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CHAPTER 6
Solutions for Exercises
E6.1
(a) The frequency of
)
2000
2
cos(
2
)
(
in
t
t
v
⋅
=
π
is 2000 Hz.
For this frequency
.
60
2
)
(
∠
=
f
H
Thus,
60
4
0
2
60
2
)
(
in
out
∠
=
∠
×
∠
=
=
V
V
f
H
and we have
).
60
2000
2
cos(
4
)
(
out
+
⋅
=
t
t
v
(b) The frequency of
)
20
3000
2
cos(
)
(
in

⋅
=
t
t
v
is 3000 Hz.
For this frequency
.
0
)
(
=
f
H
Thus,
0
0
2
0
)
(
in
out
=
∠
×
=
=
V
V
f
H
and we have
.
0
)
(
out
=
t
v
E6.2
The input signal
)
1500
2
cos(
3
)
20
500
2
cos(
2
)
(
t
t
t
v
⋅
+
+
⋅
=
has two components
with frequencies of 500 Hz and 1500 Hz.
For the 500Hz component we have:
35
7
20
2
15
5
.
3
)
500
(
in
out,1
∠
=
∠
×
∠
=
=
V
V
H
)
35
500
2
cos(
7
)
(
out,1
+
⋅
=
t
t
v
For the 1500Hz component:
45
5
.
7
0
3
45
5
.
2
)
1500
(
in
out,2
∠
=
∠
×
∠
=
=
V
V
H
)
45
1500
2
cos(
5
.
7
)
(
out,2
+
⋅
=
t
t
v
Thus the output for both components is
)
45
1500
2
cos(
5
.
7
)
35
500
2
cos(
7
)
(
out
+
⋅
+
+
⋅
=
t
t
t
v
E6.3
The input signal
)
3000
2
cos(
3
)
1000
2
cos(
2
1
)
(
t
t
t
v
⋅
π
+
⋅
π
+
=
has three components
with frequencies of 0, 1000 Hz and 3000 Hz.
For the dc component, we have
4
1
4
)
(
)
0
(
)
(
1
,
out,1
=
×
=
×
=
t
v
H
t
v
in
For the 1000Hz component, we have:
30
6
0
2
30
3
)
1000
(
in,2
out,2
∠
=
∠
×
∠
=
=
V
V
H
)
30
1000
2
cos(
6
)
(
out,1
+
⋅
=
t
t
v
For the 3000Hz component:
0
0
3
0
)
3000
(
in,3
out,3
=
∠
×
=
=
V
V
H
0
)
(
out,3
=
t
v
Thus, the output for all three components is
)
30
1000
2
cos(
6
4
)
(
out
+
⋅
π
+
=
t
t
v
1
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View Full DocumentE6.4
Using the voltagedivision principle, we have:
fL
j
R
R
π
2
in
out
+
×
=
V
V
Then the transfer function is:
B
f
jf
R
fL
j
fL
j
R
R
f
H
/
1
1
/
2
1
1
2
)
(
in
out
+
=
+
=
+
=
=
π
π
V
V
E6.5
From Equation 6.9, we have
Hz
200
)
2
/(
1
=
=
RC
f
B
, and from Equation 6.9, we have
.
/
1
1
)
(
in
out
B
f
jf
f
H
+
=
=
V
V
For the first component of the input, the frequency is 20 Hz,
,
71
.
5
995
.
0
)
(

∠
=
f
H
0
10
in
∠
=
V
, and
71
.
5
95
.
9
)
(
in
out

∠
=
=
V
V
f
H
Thus the first component of the output is
)
71
.
5
40
cos(
95
.
9
)
(
out,1

=
t
t
v
π
For the second component of the input, the frequency is 500 Hz,
,
2
.
68
371
.
0
)
(

∠
=
f
H
0
5
in
∠
=
V
, and
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 Fall '07
 Zatet
 Electrical Engineering, Frequency

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