solution 9.10.1.4 The rst step is to put the transfer function in time constant form. So we have G(s) = 100(s+10) s(s + 2)(s+50) = 1000(1+ s=10) (2)(50)s(1+ s=2)(1+ s=50) = 10(1+ s=10) s(1 + s=2)(1+ s=50) : Then the terms to be plotted are 10 ;; 1 s ;; 1 1+s=2 ;; 1 1+s=50 ;;and 1+s=10 log 10 (10) = 20 db Atlowfrequencies the only terms that contribute are the gain and 1=s. The term 1=s, whichisastraight line crossing the 0-dB line at ! =1,will, when the gain is added in, cross at 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(-2,3,200);; s=j*w;; z=10 p1 = 0 p2 = 2 p3 = 50 K=100 mag = 20.*log10( ( K*abs(s + z) )./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) );; semilogx(w,mag);; grid on axis([0.1 1000 -100 60]) print -deps 91014a.eps The phase plot, shown in Figure 2, is generated with the MATLAB statements 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;; 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 semilogx(w,phase,'k-',w1,phase1,'rd');; grid on axis([0.1 1000 -180 -80]) print -deps 91014b.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=10 p1 = 0 p2 = 2 p3 = 50 K=100 mag = 20.*log10( ( K*abs(s + z) )./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) );; semilogx(w,mag);; grid on axis([0.1 1000 -100 60]) print -deps 91014a.eps pause 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 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 91014b.eps 3