solution 9.10.1.2 The rst step is to put the transfer function in time constant form. So we have G(s) = 1000(s+1) s(s +30)(s+100) = 1000(s+1) (30)(100)s(1+ s=30)(1+ s=100) = 0:33(1+ s=1) s(1 + s=30)(1+ s=100) : Then the terms to be plotted are 0:33 ;; 1 s ;; 1 1+s=10 ;; 1 1+s=50 and 1+s: log 10 (0:33) = ;9:54 db Atlow frequencies 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 -9.54 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;; z=1 p1 = 0 p2 = 30 p3 = 100 K=1000 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 20]) print -deps 91012a.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;; 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 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 0]) print -deps 91012b.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=1 p1 = 0 p2 = 30 p3 = 100 K=1000 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 20]) print -deps 91012a.eps 2 10 -1 10 0 10 1 10 2 10 3 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 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 0]) print -deps 91012b.eps 3