solution 9.10.1.12 The rst step is to put the transfer function in time constant form. So we have G(s) = 18(s+2) s(s 2 +6s +18) = 36(1+ s=2) 18s(s= p 18) 2 +(1=3)s+1) = 2(1 + s=2) s(s= p 18) 2 +(1=3)s+1) : Then the terms to be plotted are 2 ;; 1 s ;;and 1 s= p 18) 2 +(1=3)s+1 and 1 + s=2: 20log 10 (2) = 6:0206 dB Atlowfrequencies the only terms that contribute are the gain and 1=s. The term 1=s, whichisastraightline crossing the 0-dB line at ! =1,withslope ;20 dB/dec. When the gain is added in, it will cross ! =1rad/s at 6 dB. The quadratic term is a straightlineout to the vicinityofthe natural frequency ! n ,and a straightline for frequencies muchlarger than ! n . The twostraightline asymptotes shown in Figure 1 capture the behavior away from the vicinityof! n . Note that the slope changes to ;60 dB when the plot crosses the 0-dB line. The damping ratio of the complex poles is about 0.8. Referring to Figure 9.11(a) weseethat there will be no hump at the resonantfrequency ( whichisvery close to ! n ). The accurate magnitude plot, also shown in Figure ??,was generated with the MATLAB statements w=logspace(-2,3,200);; s=j*w;; z=2 p1 = 0 p2 = 3 -j*3 p3 = 3 + j*3 K=18 mag = 20.*log10( ( K*abs(s + z)) ./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) );; semilogx(w,mag);; grid on axis([0.1 100 -80 40]) print -deps 910112a.eps The phase plot, shown in Figure 2, is generated with the MATLAB statements 1 10 -1 10 0 10 1 10 2 -80 -60 -40 -20 0 20 40 Figure 1: Accurate and asymptotic Bode magnitude plots w1 = logspace(-2,3,20);; s1 = j*w1;; phase = (angle(s + z) - angle(s + p1)-angle(s + p2) - angle(s + p3) )*180/pi;; phase1 =(angle(s1 + z) - angle(s1 + p1)-angle(s1 + p2) - angle(s1 + p3))*180/pi;; semilogx(w,phase,'k-',w1,phase1,'rd');; grid on axis([0.1 1000 -180 -80]) print -deps 910112b.eps Note that twentypoint spread over vedecades will giveafairly accurate phase plot. The complete MATLAB program to drawbothplots is w=logspace(-2,3,200);; s=j*w;; z=2 p1 = 0 p2 = 3 -j*3 p3 = 3 + j*3 K=18 mag = 20.*log10( ( K*abs(s + z)) ./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) );; 2 10 -1 10 0 10 1 10 2 10 3 -180 -170 -160 -150 -140 -130 -120 -110 -100 -90 -80 Figure 2: Accurate and approximate Bode phase plots semilogx(w,mag);; grid on axis([0.1 100 -80 40]) print -deps 910112a.eps pause w1 = logspace(-2,3,20);; s1 = j*w1;; phase = (angle(s + z) - angle(s + p1)-angle(s + p2) - angle(s + p3) )*180/pi;; phase1 =(angle(s1 + z) - angle(s1 + p1)-angle(s1 + p2) - angle(s1 + p3))*180/pi;; semilogx(w,phase,'k-',w1,phase1,'rd');; grid on axis([0.1 1000 -180 -80]) print -deps 910112b.eps 3