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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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