2.003 Fall 2002
Final - Sample problems
Problem 1
Match each of the following pole-zero
diagrams to the corresponding Bode
plot and step response from the
following two pages. For example is
you think that pole-zero diagram (1)
corresponds to step response (q) and
Bode plot (r), write ”1,q,r” on your
exam paper.
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5
5
5
4
4
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4
−3
−2
−1
0
1
2
3
4
−5
−5
5
5
5
4
4
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4
−5
−5
−2
−1
0
1
2
3
4
5
−4
−3
−2
−1
0
1
2
3
4
5
−4
−3
−2
−1
0
1
2
3
4
5
−4
−3
−2
−1
0
1
2
3
4
5
−4
−3
−2
−1
0
1
2
3
4
5
−4
−4
−3
−2
−1
0
1
2
3
4
−5
−5
5
5
5
4
4
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4
−5
−5
−3
−4
−4
−5
−5
−4
−4
−4
−3
−2
−1
0
1
2
3
4
−5
−5
5
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5
1
2.003 Fall 2002
Final - Sample problems
Step Response
4
3.5
Amplitude
3
Step Response
0.25
2.5
0.2
2
0.15
1.5
0
1
2
3
4
5
6
7
8
9
Amplitude
0.1
1
10
0.05
Time (sec)
0
−0.05
Step Response
1
−0.1
0
1
2
3
4
5
6
7
8
9
10
8
9
10
8
9
10
Time (sec)
0.95
0.9
0.85
Amplitude
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Step Response
2
1.8
0.08
1.6
0.07
1.4
0.06
1.2
Amplitude
Amplitude
Step Response
0.1
0.09
0.05
1
0.04
0.8
0.03
0.6
0.02
0.4
0.01
0
0.2
0
1
2
3
4
5
6
7
8
9
0
10
Time (sec)
0
1
2
3
4
5
6
7
Time (sec)
Step Response
10
8
6
Amplitude
4
2
0
−2
−4
Step Response
−6
0
1
2
3
4
5
6
7
8
9
0.25
10
Time (sec)
0.2
0.15
0.1
0.05
Amplitude
Step Response
1
0
−0.05
0.9
−0.1
0.8
−0.15
−0.2
Amplitude
0.7
−0.25
0
1
2
3
4
5
Time (sec)
0.6
0.5
0.4
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
2
6
7
2.003 Fall 2002
Final - Sample problems
Bode Diagram
0
−5
Bode Diagram
10
−15
5
−20
Magnitude (dB)
Magnitude (dB)
−10
−25
−30
−35
0
−5
−40
90
−10
90
0
Phase (deg)
Phase (deg)
45
−45
−90
0
1
10
10
45
2
10
Frequency (rad/sec)
0
−1
10
0
1
10
10
2
10
Frequency (rad/sec)
Bode Diagram
0
−10
Bode Diagram
12
−30
10
−40
Magnitude (dB)
Magnitude (dB)
−20
−50
−60
−70
8
6
4
−80
2
0
0
0
−90
−10
Phase (deg)
Phase (deg)
−45
−135
−180
0
10
1
10
−20
−30
2
10
Frequency (rad/sec)
−40
−1
0
10
1
10
10
Frequency (rad/sec)
Bode Diagram
0
−10
Bode Diagram
0
−30
−1
−40
−2
−50
Magnitude (dB)
Magnitude (dB)
−20
−60
−70
−80
−3
−4
−5
−6
180
−7
−8
135
Phase (deg)
15
45
1
10
2
10
5
10
Frequency (rad/sec)
0
−2
10
−1
0
10
10
1
10
Frequency (rad/sec)
Bode Diagram
150
Magnitude (dB)
100
50
0
−50
90
45
Phase (deg)
0
10
0
−45
−90
0
1
10
10
Frequency (rad/sec)
Bode Diagram
10
5
0
Magnitude (dB)
0
−5
−10
−15
−20
−25
−30
0
Phase (deg)
Phase (deg)
20
90
−45
−90
−1
10
0
10
Frequency (rad/sec)
3
1
10
2.003 Fall 2002
Final - Sample problems
Problem 2
x
k
M
b2
v2
b1
v(t)=1-u(t)
v1
The figure above shows a system which consist of a mass (M), spring (k), and
two dampers (b1, b2). The spring is attached at one end to the mass and to
the damper (b2), which is massless at the other. The mass is pulled by a
velocity source (v(t)). x denotes the extension of the spring. v1 is the velocity
of the mass and v2 is the velocity of the second damper. M = 1 kg, b1 = 1
Ns/m, b2 = 9 Ns/m, k = 9 N/m.
a) find v1 (0− ), v2 (0− ), and x(0− )
b) find x(0+ ) and ẋ(0+ )
c) write the differential equation for the system in terms of x.
d) find x(t) given the initial conditions in c.
4
2.003 Fall 2002
Final - Sample problems
Problem 3
R2
V� i
R1
-a(s)
+
C
Vo
a) If a(s) = 106 , find the transfer function of Op-Amp circuit above.
b) If this circuit were used as a controller, what type of controller would it be?
6
c) If a(s) = 10s what is the transfer function of the circuit?
d) C = 1 µF , R1 = 1e4 �, R2 = 5e4 �. Plot M (�) and �(�)
5