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final 3.pdf-ECE 232 - Circuits and ...
final_3.pdf-ECE 232 - Circuits and Systems
##### Page 1
ECE 232 - Circuits and Systems II
Final (Spring 2011)
Please provide clear and complete answers. Don’t forget to specify the units of measure!
1. (2 points)
Consider the transfer function
(
) = 100
(
+ 1)
(
+ 10)(
+ 100)(
+ 1000)
a.
Draw the amplitude Bode plot for the transfer function.
Specify clearly the relevant
values on the two axes. In particular, what is (approximately)
max
?
b. What type of
lter is it? Using Bode’s approximation,
nd the
3

cut-o
frequency (or
frequencies) (Hint: a linear increase of
20

for every decade is the same as an increase of
3

for every multiplicative increase by
2
).
t
[s]
t
[s]
1
2
2
x(t)
h(t)
1
2
1
2. (2 points)
Perform the convolution
(
)=
(
)
(
)
of the two
signals
(
)
and
(
)
shown in the
gure on the left.
)
(
t
v
i
R
+
-
+
-
)
(
t
v
o
L
C
L
R
3. (3 points)
Consider the circuit
in the
gure on the left with
=1

= 100
 
= 100

and
= 10
a. Find the transfer function between
(
)
and
(
)
.
b. Find and sketch the amplitude response. What type of
lter is it? What is
max
and
what are the
3

cut-o
frequencies?
c. Discuss the e
ect of
d.
Calculate the quality factor and relate this result to the position of the poles in the
complex plane.
1

##### Page 2
)
(
t
v
i
R
+
-
+
-
)
(
t
v
o
L
L
R
4. (3 points)
Consider the circuit in
the
gure on the left with
=1
,
=1

,
=1
a. Find the transfer function between
(
)
and
(
)
. What are the poles and zeros?
b. Sketch the amplitude and phase response. What type of
lter is it? What is
max
and
what is the
3

cut-o
frequency?
c. Assume that the inductor has a current at time zero equal to
1

(with direction from
top to bottom) and that
(
)=
1000
(
)
. Find the Laplace transform of the output
(
)
Sol
.:
1.
We
st write
(
)
=
100
(1 +
)
10
·
100
·
1000(1 +
10
)(1 +
100
)(1 +
1000
)
=
1
10
4
(1 +
)
(1 +
10
)(1 +
100
)(1 +
1000
)
a. Using the approach seen in class, the Bode plots starts with a
20

slope until fre-
quency
1

, then the slope is
40

until
10

, where it becomes
20

,
to become
0

at frequency
100

and
nally
20

at
1000

. Note
that

is approximately obtained between
100

and
1000

, so we get
max
'|
(
100)
|'
1
10
4
100
·
100
10
=0
1
b. The
lter is bandpass with central frequency between
100

and
1000

. The
3

cut-o
frequencies are approximately
1
=
100
2
= 70
71

2
=
1000
2 = 1414
2

2.
We perform the convolution between
(
)
and
(
+ 1)
(that is,
(
)
shifted left by
1
second) as
˜
(
)=
(
)
(
+1)
. The
nal result
(
)
will be obtained by moving the obtained
solution
˜
(
)
to the right by
1
second.
2

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