Alternating Current Analysis
249
The matrix loop equation is indicated as follows:
8
2
3
0
3
8
5
5
0
5 7
2
10
0
0
1
2
3
Ϫ
Ϫ
Ϫ
ϩ
Ϫ
Ϫ
Ϫ
ϭ
j
j
j
I
I
I
⋅
and the currents I 1 , I 2 , and I 3 are evaluated as indicated as follows:
I
V
I
Z
I
V
I
Z
I
V
I
1
1
1
2
1
2
3
1
ϭ
ϭ
ϭ
det( )
det( )
det( )
det( )
det(
3 3 )
det( )
Z
where
det( )
.
Z
j
j
j
ϭ
Ϫ
Ϫ
Ϫ
ϩ
Ϫ
Ϫ
Ϫ
ϭ
Њ
8
2
3
0
3 8
5
5
0
5 7
2
315 16 2
∠
det( )
.
.
I
j
j
1
8
5
5
5 7
2
41 1 24 9
ϭ
ϩ
Ϫ
Ϫ
Ϫ
ϭ
Њ
∠
det( ) ( )
.
I
j
2
1
3
5
0 7
2
21 8
16
ϭ Ϫ
Ϫ
Ϫ
Ϫ
ϭ
Ϫ Њ
∠
det( )
I
j
3
3 8
5
0
5
15 0
ϭ
Ϫ
ϩ
Ϫ
ϭ
Њ
∠
I 1 10 0
41 1 24 9
315 16 2
1 43 8 7
ϭ
Њ
Њ
Њ
ϭ
Њ
∠
∠
∠
∠
.
.
.
.
. A
I 2 10 0
21 8
16
315 16 2
0 693
32 2
ϭ
Њ
Ϫ Њ
Њ
ϭ
Ϫ
Њ
∠
∠
∠
∠
.
.
.
. A
I 3 10 0
15 0
315 16 2
0 476
16 2
ϭ
Њ
Њ
Њ
ϭ
Ϫ
Њ
∠
∠
∠
∠
.
.
. A
I 1
I 2
I 3
3 Ω
5 Ω
2 Ω
−j 2 Ω
−j 2 Ω
5 Ω
j 5 Ω
+
–
V 1 = 10∠0°V
FIGURE 3.22
Phasor network of R.3.57.
CRC_47760_CH003.indd 249
CRC_47760_CH003.indd 249
7/23/2008 1:27:37 PM
7/23/2008 1:27:37 PM
249
The matrix loop equation is indicated as follows:
8
2
3
0
3
8
5
5
0
5 7
2
10
0
0
1
2
3
Ϫ
Ϫ
Ϫ
ϩ
Ϫ
Ϫ
Ϫ
ϭ
j
j
j
I
I
I
⋅
and the currents I 1 , I 2 , and I 3 are evaluated as indicated as follows:
I
V
I
Z
I
V
I
Z
I
V
I
1
1
1
2
1
2
3
1
ϭ
ϭ
ϭ
det( )
det( )
det( )
det( )
det(
3 3 )
det( )
Z
where
det( )
.
Z
j
j
j
ϭ
Ϫ
Ϫ
Ϫ
ϩ
Ϫ
Ϫ
Ϫ
ϭ
Њ
8
2
3
0
3 8
5
5
0
5 7
2
315 16 2
∠
det( )
.
.
I
j
j
1
8
5
5
5 7
2
41 1 24 9
ϭ
ϩ
Ϫ
Ϫ
Ϫ
ϭ
Њ
∠
det( ) ( )
.
I
j
2
1
3
5
0 7
2
21 8
16
ϭ Ϫ
Ϫ
Ϫ
Ϫ
ϭ
Ϫ Њ
∠
det( )
I
j
3
3 8
5
0
5
15 0
ϭ
Ϫ
ϩ
Ϫ
ϭ
Њ
∠
I 1 10 0
41 1 24 9
315 16 2
1 43 8 7
ϭ
Њ
Њ
Њ
ϭ
Њ
∠
∠
∠
∠
.
.
.
.
. A
I 2 10 0
21 8
16
315 16 2
0 693
32 2
ϭ
Њ
Ϫ Њ
Њ
ϭ
Ϫ
Њ
∠
∠
∠
∠
.
.
.
. A
I 3 10 0
15 0
315 16 2
0 476
16 2
ϭ
Њ
Њ
Њ
ϭ
Ϫ
Њ
∠
∠
∠
∠
.
.
. A
I 1
I 2
I 3
3 Ω
5 Ω
2 Ω
−j 2 Ω
−j 2 Ω
5 Ω
j 5 Ω
+
–
V 1 = 10∠0°V
FIGURE 3.22
Phasor network of R.3.57.
CRC_47760_CH003.indd 249
CRC_47760_CH003.indd 249
7/23/2008 1:27:37 PM
7/23/2008 1:27:37 PM