To find the Thévenin equivalent resistance $R_{th}$ of the circuit to the left of $R_L$, follow these steps:

Step 1: Remove $R_L$

Since we are calculating the Thévenin resistance, we temporarily remove $R_L$ from the circuit. This isolates the network to the left of $R_L$, so we can determine the equivalent resistance of that part of the circuit.

Step 2: Deactivate the Voltage Source

Replace the 3V voltage source with a short circuit (effectively making it a wire), because in Thévenin resistance calculations, we deactivate all independent sources.

Step 3: Identify Resistor Combinations

Now, analyze the resistor network with the voltage source shorted:

1. The 20 $\Omega$ and 30 $\Omega$ resistors at the top are in series: $R_{top} = 20 \Omega + 30 \Omega = 50 \Omega$

2. The 20 $\Omega$ and 30 $\Omega$ resistors at the bottom are also in series: $R_{bottom} = 20 \Omega + 30 \Omega = 50 \Omega$

3. Now, $R_{top}$, $R_{bottom}$, and the 50 $\Omega$ resistor in the middle are all in parallel. The equivalent resistance $R_{th}$ is found using the parallel resistance formula: $\frac{1}{R_{th}} = \frac{1}{R_{top}} + \frac{1}{50 \Omega} + \frac{1}{R_{bottom}}$ Substituting the values: $\frac{1}{R_{th}} = \frac{1}{50 \Omega} + \frac{1}{50 \Omega} + \frac{1}{50 \Omega} = \frac{3}{50 \Omega}$ $R_{th} = \frac{50 \Omega}{3} \approx 16.67 \Omega$

Final Answer

The Thévenin equivalent resistance $R_{th}$ is approximately: $R_{th} \approx 16.67 \Omega$

Would you like further explanation on any part of the calculation?

Further Questions to Explore:

1. How would you find the Thévenin equivalent voltage across the terminals of $R_L$?
2. How does removing $R_L$ affect the calculation of the Thévenin equivalent?
3. What changes if we have a current source instead of a voltage source?
4. Why do we deactivate independent sources when calculating Thévenin resistance?
5. How would you find the Norton equivalent resistance from this circuit?

Tip: In circuits with multiple resistors, simplify by combining series and parallel resistors step-by-step.