solution 9.10.1.15 The rst step is to put the transfer function in time constant form. So we have G(s) = 26(s+5) s(s 2 +2s +26) = (26)(5)(1+ s=5) 26s(s= p 26) 2 +(1=13)s+1) = 5 s(s= p 26) 2 +(1=13)s+1) : Then the terms to be plotted are 5 ;; 1 s ;;and 1 s= p 26) 2 +(1=13)s+1 : 20log 10 (5) = 13:98 dB Atlowfrequencies the only terms that contribute are the gain and 1=s. The term 1=s, whichisastraightline crossing the 0-dB line at ! =1,withslope ;20 dB/dec. When the gain is added in, it will cross ! =1rad/s at 13.98 dB, since the gain, in dB, is 13.98. The quadratic term is a straightlineout to the vicinityofthe natural frequency ! n ,and a straightline for frequencies muchlargerthan ! n . The twostraight line asymptotes shown in Figure 1 capture the behavior awayfrom the vicinityof! n . Note that the slope should change to zero when the zero at s = ;5 breaks in, but the resonant frequency of the complex poles is also around 5 rad/s, so there is no at" part to the asymptotic plot. The slope if the high frequency asymptote is ;40 dB since wehavethree poles and one zero and hence a pole zero excess of two. The damping ratio of the complex poles is about 0.196. Referring to Figure 9.11(a) wesee that there will be signi canthump at the resonant frequency ( whichisvery close to ! n ). The accurate magnitude plot, also shown in Figure ??,was generated with the MATLAB statements w=logspace(-2,3,200);; s=j*w;; z=5 p1 = 0 p2 = 1 -j*5 p3 = 1 + j*5 K=26 mag = 20.*log10( ( K*abs(s + z)) ./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) );; semilogx(w,mag);; 1 10 -1 10 0 10 1 10 2 -80 -60 -40 -20 0 20 40 Figure 1: Accurate and asymptotic Bode magnitude plots grid on axis([0.1 100 -80 40]) print -deps 910115a.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;; semilogx(w,phase,'k-',w1,phase1,'rd');; grid on axis([0.1 1000 -200 -60]) print -deps 910115b.eps zeta = cos(atan(5/1)) Note that twentypoint spread over vedecades will giveafairly accurate phase plot. The complete MATLAB program to drawbothplots is w=logspace(-2,3,200);; 2 10 -1 10 0 10 1 10 2 10 3 -200 -180 -160 -140 -120 -100 -80 -60 Figure 2: Accurate and approximate Bode phase plots s=j*w;; z=5 p1 = 0 p2 = 1 -j*5 p3 = 1 + j*5 K=26 mag = 20.*log10( ( K*abs(s + z)) ./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) );; semilogx(w,mag);; grid on axis([0.1 100 -80 40]) print -deps 910115a.eps pause 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 -200 -60]) print -deps 910115b.eps zeta = cos(atan(5/1)) 3