REGENERATION THEORY 

 For large values of t this approaches 



s{t) = - i5e(«/^^"-«)' sin {BlH'l - 7r/4). 



141 



(56) 



This example is interesting because it shows a case of instability 

 although there is a reversal. Fig. 6 shows the admittance diagram for 



5V2 - a < and for B^ - a > 0. 



7. Let 



AGiS) = Mt) = Ail -t), < ^ < 1. (57) 



AG(t) = Mt) = 0, -^</<0, l</<oo. (57') 

 We have 



AJ{ico) = A \ (1 - t)e-'"'dt 



Fig. 7 is a plot of this case for ^ = 1. 



V 



W -PLANE 



8. Let 



Fig. 7 — Illustrating Example 7. 

 A{\ -l-fco) 



AJ{iw) = 



(1 + Hl 



(59) 



This is plotted on Fig. 8 for ^ = 3. It will be seen that the point 1 

 lies outside of the locus and for that reason we should expect that the 

 system would be stable. We should expect from inspecting the dia- 

 gram that the system would be stable for vl < 1 and A > 2 and that it 

 would be unstable for 1 < ^ < 2. We have overlooked one fact, 

 however; the expression for AJ{iw) does not approach zero as w 



