98 BELL SYSTEM TECHNICAL JOURNAL 



occur to the designer who requires special arrangements to fit special 

 cases. The generalized circuit of Fig. 18 is suggested as being adapt- 

 able to meet the most widely varying conditions. This is particularly 

 true at very high frequencies, since all the interelectrode capacities are 

 included in the circuit of that figure. 



The popular "push-pull" type of circuit may likewise be generalized 

 to correspond to several of the fundamental circuits illustrated in the 

 figures, and may be stabilized by the methods indicated. However, 

 because of the nonuniformity of vacuum tubes and the added com- 

 plication of the circuit, no advantage has been obtained by its use, so 

 that the single tube circuits are to be preferred wherever special con- 

 ditions do not require the push-pull type. 



Appendix 



The complete and rigorous mathematical relations for oscillation 

 circuits containing vacuum tubes have seldom been discussed in con- 

 nection with their practical application to useful circuits. In the case 

 of the stabilization of oscillators against changes in battery voltages it 

 is important to base the theory upon as strictly rigorous a mathemati- 

 cal foundation as possible, yet at the same time to be able to express 

 the results in readily useful terms. It will be shown that this desirable 

 result may be attained by a proper interpretation of the meaning of 

 the internal impedances, Tj, and Vg, of the vacuum tube. 



To show this in the shortest and most obvious way, a simple series 

 circuit will be considered. Let the circuit consist of a resistance, R, a 

 condenser, C, and an inductance, L, all connected in series with a 

 vacuum tube which may be taken as having a "negative resistance" 

 characteristic. In order to increase the generality of the demonstra- 

 tion, a sinusoidal driving voltage, E, of angular frequency, w, is also 

 allowed to act on the circuit. By Kirchkoff's Law, the current in the 

 circuit is expressed by the equation 



E = RI + Lj^+~^ Idt-\- V (1) 



where V is the drop across the vacuum tube. As a general expression 

 for V in terms of the current the following expression may be used: 



V = Fo + A,I -f A'.P + A,P -f • • •. (2) 



We are interested in the "steady state" solution, and accordingly a 

 Fourier series will be the most general form which can be assumed for 



