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BELL SYSTEM TECHNICAL JOURNAL 



margin against singing, at the cost of a less rapid cutoff. For example, 

 if we choose ^ = 1.5 the limiting phase shift in the m/5 loop becomes 

 135°, which provides a margin of 45° against instability, while the rate 

 of cutoff is reduced to 9 db per octave. The value k = 1.67, which cor- 

 responds to a cutoff rate of 10 db per octave and a phase margin of 30°, 

 has been chosen for illustrative purposes in preparing Fig. 10. The loss 

 margin depends upon considerations which will appear at a later point. 

 Although characteristics of the type shown by Fig. 5 are reasonably 

 satisfactory as amplifier cutoffs they evidently provide a greater phase 



o -10 



-20 



-40 

 </) -180 



-120 



\ A 



K,^ 



0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 2 3 4 5 6 8 10 



X 



fo 

 Fig. 10 — Ideal loop cutoff characteristics. Drawn for a 30° phase margin. 



margin against instability in the region just beyond the useful band 

 than they do at high frequencies. In virtue of the phase area law this 

 must be inefficient if, as is supposed here, the optimum characteristic 

 is one which would provide a constant margin throughout the cutoff 

 interval. The relation between the phase and the slope of the attenua- 

 tion suggests that a constant phase margin can be obtained by increas- 

 ing the slope of the cutoff characteristic near the edge of the band, 

 leaving its slope at more remote frequencies unchanged, as shown by 

 the solid lines in Fig. 10. The exact expression for the required curve 

 can be found from (6), where the problem of determining such a char- 

 acteristic appeared as an example of the use of the general formula (4). 



