solution 9.10.1.3 The rst step is to put the transfer function in time constant form. So we have G(s) = 50 s(s + 5)(s+50) = 50 (5)(50)s(1+ s=5)(1+ s=50) = 0:2 s(1 + s=5)(1+ s=50) : Then the terms to be plotted are 0:2 ;; 1 s ;; 1 1+s=5 ;; 1 1+s=50 log 10 (0:2) = ;14 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 -14 dB. The other terms are straight lines. 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;; p1 = 0;; p2 = 5;; p3 = 50;; K=50;; mag = 20.*log10( K./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) );; semilogx(w,mag);; grid on axis([0.1 1000 -100 20]) print -deps 91013a.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 + p1)-angle(s + p2) - angle(s + p3))*180/pi;; phase1 =( - angle(s1 + p1)-angle(s1 + p2) - angle(s1 + p3))*180/pi;; semilogx(w,phase,'k-',w1,phase1,'rd');; grid on axis([0.1 1000 -270 -80]) print -deps 91013b.eps 1 10 -1 10 0 10 1 10 2 10 3 -100 -80 -60 -40 -20 0 20 Figure 1: Accurate and asymptotic Bode magnitude plots 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 = 5;; p3 = 50;; K=50;; mag = 20.*log10( K./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) );; semilogx(w,mag);; grid on axis([0.1 1000 -100 20]) print -deps 91013a.eps w1 = logspace(-2,3,20);; s1 = j*w1;; phase = (- angle(s + p1)-angle(s + p2) - angle(s + p3))*180/pi;; phase1 =( - angle(s1 + p1)-angle(s1 + p2) - angle(s1 + p3))*180/pi;; semilogx(w,phase,'k-',w1,phase1,'rd');; grid on 2 10 -1 10 0 10 1 10 2 10 3 -260 -240 -220 -200 -180 -160 -140 -120 -100 -80 Figure 2: Accurate and approximate Bode phase plots axis([0.1 1000 -270 -80]) print -deps 91013b.eps 3