86 BELL SYSTEM TECHNICAL JOURNAL 



Equation (24) contains neither Vp, Vg, nor n, so that the important 

 conclusion can be drawn that the frequency of an oscillator with unity 

 coupling between the plate and grid circuits depends only upon the 

 inductances and capacities in the circuit, and not at all upon the tube 

 parameters, r,„ Vg, and ix; provided, however, that the losses in the ex- 

 ternal circuit are small, and the harmonic voltages across the tube are 

 small enough to allow rp and Vg to be considered as pure resistances. 

 In this connection, the interelectrode capacities may be grouped with 

 the external circuit elements forming Xz, A'4, and A^s, so that no high- 

 frequency difficulty is to be anticipated from them. 



Equation (25) contains the relation between r^, Vg, and /x necessary 

 to insure the presence of oscillation. In practice, the amplitude of the 

 oscillations builds up until this relation is satisfied. 



The foregoing theory of the action of a unity coupled oscillator has 

 led to an extremely useful and desirable result, namely, the independ- 

 ence of frequency and operating voltages. The point now remaining 

 to be shown is how to get the unity coupling. 



In attacking this question, the first thing to notice is that our theory 

 does not require that the unity coupling condition, 



M = VL1L2 



should be obtained. What actually is required is the much less rigid 

 condition: 



X,„ = ^Y^Xl (26) 



where Xi and X2 are not limited to inductance alone. 



Thus, imagine one of the impedances, say X^, to consist of a coil 

 Li, in series with a condenser, Co. We have then, by (26): 



oiHP = coLi ( C0L2 - -^ ^ (27) 



\ C0C2/ 



or, writing 



M = kyJLiLi 



where k may now be less than one, we have from (27) 



which gives the value of C2 necessary to provide "unity coupling" at 

 the operating frequency. 

 The value of X^/Xi is thus 



C0L2 - — 

 Xi _ C0C2 _ C0L2 — a;L2(l — k"^) _ 7,2 :^ . (jQ\ 



X\ coLi o}L\ Li 



