solution 9.10.1.16 The rst step is to put the transfer function in time constant form. So we have G(s) = 101(s +1) s(s 2 +2s +101) = 101(1+ s) 101s(s= p 101) 2 +(2=101)s+1) = (s +1) s(s= p 26) 2 +(1=13)s+1) : Then the terms to be plotted are 1 ;; 1 s ;; 1 s= p 26) 2 +(1=13)s+1 ;; and 1 + s: 20log 10 (1) = 0 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 0 dB, since the gain, in dB, is 0. The quadratic term is a straight line out to the vicinityofthe natural frequency ! n ,and a straightlineforfrequencies much larger than ! n .Thetwo straightline asymptotes shown in Figure 1 capture the behavior awayfrom the vicinityof! n .Note that the slope changes to zero when the zero at s = ;1breaks in, and does, beccausethe resonant frequency of the complex poles is a decade higher at 10 rad/s, so there is a distinct 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 0.0995. Referring to Figure 9.11(a) we see that there will be signi canthump at the resonant frequency ( whichisvery close to ! n ). The hump is accentuated by the presence of the zero. The accurate magnitude plot, also shown in Figure ??,was generated with the MATLAB statements w=logspace(-2,3,200);; s=j*w;; z=1 p1 = 0 p2 = 1 -j*10 1 10 -2 10 -1 10 0 10 1 10 2 -40 -30 -20 -10 0 10 20 30 40 Figure 1: Accurate and asymptotic Bode magnitude plots p3 = 1 + j*10 K=101 mag = 20.*log10( ( K*abs(s + z)) ./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) semilogx(w,mag);; grid on axis([0.01 100 -40 40]) print -deps 910116a.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([0.01 1000 -200 0]) 2 print -deps 910116b.eps zeta = cos(atan(5/1)) 10 -2 10 -1 10 0 10 1 10 2 10 3 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 Figure 2: Accurate and approximate Bode phase 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;; z=1 p1 = 0 p2 = 1 -j*10 p3 = 1 + j*10 K=101 mag = 20.*log10( ( K*abs(s + z)) ./( abs( s + p1) .* abs(s + p2).*abs(s + p3) ) semilogx(w,mag);; grid on axis([0.01 100 -40 40]) print -deps 910116a.eps 3 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([0.01 1000 -200 0]) print -deps 910116b.eps zeta = cos(atan(10/1)) 4