Solution 10.8.1.13 The Bode magnitude and phase plots are shown in gure 1 As can be seen -275 -250 -225 -200 -175 Phase in Degrees -60 -40 -20 0 20 40 60 Magnitude in Decibels 0.1 110100 1000 Frequency in Radians/Sec. nyq10abode.dat Phase in Degrees Magnitude in Decibels Figure 1: Bode Plots of GH = K(s+4) s 2 (s+6)(s+20) For K =100 from the phase plot the Nyquist plot in the GH just barely gets into the third quadrantbefore crossing into the third quadrant. The pieces of 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(j3) = 10 ;12=20 6 ;180  =0:2511 6 ;180  : 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 Thus for K> 100 0:2511 =398;; point `a' is to the left of the point ; in the GH-plane. Thus wehavetwo stability cases. For K<398 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. 2 For K>398 There are twoclockwise encirclements of the point ;1in the GH-plane and the Nyquist equation becomes Z = N +P = 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 -6 -4 Re(s) Im(s) j 3 for K = 398 Figure 3: Root Locus 3