solution 9.10.1.7 The rst step is to put the transfer function in time constant form. So we have G(s) = 500 s 2 (s 2 +10s +50) = 500 50s 2 (s= p 50) 2 + s=25 + 1) = 10 s 2 (s= p 50) 2 + s=25+ 1) : Then the terms to be plotted are 10 ;; 1 s 2 ;;and 1 s= p 50) 2 + s=25+ 1 : 20log 10 (9:90099) = 20 dB Atlow frequencies the only terms that contribute are the gain and 1=s 2 . The term 1=s, whichisastraight line crossing the 0-dB line at ! =1, with slope ;40 dB/dec. When the gain is added in, it will cross at 20 dB. The quadratic term is a straightline oout to the vicinityofthenatural 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 0.7. 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;; p1 = 0 p2 = 0 p3 = 5 -j*5 p4 = 5 + j*5 K=500 mag = 20.*log10( K ./( abs( s + p1) .* abs(s + p2).*abs(s + p3).*abs(s + p4) ) );; semilogx(w,mag);; grid on axis([0.1 1000 -100 60]) print -deps 91017a.eps The phase plot, shown in Figure 2, is generated with the MATLAB state- ments 1 10 -1 10 0 10 1 10 2 10 3 -100 -80 -60 -40 -20 0 20 40 60 Figure 1: Accurate and asymptotic Bode magnitude plots w1 = logspace(-2,3,20);; s1 = j*w1;; phase = (- angle(s + p1)-angle(s + p2) - angle(s + p3) -angle(s +p4))*180/pi;; phase1 =( - angle(s1 + p1)-angle(s1 + p2) - angle(s1 + p3)-angle(s1 + p4))*180/pi;; semilogx(w,phase,'k-',w1,phase1,'rd');; grid on axis([0.1 1000 -360 -180]) print -deps 91017b.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;; p1 = 0 p2 = 0 p3 = 5 -j*5 p4 = 5 + j*5 K=500 mag = 20.*log10( K ./( abs( s + p1) .* abs(s + p2).*abs(s + p3).*abs(s + p4) ) );; semilogx(w,mag);; 2 10 -1 10 0 10 1 10 2 10 3 -360 -340 -320 -300 -280 -260 -240 -220 -200 -180 Figure 2: Accurate and approximate Bode phase plots grid on axis([0.1 1000 -100 60]) print -deps 91017a.eps pause w1 = logspace(-2,3,20);; s1 = j*w1;; phase = (- angle(s + p1)-angle(s + p2) - angle(s + p3) -angle(s +p4))*180/pi;; phase1 =( - angle(s1 + p1)-angle(s1 + p2) - angle(s1 + p3)-angle(s1 + p4))*180/pi;; semilogx(w,phase,'k-',w1,phase1,'rd');; grid on axis([0.1 1000 -360 -180]) print -deps 91017b.eps 3