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Chapter 3, Solution 1.
8
Ω
v
2
2
Ω
40
Ω
v
1
6 A
10 A
At node 1,
6 = v
1
/(8) + (v
1
- v
2
)/4 48 = 3v
1
- 2v
2
(1)
At node 2,
v
1
- v
2
/4 = v
2
/2 + 10 40 = v
1
- 3v
2
(2)
Solving (1) and (2),
v
1
= 9.143V, v
2
= -10.286 V
P
8Ω
=
()
==
8
143.9
8
v
2
2
1
10.45 W
P
4Ω
=
()
=
−
4
vv
2
21
94.37 W
P
2Ω
=
()
=
=
=
2
286.10
2
v
2
1
2
52.9 W
Chapter 3, Solution 2
At node 1,
2
vv
6
5
v
10
v
2111
−
+=−
−
60 = - 8v
1
+ 5v
2
(1)
At node 2,
2
vv
63
4
v
212
−
++=
36 = - 2v
1
+ 3v
2
(2)
Solving (1) and (2),
v
1
= 0 V, v
2
= 12 V
Chapter 3, Solution 3
Applying KCL to the upper node,
10 =
60
v
2
30
v
20
v
10
v
0oo
+++
0
+ v
0
= 40 V
i
1
= =
10
0
v
4 A , i
2
= =
20
v
0
2 A, i
3
= =
30
v
0
1.33 A, i
4
= =
60
v
0
67 mA
Chapter 3, Solution 4
At node 1,
4 + 2 = v
1
/(5) + v
1
/(10) v
1
= 20
At node 2,
5 - 2 = v
2
/(10) + v
2
/(5) v
2
= 10
i
1
= v
1
/(5) = 4 A, i
2
= v
1
/(10) = 2 A, i
3
= v
2
/(10) = 1 A, i
4
= v
2
/(5) = 2 A
Chapter 3, Solution 5
Apply KCL to the top node.
k
4
v
k
6
v20
k
2
v30
000
=
−
+
−
v
0
= 20 V
i
1
10
Ω
10
Ω
2A
i
4
i
3
v
1
v
2
i
2
4 A
5 Ω 5
Ω
5 A
Chapter 3, Solution 6
i
1
+ i
2
+ i
3
= 0 0
2
10v
6
v
4
12v
00
2
=
−
++
−
or v
0
= 8.727 V
Chapter 3, Solution 7
At node a,
ba
baaa
VV
VVVV
3610
101530
10
−=→
−
+=
−
(1)
At node b,
ba
bbba
VV
VVVV
72240
5
9
20
12
10
−=→=
−−
+
−
+
−
(2)
Solving (1) and (2) leads to
V
a
= -0.556 V, V
b
= -3.444V
Chapter 3, Solution 8
i
2
5
Ω
2 Ω
i
1
3 Ω
v
1
i
3
1
Ω
+
–
4V
0
3V
–
+
+
V
0
–
i
1
+ i
2
+ i
3
= 0 0
5
v4v
1
3v
5
v
01
11
=
−
+
−
+
But
10
v
5
2
v = so that v
1
+ 5v
1
- 15 + v
1
- 0v
5
8
1
=
or v
1
= 15x5/(27) = 2.778 V, therefore v
o
= 2v
1
/5 = 1.1111 V
Chapter 3, Solution 9
+
v
1
–
i
2
6
Ω
i
1
3 Ω
v
1
i
3
8
Ω
+
–
+ v
0
–
2v
0
12V
–
+
At the non-reference node,
6
v2v
8
v
3
v
01
11
12
−
+=
−
(1)
But
-12 + v
0
+ v
1
= 0 v
0
= 12 - v
1
(2)
Substituting (2) into (1),
6
24v3
8
v
3
v
111
−
+=
−12
v
0
= 3.652 V
Chapter 3, Solution 10
At node 1,
8
v
4
1
vv
112
+=
−
32 = -v
1
+ 8v
2
- 8v
0
(1)
1
Ω
4A
2i
0
i
0
v
1
v
0
8 Ω 2
Ω
v
2
4
Ω
At node 0,
0
0
I2
2
v
+=4 and
8
v
I
1
0
= 16 = 2v
0
+ v
1
(2)
At node 2,
2I
0
=
4
v
1
v
212
+
−
v
and
8
v
1
0
=
I
v
2
= v
1
(3)
From (1), (2) and (3), v
0
= 24 V, but from (2) we get
i
o
= 62
4
24
2
2
2
v
4
o
−=−=
−
= - 4 A
Chapter 3, Solution 11
i
3
6
Ω
v
i
1
i
2
4 Ω 3
Ω
–
+
10 V
5 A
Note that i
2
= -5A. At the non-reference node
6
v
5
4
v10
=+
−
v = 18
i
1
=
=
−
4
v10
-2 A, i
2
= -5 A
Chapter 3, Solution 12
i
3
40 Ω
v
1
v
2
10 Ω 20
Ω
–
+
5 A
50
Ω
24 V
At node 1,
40
0v
20
vv
10
v24
1211
−
+
−
=
−
96 = 7v
1
- 2v
2
(1)
