REGENERATION THEORY 



137 



changes in the same direction with increasing w, where w is real, the 

 rule can be stated as follows: The system is stable or unstable according 

 to whether or not a real frequency exists for which the feed-back ratio is 

 real and equal to or greater than unity. 



In case ddjdw changes sign we may have the case illustrated in Figs. 3 

 and 4. In these cases there are frequencies for which w is real and 



W-PLANE 



Fig. 3 — Illustrating case where amplifying ratio is real and greater than unity 

 for two frequencies, but where nevertheless the path of integration does not include 

 the point 1, 0. 



greater than 1. On the other hand, the point (1, 0) is outside of the 

 locus X = and, therefore, according to the rule there is a stable 

 condition. 



W- PLANE 



Fig. 4 — Illustrating case where amplifying ratio is real and greater than unity 

 for two frequencies, but where nevertheless the path of integration does not include 

 the point 1, 0. 



If networks of this type were used we should have the following 

 interesting sequence of events: For low values of A the system is in a 

 stable condition. Then as the gain is increased gradually, the system 

 becomes unstable. Then as the gain is increased gradually still 

 further, the system again becomes stable. As the gain is still further 

 increased the system may again become unstable. 



