432 BELL SYSTEM TECHNICAL JOURNAL 



traced by the vector representing the transmission around the feedback 

 loop as the frequency is assigned all possible real values. In accordance 

 with Nyquist's results a path such as II, which encircles the point 

 — 1,0, indicates an unstable circuit and must be avoided. A stable 

 amplifier is obtained if the path resembles either I or III, neither of 

 which encircles —1,0. The stability represented by Curve III, how- 

 ever, is only "Nyquist" or "conditional." The path will enclose the 

 critical point if it is merely reduced in scale, which may correspond 

 physically to a reduction in tube gain. Thus the circuit may sing 

 when the tubes begin to lose their gain because of age, and it may also 

 sing, instead of behaving as it should, when the tube gain increases 

 from zero as power is first applied to the circuit. Because of these 

 possibilities conditional stability is usually regarded as undesirable 

 and the present discussion will consequently be restricted to "abso- 

 lutely" or "unconditionally" stable amplifiers having Nyquist 

 diagrams of the type resembling Curve I. 



The condition that the amplifier be absolutely stable is evidently 

 that the loop phase shift should not exceed 180° until the gain around 

 the loop has been reduced to zero or less. A theoretical characteristic 

 which just met this requirement, however, would be unsatisfactory, 

 since it is inevitable that the limiting phase would be exceeded in fact 

 by minor deviations introduced either in the detailed design of the 

 amplifier or in its construction. It will therefore be assumed that the 

 limiting phase is taken as 180° less some definite margin. This is 

 illustrated by Fig. 9, the phase margin being indicated as yir radians. 

 At frequencies remote from the band it is physically impossible, in 

 most circuits, to restrict the phase within these limits. As a supple- 

 ment, therefore, it will be assumed that larger phase shifts are permis- 

 sible if the loop gain is x db below zero. This is illustrated by the 

 broken circular arc in Fig. 9. A theoretical loop characteristic meeting 

 both requirements will be developed for an amplifier transmitting 

 between zero and some prescribed limiting frequency with a constant 

 feedback, and cutting off thereafter as rapidly as possible. This basic 

 characteristic can be adapted to amplifiers with varying feedback in 

 the useful range or with useful ranges lying in other parts of the spec- 

 trum by comparatively simple modifications which are described at a 

 later point. It is, of course, contemplated that the gain and phase 

 margins x and y will be chosen arbitrarily in advance. If we choose 

 large values we can permit correspondingly large tolerances in the 

 detailed design and construction of the apparatus without risk of in- 

 stability. It turns out, however, that with a prescribed width of 

 cutoff interval the amount of feedback which can be realized in the 



