446 BELL SYSTEM TECHNICAL JOURNAL 



will also be assumed that the additional tubes for the low-power por- 

 tions of the circuit have an/i of 10 mc.^° The corresponding At'is Z3> 

 nepers ^^ so that equation (11) would say that the feedback would be in- 

 creased by the addition of as many as 27 tubes to the circuit. Natur- 

 ally in such an extreme case this result can be looked upon only as a 

 qualitative indication of the direction in which to proceed. If we add 

 only 4 tubes, however, the available feedback becomes 46 db while if we 

 add 10 tubes it reaches 60 db. It is to be observed that only a small 

 part of the available gain of the added tubes is used in directly increas- 

 ing the feedback. The remainder is consumed in compensating for 

 the unfortunate phase shifts introduced by the rest of the circuit. 



Amplifiers of Other Types 



The amplifier considered thus far is of a rather special type. It has 

 a useful band extending from zero up to some prescribed frequency /o, 

 constant feedback in the useful band, and it is absolutely stable. De- 

 partures from absolute stability are rather unusual in practical ampli- 

 fiers and will not be considered here. It is apparent from the phase 

 area relation that a conditionally stable amplifier may be expected to 

 have a greater feedback for a cut-off interval of given breadth than a 

 structure which is unconditionally stable, but a detailed discussion of 

 the problem is beyond the scope of this paper. 



Departures from the other assumptions are easily treated. For 

 example, if a varying feedback in the useful band is desired, as it may 

 be in occasional amplifiers, an appropriate cut-off characteristic can 

 be constructed by returning to the general formula (4), performing the 

 integrations graphically, if necessary. If the phase requirement in 

 the cut-off region is left unchanged only the first integral need be 

 modified. The most important question, for ordinary purposes, is that 

 of determining how high the varying feedback can be, in comparison 

 with a corresponding constant feedback characteristic, for any given 

 asymptote. This can be answered by observing the form to which the 

 first integral in (4) reduces when/c is made very large. It is easily seen 

 that the asymptotic conditions will remain the same provided the 



^^ In tubes operating at a high-power level ft may, of course, be quite low. It is 

 evident, however, that only the tubes added to the circuit are significant in interpret- 

 ing (11). The additional tubes may be inserted directly in the feedback path if they 

 are made substantially linear in the voice range by subsidiary feedback of their own. 

 This will not affect the essential result of the present analysis. 



" It is, of course, not to be expected that the actual asymptotic slope will be con- 

 stant from 40 kc. to 10 mc. Since only the region extending a few octaves above 40 

 kc. is of interest in the final design, however, the apparent At can be obtained by 

 extrapolating the slope in this region. 



