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Chapter 11, Solution 1.
)t50cos(160)t(v =
)9018030t50cos(2)30t50sin(20-)t(i °
−
°
+
°
−
=°−=
)60t50cos(20)t(i °+=
)60t50cos()t50cos()20)(160()t(i)t(v)t(p °
+
==
[
]
W)60cos()60t100cos(1600)t(p °
+
°+=
=)t(p W)60t100cos(1600800 °
+
+
)60cos()20)(160(
2
1
)cos(IV
2
1
P
ivmm
°=θ−θ=
=P W800
Chapter 11, Solution 2.
First, transform the circuit to the frequency domain.
°∠→ 030)t500cos(30 , 500
=
ω
150jLjH3.0 =ω→
100j-
)10)(20)(500(
j-
Cj
1
F20
6-
==
ω
→µ
I
I
1
I
2
+
−
30∠0° V
j
150
Ω
-
j
100
Ω
200 Ω
2.0j-902.0
150j
030
1
=°−∠=
°∠
=I
)t500sin(2.0)90t500cos(2.0)t(i
1
=
°
−
=
06.0j12.056.261342.0
j2
3.0
100j200
030
2
+=°∠=
−
=
−
°∠
=I
)56.25t500cos(1342.0)t(i
2
°
+
=
°
∠
=
−
=+= 49.4-1844.014.0j12.0
21
III
)35t500cos(1844.0)t(i °
−
=
For the voltage source,
])35t500cos(1844.0[])t500cos(30[)t(i)t(v)t(p
°
−
×
==
At
,
s2t
= )351000cos()1000cos(532.5p °
−
=
)935.0)(5624.0)(532.5(p
=
=p W91.2
For the inductor,
])t500sin(2.0[])t500cos(30[)t(i)t(v)t(p
×
==
At
, s2t = )1000sin()1000cos(6p =
)8269.0)(5624.0)(6(p
=
=p W79.2
For the capacitor,
°
∠
== 63.44-42.13)100j-(
2c
IV
)56.25t500cos(1342.0[])44.63500cos(42.13[)t(i)t(v)t(p °+
×
°
−
==
At , s2t = )56.261000cos()44.631000cos(18p °
+
°−=
)1329.0)(991.0)(18(p =
=p
W37.2
For the resistor,
°
∠
== 56.2584.26200
2R
IV
])56.26t500cos(1342.0[])56.26t500cos(84.26[)t(i)t(v)t(p °+
×
°
+
==
At
, s2t = )56.251000(cos602.3p
2
°+=
2
1329.0)(602.3(p =
=p
W0636.0
Chapter 11, Solution 3.
10 , °∠→°+ 3010)30t2cos( 2
=
ω
2jLjH1 =ω→
-j2
Cj
1
F25.0 =
ω
→
4
Ω
2
Ω
I I
1
I
2
+
−
10∠30° V
j
2
Ω
-
j
2
Ω
2j2
2
)2j2)(2j(
)2j2(||2j +=
−
=−
°∠=
++
°∠
= 565.11581.1
2j24
3010
I
°∠=== 565.101581.1j
2
2j
1
III
°∠=
−
= 565.56236.2
2
2j2
2
II
For the source,
)565.11-581.1)(3010(
2
1
*
°∠°∠== IVS
5.2j5.718.43905.7
+
=
°
∠
=S
The average power supplied by the source = W5.7
For the 4-Ω resistor, the average power absorbed is
=== )4()581.1(
2
1
R
2
1
P
2
2
IW5
For the inductor,
5j)2j()236.2(
2
1
2
1
2
L
2
2
=== ZIS
The average power absorbed by the inductor = W0
For the 2-Ω resistor, the average power absorbed is
=== )2()581.1(
2
1
R
2
1
P
2
2
1
IW5.2
For the capacitor,
5.2j-)2j-()581.1(
2
1
2
1
2
c
2
1
=== ZIS
The average power absorbed by the capacitor =
W0
Chapter 11, Solution 4.
20
Ω
10
Ω
I
2
I
1
+
−
-
j
10
Ω
50 V
j
5 Ω
For mesh 1,
21
10j)10j20(50 II
+
−=
21
j)j2(5 II +−= (1)
For mesh 2,
12
10j)10j5j10(0 II
+
−+=
12
2j)j2(0 II +−= (2)
In matrix form,
−
−
=
2
1
j22j
jj2
0
5
I
I
4j5 −=∆ ,
)j2(5
1
−
=
∆
, -j10
2
=
∆
°∠=
−
−
=
∆
∆
= 1.12746.1
4j5
)j2(5
1
1
I
°∠==
∆
∆
= 66.128562.1
j4-5
j10-
2
2
I
For the source,
°∠== 12.1-65.43
2
1
*
1
IVS
The average power supplied
=
°= )1.12cos(65.43 W68.42
For the 20-Ω resistor,
== R
2
1
P
2
1
IW48.30
For the inductor and capacitor,
=P W0
For the 10-Ω resistor,
== R
2
1
P
2
2
IW2.12
Chapter 11, Solution 5.
