How to solve this question by nodal analysis?

Question

I have this circuit where I have to find the Thevenin equivalent. How do I solve this problem by nodal analysis?

Answer 1

From a comment by the op: -

I wonder whether Nodal analysis applicable to this circuit.

It is applicable in that you can apply nodal analysis to the circuit but, this site is about circuits and understanding circuits and transforming circuits so, I would go down the route of source transformation to solve this. Here are a few simple pictorial steps that should help you get to the answer you want: -

You should be able to see that the Thevenin voltage is 22 volts and, the Thevenin resistance is 9 Ω (with hardly any math involved at all).

Answer 2

Please see the answer below using nodal analysis

Answer 3

Yes I am self learning.The question asks for Maximum power for the load resistance.So,using maximum power transfer theorem we have to find the load resistance which should be equal to thevenins resistance

Hmm. But your title says:

How to solve this question by nodal analysis?

So I will take it that you understand already that the Thevenin impedance must be matched by the load. That's good. I'm glad you already understand that fact.

Here's an updated diagram for nodal analysis:

From this, we can form the following nodal equations:

$$\begin{align*} \frac{V_1}{R_1}+\frac{V_1}{R_2}+\frac{V_1}{R_3} &=\frac{V_0}{R_1}+\frac{V_2}{R_3} \\\\ \frac{V_2}{R_3}+\frac{V_2}{R_4}&=\frac{V_1}{R_3}+\frac{V_3}{R_4}+I_0 \\\\ \frac{V_3}{R_4}&=\frac{V_2}{R_4}+I_1 \end{align*}$$

We will use two values for an injected current, \$I_1\$: at \$0\:\text{A}\$ and \$1 \:\text{A}\$:

EQ1 = Eq( v1/r1 + v1/r2 + v1/r3, v0/r1 + v2/r3 )
EQ2 = Eq( v2/r3 + v2/r4, v1/r3 + v3/r4 + i0 )
EQ3 = Eq( v3/r4, i1 + v2/r4 )
solve([ EQ1, EQ2, EQ3 ], [ v1, v2, v3 ])[v3].subs(
    { v0:12, i0:2, r1:6, r2:12, r3:3, r4:2, i1:0 })
22
solve([ EQ1, EQ2, EQ3 ], [ v1, v2, v3 ])[v3].subs(
    { v0:12, i0:2, r1:6, r2:12, r3:3, r4:2, i1:1 })
31

From the above, you can see that the unloaded voltage is \$22\:\text{V}\$ and that after injecting \$1\:\text{A}\$ that the voltage moves to \$31\:\text{V}\$. So this says that the Thevenin impedance is \$9\:\Omega\$.

So that is the load impedance needed to maximize the power delivery into the load.

The above uses nodal analysis to reach this result, as requested.

As proof, here's the result from LTspice:

You can readily see that the peak power in the load occurs when the load impedance is \$9\:\Omega\$.