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BELL SYSTEM TECHNICAL JOURNAL 



In setting up (7) and (8) it has been assumed that the amplifier will, 

 if necessary, be built with zero margins against singing. Any surplus 

 which the equations indicate over the actual feedback required can, of 

 course, be used to provide a cutoff characteristic having definite phase 

 and gain margins. For example, if we begin with a lower feedback in 

 the useful band the derivative of the attenuation between this region 

 and the crossover can be proportionately reduced, with a corresponding 

 decrease in phase shift. We can also carry the flat portion of the 

 characteristic below the zero gain axis, thus providing a gain margin 

 when the phase characteristic crosses 180°. In reproportioning the 

 characteristic to suit these conditions, use may be made of the ap- 

 proximate formula 



Am- A = {A,„ + 17.4)3; + 



- 2 2 



X H — xy, 



n n 



(9) 



where Am is the maximum obtainable feedback (in db), A is the actual 

 feedback, and x and y are the gain and phase margins in the notation of 

 Fig. 9. Qnce the available margin has been divided between the x and 



C2. 



-20 



0.5 0.6 0.8 1.0 



3 4 5 6 8 10 

 J_ 

 ^0 



20 30 40 50 



Fig. 16 — Modified cutoff permitting 30° phase margin and 9 db gain margin. 



y components by means of this formula the cutoff characteristic is, of 

 course, readily drawn in. An example is furnished by Figs. 16 and 17, 



