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$$
\begin{array}{l}{\text { a) Find the value of } R \text { that enables the circuit }} \\ {\text { shown to deliver maximum power to the }} \\ {\text { terminals a, b. }} \\ {\text { b) Find the maximum power delivered to } R \text { . }}\end{array}
$$

 

$$
\text { Using Nodal }
$$

Node (1) 

$$
\left(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right) V_{1}-\frac{100}{4}-\frac{V_{th}}{4}=\frac{20}{4} \longrightarrow (1)
$$

Node (2)

$$
\left(\frac{1}{4}+\frac{1}{4}\right) V_{th}-\frac{V_1}{4}-\frac{100}{4}=\frac{V}{4} \longrightarrow (2)
$$

$$
V=20-V_{1} \longrightarrow (3)
$$

$$
V_{th}=120
$$

$$
\text { Mesh Current }
$$

Mesh (1) 

$$
8 I_{1}-4I_2-4I_3-100+20=0 \longrightarrow (1)
$$

Mesh (2)

$$
12I_2-4I_3-4I_1-V=0 \longrightarrow (2)
$$

Mesh (3)

$$
8I_3-4 I_{1}-4I_{2}-20=0 \longrightarrow (3)
$$

$$
V=4\left(I_{1}-I_{3}\right) \longrightarrow (4)
$$

Solve (1), (2), (3), (4) 

$$
I_{1}=4 5 A, I_{2}=30 A, I_{3}=40,
$$

$$
I_{3}=I_{N}=40
$$

$$
R_{t h}=\frac{V_{ t h}}{I_{N}}=\frac{120}{40}=3 \Omega
$$

$$
R_{t h}=R_{L}=3 \Omega
$$

$$
P_{max}=\frac{V_{th}^{2}}{4 R_{ L}}=\frac{(120)^{2}}{4*3}
$$

$$
P_{max}=1200 \mathrm{w}
$$

 

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