128 BELL SYSTEM TECHNICAL JOURNAL 



The fundamental difficulty with this method can be made apparent 

 by considering the nature of the current expressed by {a) above. 

 Does the expression cos co/ indicate a current which has been going on 

 for all time or was the current zero up to a certain time and cos wt 

 thereafter? In the former case we introduce infinities into our 

 expressions and make the equations invalid; in the latter case there will 

 be transients or building-up processes whose importance may increase 

 as n increases but which are tacitly neglected in equations (h) — {e). 

 Briefly then, the difficulty with this method is that it neglects the 

 building-up processes. 



Another method is as follows: Let the voltage (or current) at any 

 point be made up of two components 



V = Fi + V,, (/) 



where Fis the total voltage, Vi is the part due directly to the impressed 

 voltage, that is to say, without the feed-back, and F2 is the component 

 due to feed-back alone. We have 



F2 = AJV. {g) 



Eliminating F2 between (/) and (g) 



F= Fi/(1 - AJ). Qi) 



This result agrees with experiment when |^/[< 1 but does not 

 generally agree when ^/ is positive and greater than unity. The 

 difificulty with this method is that it does not investigate whether or 

 not a steady state exists. It simply assumes tacitly that a steady 

 state exists and if so it gives the correct value. When a steady state 

 does not exist this method yields no information, nor does it give any 

 information as to whether or not a steady state exists, which is the 

 important point. 



The experimental facts do not appear to have been formulated 

 precisely but appear to be well known to those working with these 

 circuits. They may be stated loosely as follows: There is an unstable 

 condition whenever there is at least one frequency for which AJ \s 

 positive and greater than unity. On the other hand, when ^1/ is 

 negative it may be very much greater than unity and the condition is 

 nevertheless stable. There are instances of \AJ\ being about 100 

 without the conditions being unstable. This, as will appear, accords 

 closely with the rule deduced below. 



