Solution 10.8.1.17 The Bode magnitude and phase plots are shown in gure 1 The pieces of -275 -250 -225 -200 -175 -150 Phase in Degrees -100 -80 -60 -40 -20 0 20 40 60 80 Magnitude in Decibels 0.001 0.01 0.1 1 10 100 1000 Frequency in Radians/Sec. Phase in Degrees Magnitude in Decibels Figure 1: Bode Plots of GH(s)= 100(s+3)(s+7) s 2 (s+5)(s+8)(s+20) the Nyquist plot and the completed plot are shown in gure 2. From the Bode Magnitude and phase plots weseethat for a gain of K =100 GH(j6) = 10 ;20=20 6 ;180  =10 ;1 6 ;180  : Thus for K> 100 10 ;1 =1000;; 1 Re(GH) Im(GH) GH(I) Re(GH) Im(GH) GH(I,II,I*) (a) (b) Re(GH) Im(GH) I II a (c) Figure 2: Nyquist Plot, By Stages and Final point `a' is to the left of the point ;1inthe GH-plane. Thus wehavetwo stability cases. For K<10;;000 the point ;1isregion I and there are no encirclements. The Nyquist equation is Z = N +P = 0+0 = 0 There are no closed loop poles in the righthalfofthe s-plane. For K>10;;000 There are two clockwise encirclements of the point ;1 in the GH-plane and the Nyquist equation becomes Z = N +P 2 = 2+0 = 2;; and there are twoclosed loop poles in the right half of the s-plane. The root locus is shown in gure 3. 2 poles -20 Re(s) Im(s) ω = 6 for K = 1000 -3-5-7-8 Figure 3: Root Locus 3