Converting the circuit into the frequency domain, we get:
1
Ω
2
Ω
+
−
j6
–j2
8∠–40˚
W4159.11
2
6828.1
P
38.256828.1
2j26j
)2j2(6j
1
408
I
2
1
1
==
°−∠=
−+
−
+
°−∠
=
Ω
Ω
P
3H
= P
0.25F
= 0
W097.52
2
258.2
P
258.238.256828.1
2j26j
6j
I
2
2
2
==
=°−∠
−+
=
Ω
Ω
Chapter 11, Solution 6.
20
Ω
10
Ω
I
2
I
1
+
−
-
j
10
Ω
50 V
j
5 Ω
For mesh 1,
04)604(2j)2j4(
o1
=
+
°∠−+ VI (1)
)604(2
2o
IV −°∠= (2)
For mesh 2,
04)604(2)j2(
o2
=
−
°∠−− VI (3)
Substituting (2) into (3),
0)604(8608)j2(
22
=
−
°
∠
−
°∠−− II
j10
6040
2
−
°∠
=I
Hence,
j10
608j-
j10
6040
6042
o
−
°∠
=
−
°∠
−°∠=V
Substituting this into (1),
−
−
°∠=
−
°∠
+°∠=+
j10
j14
)608j(
j10
6032j
608j)2j4(
1
I
°∠=
+
+°∠
= 125.06498.2
8j21
)14j1)(604(
1
I
=== )4()498.2(
2
1
R
2
1
P
2
2
14
IW48.12
Chapter 11, Solution 7.
20
Ω
10
Ω
I
2
I
1
+
−
-
j
10
Ω
50 V
j
5 Ω
Applying KVL to the left-hand side of the circuit,
oo
1.04208 VI +=°∠ (1)
Applying KCL to the right side of the circuit,
0
5j105j
8
11
o
=
−
++
VV
I
But,
o11o
10
5j10
5j10
10
VVVV
−
=→
−
=
Hence,
0
1050j
5j10
8
o
oo
=+
−
+
V
VI
oo
025.0j VI = (2)
Substituting (2) into (1),
)j1(1.0208
o
+
=°∠ V
j1
2080
o
+
°∠
=V
°∠== 25-
2
10
10
o
1
V
I
=
== )10(
2
100
2
1
R
2
1
P
2
1
IW250
Chapter 11, Solution 8.
We apply nodal analysis to the following circuit.
At node 1,
I
o
V
2
V
1
6∠0° A
0.5 I
o
j
10
Ω
-
j
20
Ω
I
2
40 Ω
20j-10j
6
211
VVV −
+=
21
120j VV
−
=
(1)
At node 2,
40
5.0
2
oo
V
II =+
But,
j20-
21
o
VV
I
−
=
Hence,
40j20-
)(5.1
221
VVV
=
−
21
)j3(3 VV −= (2)
Substituting (1) into (2),
0j33360j
222
=
+
−
− VVV
j6)-1(
37
360
j6
360j
2
+=
−
=V
j6)-1(
37
9
40
2
2
+==
V
I
=
== )40(
37
9
2
1
R
2
1
P
2
2
2
I W78.43
Chapter 11, Solution 9.
rmsV8)2)(4(V
2
6
1V
so
==
+=
=== mW
10
64
R
V
P
2
o
10
mW4.6
The current through the 2 -kΩ resistor is
mA1
k2
V
s
=
== RIP
2
2
mW2
Similarly,
== RIP
2
6
mW6
Chapter 11, Solution 10.
No current flows through each of the resistors. Hence, for each resistor,
=P W0 .
Chapter 11, Solution 11.
, , 377=ω
4
10R =
-9
10200C ×=
754.0)10200)(10)(377(RC
-94
=×=ω
°=ω 02.37)RC(tan
-1
Ω°∠=°∠
+
= k37.02-375.637.02-
)754.0(1
k10
Z
2
ab
mA)68t377cos(2)22t377sin(2)t(i
°
−
=
°+=
°∠= 68-2I
3
2
-3
ab
2
rms
10)37.02-375.6(
2
102
ZIS ×°∠
×
==
mVA37.02-751.12S °∠=
== )02.37cos(SP mW181.10
Chapter 11, Solution 12.
(a)
We find using the circuit in Fig. (a).
Th
Z
Z
th
8 Ω
-
j
2
Ω
(a)
882.1j471.0)4j1(
17
8
j28
(8)(-j2)
-j2||8
Th
−=−=
−
==Z
==
*
ThL
ZZ
Ω
+
882.1j471.0
We find using the circuit in Fig. (b).
Th
V
I
o
+
V
th
−
-
j
2
Ω
4
∠
0° A
8 Ω
(b)
)04(
2j8
2j-
o
°∠
−
=I
j28
j64-
I8
oTh
−
==V
=
==
)471.0)(8(
68
64
R8
P
2
L
2
Th
max
V
W99.15
(b)
We obtain from the circuit in Fig. (c).
Th
Z
5 Ω -
j
3
Ω
Z
th
j
2
Ω
4
Ω
(c)
167.1j5.2
3j9
)3j4)(5(
2j)3j4(||52j
Th
+=
−
−
+=−+=Z
==
*
ThL
ZZ
Ω
−
167.1j5.2
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