REGENERATION THEORY 127 



Steady-State Tiii'ORiits and Experience 



First, a discussion will be made of certain steady-state theories; and 

 reasons why they are unsatisfactory will be pointed out. The most 

 obvious method may be referred to as the series treatment. Let the 

 complex quantity ^/(7co) represent the ratio by which the amplifier and 

 feed-back circuit modify the current in one round trip, that is, let 

 the magnitude oi AJ represent the ratio numerically and let the angle 

 of ^/ represent the phase shift. It will be convenient to refer to A J 

 as an admittance, although it does not have the dimensions of the 

 quantity usually so called. Let the current 



/o = cos co/ = real part of g*"' (a) 



be impressed on the circuit. The first round trip is then represented by 



1 1 = real part of AJe^'^^ (b) 



and the nth by 



Im = real part of yl"/"e*"'. (c) 



The total current of the original impressed current and the first n 

 round trips is 



In = real part of (1 -\- AJ + AU- + • • • ^"/")g*"'. (d) 



If the expression in parentheses converges as n increases indefinitely, 

 the conclusion is that the total current equals the limit of (d) as n 

 increases indefinitely. Now 



1 — A n+l rn+l 



If \AJ\< 1 this converges to 1/(1 - AJ) which leads to an answer 

 which accords with experiment. When \AJ\ > 1 an examination of 

 the numerator in (e) shows that the expression does not converge but 

 can be made as great as desired by taking n sufificiently large. The 

 most obvious conclusion is that when \AJ\ > 1 for some frequency 

 there is a runaway condition. This disagrees with experiment, for 

 instance, in the case where ^/ is a negative quantity numerically 

 greater than one. The next suggestion is to assume that somehow the 

 expression 1/(1 - AJ) may be used instead of the limit of (e). This, 

 however, in addition to being arbitrary, disagrees with experimental 

 results in the case where ^/ is positive and greater than 1, where the 

 expression 1/(1 - AJ) leads to a finite current but where experiment 

 indicates an unstable condition. 



