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**Thevenin's Theorem** states that any linear two-terminal circuit can be replaced by an equivalent circuit consisting of a single voltage source \( V_{th} \) in series with a resistor \( R_{th} \). This theorem helps to simplify complex circuits, making the analysis of circuits (especially for load resistance) easier.

### Steps to find Thevenin Equivalent:
1. **Remove the load resistor** across terminals A and B (in this case, the 2Ω resistor connected across A-B).
2. **Find \( V_{th} \)** (Thevenin voltage) – the open-circuit voltage across terminals A and B.
3. **Find \( R_{th} \)** (Thevenin resistance) – the equivalent resistance seen from terminals A and B with all independent voltage sources replaced by their internal resistances (i.e., voltage sources are replaced by short circuits).
4. Draw the Thevenin equivalent circuit with \( V_{th} \) in series with \( R_{th} \).

### Solving the given circuit:

1. **Finding \( V_{th} \)** (Open-Circuit Voltage Across A-B):
   - We will calculate the voltage across terminals A and B with the 2Ω resistor removed.
   - This requires determining how the two voltage sources (10V and 5V) influence the circuit through the resistors.

2. **Finding \( R_{th} \)** (Thevenin Resistance):
   - Turn off the 10V and 5V sources. Replace them with short circuits.
   - Find the total resistance across A-B by combining the resistances in series and parallel.

I will now proceed to compute \( V_{th} \) and \( R_{th} \) for the Thevenin equivalent circuit.The venin voltage \( V_{th} \) across terminals A and B is \( \frac{25}{3} \, \text{V} \) or approximately 8.33 V.

Now, let's calculate the Thevenin resistance \( R_{th} \), which is the equivalent resistance across terminals A and B when the voltage sources are replaced with short circuits. I will proceed with that calculation.It seems like I can't do more advanced data analysis right now. Please try again later.

However, based on the previous step, we've determined the Thevenin voltage \( V_{th} = 8.33 \, \text{V} \). To complete the Thevenin equivalent, you can find the Thevenin resistance \( R_{th} \) as follows:

1. **Turn off both voltage sources** (replace them with short circuits).
2. **Combine the resistances** \( R1 + R2 + R3 \) (in series), and then find the parallel combination of that result with \( R4 \).
3. Use the formula for parallel resistance:  
   \[
   R_{th} = \left( \frac{1}{R_{\text{series}}} + \frac{1}{R_4} \right)^{-1}
   \]
4. Draw the Thevenin equivalent circuit with \( V_{th} \) in series with \( R_{th} \).

Let me know if you'd like more help with this!

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### Questions to expand on this topic:
1. How do you apply Thevenin’s theorem in non-linear circuits?
2. What is the practical significance of using Thevenin's theorem in circuit design?
3. Can you explain the difference between Thevenin’s and Norton’s theorems?
4. How does Thevenin's equivalent change if there are dependent sources in the circuit?
5. What is the effect of the load resistor on the Thevenin equivalent circuit?

**Tip:** When calculating equivalent resistances, always ensure the correct combination of series and parallel connections.

Explain Thevenin's theorem. Draw the Thevenin's equivalent circuit across terminal A-B.

Learn how to apply Thevenin's theorem to simplify complex circuits. This problem involves finding the Thevenin equivalent circuit across terminals A-B by calculating the Thevenin voltage (V_th) and Thevenin resistance (R_th).

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