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$$
\begin{array}{l}{\text { Use the node-voltage method to find } v \text { in the }} \\ {\text { circuit shown. }}\end{array}
$$

خطوات الحل

(1) $$
\text { reference node }
$$

(2) $$
V_{3}, V_{2}, V_{1}
$$ 
non refernce node 

(3) $$
\left[\sum\left(\frac{1}{R}\right)\right] V_1-\left[\sum\frac{1}{R}\right]V_2=\sum I
$$

$$
\text { at  } V_1
$$

$$
\left[\frac{1}{1}+\frac{1}{8}\right] V_{1}-\frac{1}{8} V_2=4.5 \longrightarrow (1)
$$

$$
\text {at} V_2
$$

$$
\left(\frac{1}{8}+\frac{1}{12}+\frac{1}{4}\right)V_2-\frac{1}{8}V_1=75\longrightarrow (2)
$$

Solve (1) , (2) $$
V_{1}=6 V
$$  , $$
V_{2}=18 \mathrm{V}
$$

$$
\text { KVL }
$$

$$
c l \cdot 1
$$

$$
-V+I*6+V_1=0
$$

$$
-V+? ?*6+6
$$

$$
I=\frac{V_{2}-V_{1}}{2+6}
$$

$$
I=\frac{18-6}{8}=1.5 \mathrm{A}
$$

$$
-V+1.5 * 6+6=0\Rightarrow V=15V
$$

 

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