Solution: 10.8.3.4 The angle contributions of eachofthe poles and zeros can be determined from the vector diagram of Figure . For ! =0, =  2 =0  while  1 is 180  . Im(s) Re(s)x x -1 1-3 α θ 1 θ 2 Figure 1: Vector Components of GH Thus 6 GH(j0) = ;180  =0  : For K =10, jGH(j0)j=30: As ! !1, ,  1 ,and 2 all appproch90  ,Sothat lim !!1 6 GH(j!)=;90  : At the same time the magnitude of eachofthevectors in Figure becomes in nite in length. Since one of the vectors is in the numerator of GH and two are in the denominator, clearly lim !!1 jGH(j!)j=0 This can also be seen from the Bode plots in Figure 2, where 6 GH ap- proaches ;90  and the magnitude plot is descending at -20 db/decade for large !. The the Nyquist plot is shown in Figure 3. These values are obtained bysimply evaluating GH at selected values of s, namely s = j! for 0 <!<1.ABode plot is simply this information in logarithmic form. -200 -175 -150 -125 -100 -75 Phase in Degrees -40 -20 0 20 40 Magnitude in dB 0.01 0.1 1 10 100 1000 Frequency in Radians/sec. Phase in Degrees Magnitude in dB Figure 2: Bode Magnitude and Phase Plots of GH Im(GH) Re(GH) III a Figure 3: Nyquist Plot From the Nyquist plot, weseethat for a gain of 10, point `a' is at s = ;30. Thus for a gain of K = 1 3 ;; the Nyquist plot passes through ;1. For gains below 0.333, the number of encirclements is N =0,and the Nyquist equation is Z = N +P = 0+1 = 1 This means there is one closed loop pole inside in the s-plane. Since encloses the left half of the s-plane, there is one unstable pole for K<0:333. For K>0:333 The point ;1inthe GH-plane is in region II.Inthis case N =1.The sign of N is positivebecause the encirclementof;1isin the counterclockwise (ccw) direction, and in our evaluation of GH along we traveled counterclockwise(cw). Thus, Z = N +P = 1+1 = 2 This means that for K>0:333 there are twoclosed loop poles inside the contour , that is twoclosed looppoles in the left half of the s-plane. The root locus is shown in Figure 4. As can be seen the unstable closed loop pole reaches the orgin for K =0:333. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Figure 4: Root Locus