REGENERATION THEORY AND EXPERIMENT 



683 



This may be shown as follows with reference to the particular circuit 

 of Fig. 1 : If the feed-back circuit is broken and then properly termi- 

 nated, the voltage existing across the input is taken as e. Now suppose 

 the feed-back path to be restored. Designating the voltage existing 

 across the input in the presence of feed-back as ei we have ei = e -\- Aci, 

 from which the above equation follows. 



Fig. 1 — Series type feed-back; loop broken and terminated at left, normal feed-back 



circuit at right. 



If we let Fi{p) represent the discriminant of the system when the 

 loop is broken and terminated, then the roots of the equation formed 

 by setting the discriminant equal to zero are assumed to have positive 

 real parts. Now for the corresponding discriminant when the loop is 

 restored, we have in accordance with the above considerations 



F{p) = [(1 - A{p):[F,{p). 



In setting this discriminant equal to zero to obtain the roots, the 

 only ones which have nonnegative real parts are those corresponding to 

 the feed-back term 



f{p)=\-A{p). (2) 



The above-mentioned criterion may be deduced from this expression. 

 For steady oscillations to exist the output potential must be identical 

 in amplitude and in phase with that existing across the input at the fre- 

 quency of oscillation {p = jco), in which case the transfer factor is unity. 

 This seems reasonable on the basis that when the input and output 

 terminals are connected through, the oscillation will neither increase 

 nor decrease with time. It may be demonstrated by direct analysis 

 that these several criteria, framed for the critical case of undamped 

 oscillations, all lead to the same correct conclusion. 



Of course in any actual oscillating circuit it is practically impossible 

 to get these conditions fulfilled exactly, and what is ordinarily done 

 in the practical design of oscillating circuits is to ensure that the 

 voltage fed back will be greater than that required to produce oscilla- 

 tion. This evidently goes a step further than the above criteria, and 



