440 



BELL SYSTEM TECHNICAL JOURNAL 



proximately linear phase characteristics of opposite sign. If the 

 frequencies B and C at which the slopes begin are in the same ratio, 

 12 : 18, as the slopes themselves the contributions of the added slopes 

 will substantially cancel each other and the net phase shift throughout 

 the cutoff interval will be almost the same as that of the ideal curve 

 alone. The exact phase characteristic is shown by Fig. 15. It dips 



0.5 0.6 0.8 1.0 



2 3 4 5 6 8 10 



f 



fo 



20 30 40 50 



Fig. 15 — Phase characteristic corresponding to gain characteristic of Fig. 14. 



slightly below 180° at the point at which the characteristic reaches the 

 zero gain axis, so that the circuit is in fact stable. 



The same analysis can evidently be applied to asymptotes of any 

 other slope. This makes it easy to compute the maximum feedback 

 obtainable under any asymptotic conditions. If /o and /„ are respec- 

 tively the edge of the useful band and the intercept (C in Figs. 12 and 

 14) of the asymptote with the zero gain axis, and n is the asymptotic 

 slope, in units of 6 db per octave, the result appears as 



^„. = 401ogio^", 

 where Am\s the maximum feedback in db.^ 



(7) 



^ The formulae for maximum feedback given here and in the later equation (8) 

 are slightly conservative. It follows from the phase area law that more feedback 

 should be obtained if the phase shift were exactly 180° below the crossover and rose 



