422 BELL SYSTEM TECHNICAL JOURNAL 



feedback amplifier design as long as the problem is attacked blindly. 

 To avoid it, we must have some way of determining in advance when 

 we are either attempting something which is beyond our resources, 

 like the young man on the way to the theater, or something which is 

 literally impossible, like the perpetual motion enthusiast. 



This paper is written to call attention to several simple relations 

 between the gain around an amplifier loop, and the phase change 

 around the loop, which impose limits to what can and cannot be done 

 in a feedback design. The relations are mathematical laws, which in 

 their sphere have the same inviolable character as the physical law 

 which forbids the building of a perpetual motion machine. They show 

 that the attempt to build amplifiers with certain types of loop char- 

 acteristics must fail. They permit other types of characteristic, but 

 only at the cost of certain consequences which can be calculated. In 

 particular, they show that the loop gain cannot be reduced too ab- 

 ruptly outside the frequency range which is to be transmitted if we 

 wish to secure an unconditionally stable amplifier. It is necessary to 

 allow at least a certain minimum interval before the loop gain can be 

 reduced to zero. 



The question of the rate at which the loop gain is reduced is an im- 

 portant one, because it measures the actual magnitude of the problem 

 confronting both the designer and the manufacturer of the feedback 

 structure. Until the loop gain is zero, the amplifier will sing unless the 

 loop phase shift is of a prescribed type. The cutoff interval as well 

 as the useful transmission band is therefore a region in which the 

 characteristics of the apparatus must be controlled. The interval 

 represents, in engineering terms, the price of the ticket. 



The price turns out to be surprisingly high. It can be minimized 

 by accepting an amplifier which is only conditionally stable.^ For the 

 customary absolutely stable amplifier, with ordinary margins against 

 singing, however, the price in terms of cutofif interval is roughly one 

 octave for each ten db of feedback in the useful band. In practice, 

 an additional allowance of an octave or so, which can perhaps be re- 

 garded as the tip to the hat check girl, must be made to insure that the 

 amplifier, having once cut off, will stay put. Thus in an amplifier 

 with 30 db feedback, the frequency interval over which effective con- 

 trol of the loop transmission characteristics is necessary is at least four 

 octaves, or sixteen times, broader than the useful band. If we raise 

 the feedback to 60 db, the effective range must be more than a hundred 

 times the useful range. If the useful band is itself large these factors 



^ Definitions of conditionally and unconditionally stable amplifiers are given on 

 page 432. 



