To determine the Thevenin equivalent circuit at terminals $a$-$b$, we need to find two key parameters:

1. Thevenin Voltage $V_{th}$: The open-circuit voltage across terminals $a$-$b$.
2. Thevenin Resistance $R_{th}$: The equivalent resistance seen from terminals $a$-$b$ when all independent sources are turned off (voltage sources are short-circuited, and current sources are open-circuited).

Let's go through these steps for this specific circuit.

Step 1: Finding $V_{th}$ (Open-Circuit Voltage)

To find $V_{th}$:

1. The 15V voltage source and the 4A current source will contribute to the voltage across $a$-$b$.
2. Perform a node or mesh analysis (such as KVL or KCL) at key points to find the voltage difference between nodes $a$ and $b$ (open circuit).

In this case, the voltage at node $c$ can be influenced by the voltage source (15V) and the current source (4A). Calculate the node voltage across each element in the loop to determine $V_{th}$.

Step 2: Finding $R_{th}$ (Thevenin Resistance)

To find $R_{th}$:

1. Turn off the 15V voltage source (replace it with a short circuit).
2. Turn off the 4A current source (replace it with an open circuit).
3. Determine the resistance looking into terminals $a$-$b$ with these source changes.

After turning off the sources:

• The resistors $3\Omega$ (left of node $c$) and $3\Omega$ (right of node $c$) are in series.
• Their equivalent is in parallel with the $6\Omega$ resistor.

Calculate the total resistance using the formula for resistors in parallel.

Solution Outline:

1. Calculate $V_{th}$ using node or mesh analysis.
2. Calculate $R_{th}$ by finding the equivalent resistance at terminals $a$-$b$ after turning off the sources.

Would you like a detailed walkthrough of the calculations for each step?

Here are five questions to expand on this topic:

1. How do we apply mesh analysis to circuits with both voltage and current sources?
2. How does the presence of a current source affect the calculation of Thevenin resistance?
3. What is the significance of Thevenin’s theorem in simplifying complex circuits?
4. How would we find the Norton equivalent of this circuit?
5. Can Thevenin's theorem be applied to nonlinear circuits?

Tip: When calculating equivalent resistance for Thevenin, always remember to turn off all independent sources.