6 BELL SYSTEM TECHNICAL JOURNAL 



and feedback circuits equals cos~^ |A'i3|/2 and, therefore, lies between 

 — 90° and + 90°. For all conditions inside or above this boundary, 

 the gain with feedback is increased; outside or below, the gain is 

 decreased. 



Stability 



From equation (2), mV(1 ~ Mi3) is the amplified signal with feedback 

 and, therefore, ^/(l — m/3) is an index of the amplification. It is of 

 course a complex ratio. It will be designated Ap and referred to as 

 the amplification with feedback. 



To consider the effect of feedback upon stability of amplification, 

 the stability will be viewed as the ratio of a change, hAp, to Af where 

 hAp is due to a change in either a* or j3 and the effects may be derived 

 by assuming the variations are small. 



Ap = 



1 -;z/3' 



(5) 



(6) 

 (7) 



If /i/3 :^ 1, it is seen that p, or the ^t-circuit is stabilized by an amount 

 corresponding to the reduction in amplification and the effect of intro- 

 ducing a gain or loss in the /^-circuit is to produce no material change 

 in the overall amplification of the system ; the stability of amplification 

 as affected by j8 or the jS-circuit is neither appreciably improved nor 

 degraded since increasing the loss in the /S-circuit raises the gain of 

 the amplifier by an amount almost corresponding to the loss intro- 

 duced and vice-versa. If /x and /3 are both varied and the variations 

 sufficiently small, the effect is the same as if each were changed sepa- 

 rately and the two results then combined. 



In certain practical applications of amplifiers it is the change in 

 gain or ammeter or voltmeter reading at the output that is a measure 

 of the stability rather than the complex ratio previously treated. The 

 conditions surrounding gain stability may be examined by considering 

 the absolute value of ^f- This is shown as follows: Let {dh) represent 

 the gain in decibels corresponding to A p. Then 



{dh) = 20 1ogio \Ap\, 

 b\Ap\ 



8(db) = 8.686 



\A. 



(8) 



