EE422

Solution to HW#7

Answers to questions:

1a) i) Bandwidth = 1200 rad/sec = 191 Hz; ii) pole at s = -1200; iii) Yes! Asymptotically stable (all poles in LHP); iv) Low freq. Gain = 80dB

¹ 40dB

1b) i) Bandwidth = 121,200 rad/sec = 19,290 Hz (about 20 KHz); ii) pole at s = -1.212e5; iii) Yes! Asymptotically stable; iv) Low freq. Gain = 99.01 = 39.9 dB (about 40dB)

1c) i) The original amp has gain of about 47 dB = 223, and phase of about -87

° at w = 50,000 rad/sec Þ y(t) = 2230Acos(50,000t-87° );

ii) modified amp has gain of 19 dB = 8.9, and phase of -23

° at w = 50,000 rad/sec Þ y(t) = 89Acos(50,000t-23° );

iii) The modified amp is closer to the ideal amp because its bandwidth is about 20,000 Hz

1d) i) Pole is s = + 118,800; ii) UNSTABLE!! iii) Positive feedback occurs when the microphone picks up the P.A. output and feeds it back!!

2a) Step produces a ramp output in open-loop!!

2b) i) Yes! The root locus crosses the jw-axis at about

± j12; ii) For K=25, all points are the the left of s = -2

2c) i) Steady-state error is 0; ii) ts = 1.7 sec.; iii) Yes!! Big Bonus!!

 

Matlab Diary file:

num = 12000000;den = [1 1200];
w=logspace(2,6);[mag,phase]=bode(num,den,w);
semilogx(w,20*log10(mag)); grid; title('EE422 - HW#10: Bode plot of 12000000/(s+1200)');ylabel('|G(jw)| in dB');xlabel('w (rad/sec)')
semilogx(w,phase); grid; title('EE422 - HW#10: Bode plot of 12000000/(s+1200)');ylabel('Phase of G(jw) in deg.');xlabel('w (rad/sec)')
num2 = 1200000;
den2=[1 1200+0.01*12000000];
[mag2,phase2]=bode(num2,den2,w);
semilogx(w,20*log10(mag2)); grid; title('EE422 - HW#10: Bode plot of 1.2e7/(s+1.21e6)');ylabel('|G(jw)| in dB');xlabel('w (rad/sec)')
semilogx(w,phase2); grid; title('EE422 - HW#10: Bode plot of 1.2e7/(s+1.21e6)');ylabel('Phase of G(jw) in deg.');xlabel('w (rad/sec)')

%Positive feedback:denp = [1 1200-0.01*12000000]

denp =

 

1 -118800

%Problem #2

num=10;
den=poly([0 -8 -20]);
t=[0:.01:2];
y=step(num,den,t);
plot(t,y); grid; title('EE422 - HW#10: Unit step response of 10/(s(s+8)(s+20))');ylabel('y(t) (rad)');xlabel('time (sec)')
k=logspace(-1,3);
r=rlocus(num,den,k);
plot(r,'x'); grid; title('EE422 - HW#10: Root Locus of 10/(s(s+8)(s+20)) for k=0.1 to 1000');
ylabel('IMAG');xlabel('REAL')
k=[20:1:30];
r=rlocus(num,den,k);
plot(real(r),imag(r),'x',[-2 -2],[2 -2],'--'); grid; title('EE422 - HW#10: Root Locus of 10/(s(s+8)(s+20)) for k=20 to 30');ylabel('IMAG');xlabel('REAL')
[k' r]

ans =

20.0000 -20.7554 -5.4891 -1.7555

21.0000 -20.7898 -5.3068 -1.9034

22.0000 -20.8238 -5.1078 -2.0684

23.0000 -20.8576 -4.8850 -2.2573

24.0000 -20.8912 -4.6248 -2.4840

25.0000 -20.9244 -4.2915 -2.7840

26.0000 -20.9574 -3.5213 - 0.0819i -3.5213 + 0.0819i

27.0000 -20.9902 -3.5049 - 0.7608i -3.5049 + 0.7608i

28.0000 -21.0227 -3.4886 - 1.0716i -3.4886 + 1.0716i

29.0000 -21.0550 -3.4725 - 1.3097i -3.4725 + 1.3097i

30.0000 -21.0871 -3.4565 - 1.5098i -3.4565 + 1.5098i

 

K_best = 25;
num_closed = K_best*numnum_closed =

250

 

den_closed = den + [0 0 0 num_closed]den_closed =

1 28 160 250

 

roots(den_closed)ans =

-20.9244

-4.2915

-2.7840

 

y=step(num_closed,den_closed,t);
plot(t,y,[0 2],[.98 .98],'--'); grid; title('EE422 - HW#10: Closed-loop Step Response of G(s)=250/(s(s+8)(s+20))');ylabel('y(t) (rad)');xlabel('time (sec)')