466 BELL SYSTEM TECHNICAL JOURNAL 



and 



7^21 = - X1X3. (20) 



The condition that D and Dn, 22 be zero at the same frequency might 

 be met by making the reactance X3 a simple series resonant combina- 

 tion and proportioning Xi and X2 to be resonant at the same frequency 

 as Xz. The two determinants would then both have zeros at this 

 frequency, but oscillations could not occur since Dii would also become 

 zero and the feedback would be destroyed. The necessary condition 

 for stabilization is, therefore, that {Xi + X2 + -^3) and {X^Xi 

 + X1X2 + -^2^0) become zero at the same frequency. It is readily 

 shown that this can be achieved by making the impedance Xq such that 



X0X3 = X1X2 (21) 



at the frequency for which (Xi + X2 + X3) is zero. The addition of 

 the plate circuit reactance Xq provides a circuit configuration which 

 makes stabilization possible. The character of the several branch 

 reactances may be such that equation (21) holds at all frequencies but 

 it need hold only at zero of Z^n, 22- This minimum restriction permits 

 considerable diversification of the form of the coupling network. 



The modified Colpitts oscillator shown in Fig. 4 is a simple case of 

 the general circuit of Fig. 3. For this circuit the reactance determi- 

 nants have the values 



<j)Cz \ L2C1 LoCi J 



U( . 1 1 



-On, 22 == — cu^ — ~— — -— , (23) 



CO \ LiiL^x L,2Lz I 



^" = ^'^' ( "' " li^ - Zk ) ~ ^I ( "' " 1^3 ) ' ^^^^ 



2^21 = - -^ . (26) 



Both D and Dn, 22 have frequency variations corresponding to those 

 of simple resonant circuits but, in the case of the former, with the 

 sign reversed. The two quantities have the same sign only in the 

 interval between the two resonance frequencies or zeros and, since 

 these resonance frequencies are independently adjustable, the interval 

 may be made as small as may be desired. The interval is reduced to 



