Fundamentals of Electric Circuits
Fundamentals of Electric Circuits
Fundamentals of Electric Circuits
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3.2 Nodal Analysis 87<br />
Substituting Eq. (3.2.5) into Eq. (3.2.4) yields<br />
2v 2 v 2 2.4 1 v 2 2.4, v 1 2v 2 4.8 V<br />
From Eq. (3.2.3), we get<br />
Thus,<br />
■ METHOD 2 To use Cramer’s rule, we put Eqs. (3.2.1) to (3.2.3)<br />
in matrix form.<br />
From this, we obtain<br />
Similarly, we obtain<br />
¢ 1 <br />
¢ 2 <br />
<br />
<br />
<br />
<br />
<br />
<br />
¢ 3 <br />
<br />
<br />
<br />
v 3 3v 2 2v 1 3v 2 4v 2 v 2 2.4 V<br />
v 1 4.8 V, v 2 2.4 V, v 3 2.4 V<br />
£<br />
12 2 1<br />
0 7 1<br />
5 0 3 15<br />
12 2 1<br />
0 7 1<br />
3 12 1<br />
4 0 1<br />
5 2 0 15<br />
3 12 1<br />
4 0 1<br />
3 2 12<br />
4 7 0<br />
5 2 3 05<br />
3 2 12<br />
4 7 0<br />
3 2 1<br />
4 7 1<br />
2 3 1<br />
v 1<br />
§ £ v 2 § £<br />
v 3<br />
12<br />
0 §<br />
0<br />
(3.2.6)<br />
v 1 ¢ 1<br />
¢ , v 2 ¢ 2<br />
¢ , v 3 ¢ 3<br />
¢<br />
where ¢, ¢ 1 , ¢ 2 , and ¢ 3 are the determinants to be calculated as<br />
follows. As explained in Appendix A, to calculate the determinant <strong>of</strong><br />
a 3 by 3 matrix, we repeat the first two rows and cross multiply.<br />
3 2 1<br />
3 2 1 4 7 1<br />
¢ 3 4 7 1 3 5 2 3 15<br />
2 3 1 3 2 1 <br />
4 7 1 <br />
<br />
<br />
21 12 4 14 9 8 10<br />
84 0 0 0 36 0 48<br />
<br />
<br />
<br />
0 0 24 0 0 48 24<br />
<br />
<br />
<br />
0 144 0 168 0 0 24
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