FEEDBACK AMPLIFIER DESIGN 435 



At high frequencies the new phase and attenuation characteristics 

 merge with those obtained from the preceding straight line cutoff, as 

 Fig. 10 indicates. In this region the relation between phase margin and 

 cutoff slope is fixed by the k in the equation (6) in the manner already 

 described for the more elementary cutoff. At low frequencies, how- 

 ever, the increased slope near the edge of the band permits 6 yfe db more 

 feedback. 



It is worth while to pause here to consider what may be said, on the 

 basis of these characteristics, concerning the breadth of cutoff interval 

 required for a given feedback, or the "price of the ticket," as it was 

 expressed in the introduction. If we adopt the straight line cutoff 

 and assume the k used in Fig. 10 the interval between the edge of the 

 useful band and the intersection of the characteristic with the zero gain 

 axis is evidently exactly 1 octave for each 10 db of low frequency feed- 

 back. The increased efificiency of the solid line characteristic saves 

 one octave of this total if the feedback is reasonably large to begin with. 

 This apparently leads to a net interval one or two octaves narrower 

 than the estimates made in the introduction. The additional interval 

 is required to bridge the gap between a purely mathematical formula 

 such as (6), which implies that the loop characteristics follow a pre- 

 scribed law up to indefinitely high frequencies, and a physical amplifier, 

 whose ultimate loop characteristics vary in some uncontrollable way. 

 This will be discussed later. It is evident, of course, that the cutoff 

 interval will depend slightly upon the margins assumed. For example, 

 if the phase margin is allowed to vanish the cutoff rate can be increased 

 from 10 to 12 db per octave. This, however, is not sufficient to affect 

 the order of magnitude of the result. Since the diminished margin is 

 accompanied by a corresponding increase in the precision with which 

 the apparatus must be manufactured such an economy is, in fact, a 

 Pyrrhic victory unless it is dictated by some such compelling considera- 

 tion as that described in the next section. 



Maximum Obtainable Feedback 



A particularly interesting consequence of the relation between feed- 

 back and cutoff interval is the fact that it shows why we cannot obtain 

 unconditionally stable amplifiers with as much feedback as we please. 

 So far as the purely theoretical construction of curves such as those in 

 Fig. 10 is concerned, there is clearly no limit to the feedback which can 

 be postulated. As the feedback is increased, however, the cutoff inter- 

 val extends to higher and higher frequencies. The process reaches a 

 physical limit when the frequency becomes so high that parasitic effects 

 in the circuit are controlling and do not permit the prescribed cutoff 



