solution 9.10.1.20 The rst step is to put the transfer function in time constant form. So we have G(s) = 100(s+20) s 2 (s +200) = (100)(20)(1+ s=20) (200)s 2 (1 + s=200) = 10(1+ s=20) s 2 (1 + s=200) : Then the terms to be plotted are 10 ;; 1 s 2 ;; 1 1+s=200 ;;and 1 + s=20: log 10 (10) = 20 db Atlow frequencies the only terms that contribute are the gain and 1=s 2 . The term 1=s 2 ,whichisastraight line crossing the 0-dB line at ! =1,will, when the gain is added in, cross the vertical line through ! =1rad/sat 20 dB. The other terms are straightlines. The asymptotic magnitude plot is shown in Figure 1 along with the accurate plot. The accurate magnitude plot was generated with the MATLAB statements w=logspace(-2,4,300);; s=j*w;; z1 = 20 p1 = 0 p2 = 0 p3 = 200 K=100 mag = 20.*log10( abs( ( K*(s + z1) ) ./( ( s + p1) .*(s + p2).*(s + p3) ) ) );; semilogx(w,mag);; grid on axis([0.1 10000 -100 40]) print -deps 910120a.eps The phase plot, shown in Figure 2, is generated with the MATLAB statements w1 = logspace(-2,4,30);; s1 = j*w1;; phase = (angle(s + z1)- angle(s + p1)-angle(s + p2) - angle(s + p3) )*180/pi;; 1 10 -1 10 0 10 1 10 2 10 3 10 4 -100 -80 -60 -40 -20 0 20 40 Figure 1: Accurate and asymptotic Bode magnitude plots phase1 =(angle(s1 + z1) - angle(s1 + p1)-angle(s1 + p2) - angle(s1 + p3) )*180/pi;; semilogx(w,phase,'k-',w1,phase1,'rd');; grid on axis([0.1 10000 -200 -120]) print -deps 910120b.eps 20*log10(0.2) Note that twentypoint spread over vedecades will giveafairly accurate phase plot. The complete MATLAB program to drawbothplots is w=logspace(-2,4,300);; s=j*w;; z1 = 20 p1 = 0 p2 = 0 p3 = 200 K=100 mag = 20.*log10( abs( ( K*(s + z1) ) ./( ( s + p1) .*(s + p2).*(s + p3) ) ) );; semilogx(w,mag);; grid on axis([0.1 10000 -100 40]) 2 10 -1 10 0 10 1 10 2 10 3 10 4 -200 -190 -180 -170 -160 -150 -140 -130 -120 Figure 2: Accurate and approximate Bode phase plots print -deps 910120a.eps pause w1 = logspace(-2,4,30);; s1 = j*w1;; phase = (angle(s + z1)- angle(s + p1)-angle(s + p2) - angle(s + p3) )*180/pi;; phase1 =(angle(s1 + z1) - angle(s1 + p1)-angle(s1 + p2) - angle(s1 + p3) )*180/pi;; semilogx(w,phase,'k-',w1,phase1,'rd');; grid on axis([0.1 10000 -200 -120]) print -deps 910120b.eps 20*log10(0.2) 3