460 BELL SYSTEM TECHNICAL JOURNAL 



Since the transfer ratios ii and /S are both complex quantities, equa- 

 tion (1) expresses the two-fold requirement that the magnitude or 

 modulus of jx^ shall be unity and that its phase angle shall be zero. 

 Taking the factors separately, it follows that the modulus of ^ must be 

 the reciprocal of the modulus of ix and that the phase angles of the two 

 must be equal and of opposite sign. For the single-tube oscillator, the 

 phase angle of /3 must be 180 degrees since the phase angle of yu has 

 that value. 



While the relationships stated above are of simple character, they 

 do not by themselves sufihce for the calculation of the oscillation fre- 

 quency from the constants of the tube and the external circuit. The 

 reason for this is that the values of the tube resistances Rx and R2 enter 

 into the determination of the frequency in the general case, and, since 

 these are dependent upon the oscillation amplitude, they cannot be 

 known until the final steady amplitude of the oscillations is known. 

 What happens in an actual oscillator circuit is that, as the oscillation 

 amplitude grows, after initiation, there is a mutual adjustment of fre- 

 quency and of the resistance values until a condition is reached under 

 which both requirements are met simultaneously. In the case of a 

 stabilized oscillator, since the frequency is independent of the tube 

 resistances, the conditions are simplified and the oscillation frequency 

 can be determined directly by means of the relationships stated above. 

 The non-linearity of the resistances afifects only the amplitude of the 

 oscillations. 



The evaluation of ^ifi in terms of the impedance parameters of the 

 circuit permits the determination of the specific circuit conditions in 

 any case for the generation of steady oscillations. The determination 

 is simplified by the consideration that the factor ix has a constant 

 phase angle of 180 degrees so that the variation of the phase of /x(8 is 

 wholly that of the factor /3. 



General Formulae for /x/3 

 The feedback path, or /3 circuit, is shown separately in Fig. 2, the 



Fig. 2 — Simplified schematic of an oscillator feedback circuit. 



notations being the same as in Fig. 1. The network B may be of any 

 degree of complexity, but may be assumed to be made up of lumped 