At node 2,
50
v
20
vv
221
=
−
+
5
500 = -5v
1
+ 7v
2
(2)
Solving (1) and (2) gives,
v
1
= 42.87 V, v
2
= 102.05 V
==
40
v
i
1
1
1.072 A, v
2
= =
50
v
2
2.041 A
Chapter 3, Solution 13
At node number 2, [(v
2
+ 2) – 0]/10 + v
2
/4 = 3 or v
2
= 8 volts
But, I = [(v
2
+ 2) – 0]/10 = (8 + 2)/10 = 1 amp and v
1
= 8x1 = 8volts
Chapter 3, Solution 14
1 Ω
2
Ω
4
Ω
v
0
v
1
5 A
8
Ω
20 V
–
+
40 V
–
+
At node 1,
1
v40
5
2
vv
001
−
=+
−
v
1
+ v
0
= 70 (1)
At node 0,
8
20v
4
v
5
2
vv
0001
+
+=+
−
4v
1
- 7v
0
= -20 (2)
Solving (1) and (2), v
0
= 20 V
Chapter 3, Solution 15
1 Ω
2
Ω
4
Ω
v
0
v
1
5 A
8
Ω
20 V
–
+
40 V
–
+
Nodes 1 and 2 form a supernode so that v
1
= v
2
+ 10 (1)
At the supernode, 2 + 6v
1
+ 5v
2
= 3 (v
3
- v
2
) 2 + 6v
1
+ 8v
2
= 3v
3
(2)
At node 3, 2 + 4 = 3 (v
3
- v
2
) v
3
= v
2
+ 2 (3)
Substituting (1) and (3) into (2),
2 + 6v
2
+ 60 + 8v
2
= 3v
2
+ 6 v
2
=
11
56
−
v
1
= v
2
+ 10 =
11
54
i
0
= 6v
i
= 29.45 A
P
65
=
=
== 6
11
54
Gv
R
2
2
1
2
1
v
144.6 W
P
55
= =
−
= 5
11
56
Gv
2
2
2
129.6 W
P
35
=
()
==− 3)2(Gvv
2
2
3L
12 W
Chapter 3, Solution 16
2 S
+
v
0
–
13 V
–
+
i
0
1 S 4 S
8 S
v
1
v
2
v
3
2 A
At the supernode,
2 = v
1
+ 2 (v
1
- v
3
) + 8(v
2
– v
3
) + 4v
2
, which leads to 2 = 3v
1
+ 12v
2
- 10v
3
(1)
But
v
1
= v
2
+ 2v
0
and v
0
= v
2
.
Hence
v
1
= 3v
2
(2)
v
3
= 13V (3)
Substituting (2) and (3) with (1) gives,
v
1
= 18.858 V, v
2
= 6.286 V, v
3
= 13 V
Chapter 3, Solution 17
60 V
i
0
3i
0
2
Ω
10
Ω
4 Ω
60 V
–
+
8 Ω
At node 1,
2
vv
8
v
4
v60
2111
−
+=
−
120 = 7v
1
- 4v
2
(1)
At node 2, 3i
0
+ 0
2
vv
10
v60
212
=
−
+
−
But i
0
= .
4
v60
1
−
Hence
()
0
2
vv
10
v60
4
v603
2121
=
−
+
−
+
−
1020 = 5v
1
- 12v
2
(2)
Solving (1) and (2) gives v
1
= 53.08 V. Hence i
0
= =
−
4
v60
1
1.73 A
Chapter 3, Solution 18
+
v
3
–
+
v
1
–
10 V
– +
v
1
v
2
2 Ω 2
Ω
4 Ω
8
Ω
v
3
5 A
(a) (b)
At node 2, in Fig. (a), 5 =
2
vv
2
vv
3212
−
+
−
10 = - v
1
+ 2v
2
- v
3
(1)
At the supernode,
8
v
4
v
2
vv
2
vv
3
1
32
12
+=
−
+
−
40 = 2v
1
+ v
3
(2)
From Fig. (b), - v
1
- 10 + v
3
= 0 v
3
= v
1
+ 10 (3)
Solving (1) to (3), we obtain v
1
= 10 V, v
2
= 20 V = v
3
Chapter 3, Solution 19
At node 1,
321
121
31
4716
482
35 VVV
VVV
VV
−−=→+
−
+
−
+= (1)
At node 2,
321
32
221
270
428
VVV
VV
VVV
−+−=→
−
+=
−
(2)
At node 3,
321
3231
3
724360
428
12
3 VVV
VVVV
V
−+=−→=
−
+
−
+
−
+ (3)
From (1) to (3),
BAV
V
V
V
=→
−
=
−
−−
−−
36
0
16
724
271
417
3
2
1
Using MATLAB,
V 267.12 V, 933.4 V, 10
267.12
933.4
10
321
1
===→
==
−
VVVBAV
Chapter 3, Solution 20
Nodes 1 and 2 form a supernode; so do nodes 1 and 3. Hence
040
414
321
3
21
=++→=++ VVV
V
VV
(1)
.
V
1 .
V
2
2
Ω
V
3
4Ω 1
Ω
4 Ω