464 BELL SYSTEM TECHNICAL JOURNAL 



may be satisfied in two distinctly different ways. In accordance with 

 the first, both D and Z^n, 22 may be finite and of Hke sign, in which case 

 the frequency depends upon the resistance product RiRi. Since these 

 resistances vary with the oscillation amplitude or with the vacuum 

 tube excitation voltages, a solution of this type is indicative of insta- 

 bility of the frequency. 



The second way in which equation (14) may be satisfied depends on 

 the fact that D and Dn, 22 may each have zero values at one or more 

 frequencies according to the degree of complexity of the coupling net- 

 work. If, then, the network can be so designed that D and Dn, 22 

 each have a zero at the same frequency and are of opposite sign else- 

 where, the condition expressed by the equation will be satisfied at that 

 frequency for any values of the tube resistances and will be satisfied 

 at that frequency only. Whether or not oscillations can be sustained 

 at the frequency so determined may be ascertained readily with the 

 help of equation (10). Oscillations occurring under the above condi- 

 tion are theoretically stable. As demonstrated experimentally by 

 Llewellyn, a very high degree of constancy of the frequency is obtained 

 in actual circuits. 



The method of stabilization described above consists in establishing 

 a limited frequency interval within which the oscillation frequency 

 must necessarily lie and then reducing the width of the interval to 

 substantially zero. The establishment of the finite interval is a matter 

 of the choice of an appropriate circuit configuration and the determina- 

 tion of its limits is effected by suitable proportioning of the elements. 



Considered in the light of the image parameters of the feedback net- 

 work, the method of stabilization consists in making the image phase 

 angle of the network take the value 180 degrees at a frequency within 

 a transmission band. Referring to equations (11) and (12), it will be 

 seen that when 1^, the image phase angle of the network, takes the 

 value 180 degrees, the quantity /^/3 becomes real and positive, indicating 

 the possibility of self-oscillation. Since the result is independent of 

 the values of the tube resistances, the oscillations are theoretically 

 stable. 



In an attenuation band, the transfer constant d may include a phase 

 angle of 180 degrees which is constant with frequency, but, because of 

 the real component representing attenuation, neither cosh 6 nor sinh 6 

 in equation (6) can become zero at any frequency. To obtain an over- 

 all phase shift of 180 degrees in the feedback path it is therefore neces- 

 sary that 



K,K2 + R1R2 = 0, (15) 



