solution 9.10.1.9 The rst step is to put the transfer function in time constant form. So we have G(s) = 10(s+5) s(s +0:01)(s+50) = 50(1+ s=5) (0:01)(50)s(1+ s=0:01)(1+ s=50) = 100(1+ s=5) s(1 + s=0:01)(1+ s=50) : Then the terms to be plotted are 100 ;; 1 s ;; 1 1+s=5 ;; 1 1+s=50 and 1+s: 20log 10 (100) = 40 db Asumptotically,all the terms are 0 db except the gain and 1=s at frequencies below ! =0:01 rad/s. The term 1=s,whichisastraight line crossing the 0-dB line at ! =1,will, when the gain is added in, cross ! =1rad/s at 40 dB, establishing the low frequency asymptote. 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(-3,3,200);; s=j*w;; z=5 p1 = 0 p2 = 0.01 p3 = 50 K=10 mag = 20*log10(abs( (K.*(s + z) )./( ( s + p1) .* (s + p2).*(s + p3)) ) );; semilogx(w,mag);; grid on axis([0.001 1000 -100 100]) print -deps 91019a.eps The phase plot, shown in Figure 2, is generated with the MATLAB statements w1 = logspace(-3,3,40);; s1 = j*w1;; 1 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 -100 -80 -60 -40 -20 0 20 40 60 80 100 Figure 1: Accurate and asymptotic Bode magnitude plots 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([0.001 1000 -180 -90]) print -deps 91019b.eps Note that fortypoints spread over six decades will giveafairly accurate phase plot. The complete MATLAB program to drawbothplots is w=logspace(-3,3,200);; s=j*w;; z=5 p1 = 0 p2 = 0.01 p3 = 50 K=10 mag = 20*log10(abs( (K.*(s + z) )./( ( s + p1) .* (s + p2).*(s + p3)) ) );; semilogx(w,mag);; grid on axis([0.001 1000 -100 100]) 2 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 -180 -170 -160 -150 -140 -130 -120 -110 -100 -90 Figure 2: Accurate and approximate Bode phase plots print -deps 91019a.eps pause w1 = logspace(-3,3,40);; 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([0.001 1000 -180 -90]) print -deps 91019b.eps normalsize 3