Relations Between Attenuation and Phase in 

 Feedback Amplifier Design 



By H. W. BODE 



Introduction 



THE engineer who embarks upon the design of a feedback amplifier 

 must be a creature of mixed emotions. On the one hand, he 

 can rejoice in the improvements in the characteristics of the structure 

 which feedback promises to secure him.^ On the other hand, he 

 knows that unless he can finally adjust the phase and attenuation 

 characteristics around the feedback loop so the amplifier will not 

 spontaneously burst into uncontrollable singing, none of these ad- 

 vantages can actually be realized. The emotional situation is much 

 like that of an impecunious young man who has impetuously invited 

 the lady of his heart to see a play, unmindful, for the moment, of the 

 limitations of the $2.65 in his pockets. The rapturous comments of 

 the girl on the way to the theater would be very pleasant if they were 

 not shadowed by his private speculation about the cost of the tickets. 



In many designs, particularly those requiring only moderate amounts 

 of feedback, the bogy of instability turns out not to be serious after all. 

 In others, however, the situation is like that of the young man who 

 has just arrived at the box office and finds that his worst fears are 

 realized. But the young man at least knows where he stands. The 

 engineer's experience is more tantalizing. In typical designs the loop 

 characteristic is always satisfactory — except for one little point. When 

 the engineer changes the circuit to correct that point, however, diffi- 

 culties appear somewhere else, and so on ad infinitum. The solution 

 is always just around the corner. 



Although the engineer absorbed in chasing this rainbow may not 

 realize it, such an experience is almost as strong an indication of the 

 existence of some fundamental physical limitation as the census which 

 the young man takes of his pockets. It reminds one of the experience 

 of the inventor of a perpetual motion machine. The perpetual mo- 

 tion machine, likewise, always works — except for one little factor. 

 Evidently, this sort of frustration and lost motion is inevitable in 



^ A general acquaintance with feedback circuits and the uses of feedback is as- 

 sumed in this paper. As a broad reference, see H. S. Black, "Stabilized Feedback 

 Amplifiers," B. S. T. J., January, 1934. 



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