between them express the frequency and relative values which rp and 

 rg must assume in order for oscillations to exist. The particular aim 

 in the present case is to find whether values of X^ or X^ exist which 

 will enable the frequency to be expressed in terms of the constants of 

 the circuit external to the vacuum tube so that if rp, Vg, and ix should 

 vary, the frequency, being dependent upon the external circuit only, 

 will remain constant. 



From (5), then, the real and imaginary parts give the following two 

 equations : 



- Xo[rp(X2 + X,) + r,(Xi + X5)] - jur,(Xi + XJ(A% + XJ 



= - X,Xmnrg - (Xi + X,,)hg - (X2 + XmYrp. (6) 



XoCV, - (Xi + X5)(X2 + X4)] - 2X„,(X, + X,„)(X2 + X,n) 



= - X,XJ - (Xi + X,„j2(X2 + X4) - (X2 + XJ2(Xi + X5). (7) 



There are certain mathematical rules for finding whether the de- 

 sired constancy of frequency may be obtained from the conditions 

 given by (6) and (7). Without, however, going into detail in regard 

 to these, it is easy to see from (7) that if A'4 and A^s have such values 

 as to satisfy the equation : 



2A^„(A^ + Xm){X<i + Xm) = (Xi + X„)2(X2 + X4) 



+ (X2 + X^)2(Xi + X5) (8) 



(which is obtained by including all terms of (7) which do not contain 

 A^o), then the frequency of oscillation is exactly that which will cause 

 Xo to become zero, and will remain so, no matter what values may be 

 taken by rp, rg, and ix. In other words, the oscillation frequency is 

 equal to the series resonant frequency of the tuned circuit. 



It follows, then, that if the battery voltages were to vary, the fre- 

 quency, being determined by the circuit elements external to the vac- 

 uum tube only, would remain constant. In regard to a changing load 

 resistance, it is evident that if this were connected in parallel either 

 with rp or rg, then the combination of the two resistances could be 

 considered as a single resistance. It therefore follows that the same 

 adjustment which causes the frequency to be independent of battery 

 voltage is also the correct one to render the frequency independent of 

 a variable load impedance when this impedance is resistive, only, and 

 is connected in parallel either with the plate or grid resistance of the 

 tube. 



