FEEDBACK AMPLIFIER DESIGN 441 



For the sake of generality it is convenient to extend this formula to 

 include also situations in which there exists some further linear phase 

 characteristic in addition to those already taken into account. In 

 exceptional circuits, the final asymptotic characteristic may not be 

 completely established by the time the curve reaches the zero gain axis 

 and the additional phase characteristic may be used to represent the 

 effect of subsequent changes in the asymptotic slope. Such a situation 

 might occur in the circuit of Fig. 11, for example, if C5 or Ce were made 

 extremely small. The additional term may also be used to represent 

 departures from a lumped constant analysis in high-frequency ampli- 

 fiers, as discussed earlier, If we specify the added phase characteristic, 

 from whatever source, by means of the frequency /d at which it would 

 equal 2njT radians, if extrapolated, the general formula corresponding 

 to (7) becomes 



A„ = 40 1ogio4r¥V- (8) 



nfofa + fd 



It is interesting to notice that equations (7) and (8) take no explicit 

 account of the final external gain of the amplifier. Naturally, if the 

 external gain is too high the available ix circuit gain may not be sufficient 

 to provide it and also the feedback which these formulae promise. 

 This, however, is an elementary question which requires no further 

 discussion. In other circumstances, the external gain may enter the 

 situation indirectly, by affecting the asymptotic characteristics of the 

 /3 path, but in a well chosen /3 circuit this is usually a minor considera- 

 tion. The external gain does, however, affect the parts of the circuit 

 upon which reliance must be placed in controlling the overall loop 

 characteristic. For example, if the external gain is high the /x circuit 

 will ordinarily be sharply tuned and will drop off rapidly in gain beyond 

 the useful band. The jS circuit must therefore provide a decreasing 

 loss to bring the overall cutoff rate within the required limit. Since 

 the j8 circuit must have initially a high loss to correspond to the high 

 final gain of the complete amplifier, this is possible. Conversely, if 

 the gain of the amplifier is low the /x circuit will be relatively flexible 

 and the /3 circuit relatively inflexible. 



rapidly to its ultimate value thereafter. These possibilities can be exploited ap- 

 proximately by various slight changes in the slope of the cutoff characteristic in the 

 neighborhood of the crossover region, or a theoretical solution can be obtained by 

 introducing a prescribed phase shift of this type in the general formula (4). The 

 theoretical solution gives a Nyquist path which, after dropping below the critical 

 point with a phase shift slightly less than 180°, rises again with a phase shift slightly 

 greater than 180° and continues for some time with a large amplitude and increasing 

 phase before it finally approaches the origin. These possibilities are not considered 

 seriously here because they lead to only a few db increase in feedback, at least for 

 moderate w's, and the degree of design control which they envisage is scarcely feasible 

 in a frequency region where, by definition, parasitic effects are almost controlling. 



