Question

In the given circuit, if Thevenin's equivalent resistance is \(2~\Omega\) when seen from the open terminals, then the value of \(R\) is

• A. \(>1~\Omega\)
• B. \(2~\Omega\)
• C. \(4~\Omega\)
• D. \(5~\Omega\)

Hint:

To calculate Thevenin resistance: \[ V\text{-source} \rightarrow \text{short circuit} \] \[ I\text{-source} \rightarrow \text{open circuit} \] Then reduce the remaining resistor network using series-parallel combinations.

Answer 1

The Correct Option is C

Concept: The Thevenin resistance of a circuit is obtained by deactivating all independent sources and then calculating the equivalent resistance seen from the specified terminals. The source transformation rules are:
• An ideal voltage source is replaced by a short circuit.
• An ideal current source is replaced by an open circuit. After deactivating the sources, the remaining resistor network is reduced using series-parallel combinations.

Step 1: Deactivate all independent sources.
The given circuit contains:
• A \(5V\) voltage source
• A \(1A\) current source Replacing them by their internal resistances: \[ \text{Voltage source } \rightarrow \text{ Short circuit} \] \[ \text{Current source } \rightarrow \text{ Open circuit} \] The resulting circuit contains:
• Left \(2\Omega\) resistor connected from the node to ground.
• Vertical \(2\Omega\) resistor connected from the same node to ground.
• Resistance \(R\) connected between the output terminal and the node.

Step 2: Find the equivalent resistance of the two \(2\Omega\) resistors.
The two \(2\Omega\) resistors are connected between the same node and ground. Hence they are in parallel. \[ R_p = 2 \parallel 2 = \frac{2\times2}{2+2} = 1\Omega. \] Thus the portion of the network connected to the node is equivalent to \[ 1\Omega. \]

Step 3: Determine the Thevenin resistance seen from the open terminals.
From the output terminals, current must pass through: \[ R \] and then through the equivalent \[ 1\Omega. \] Hence, \[ R_{th} = R+1. \] The problem states that \[ R_{th}=5\Omega. \] Therefore, \[ R+1=5. \] \[ R=4\Omega. \]

Step 4: Write the final answer.
Hence, \[ \boxed{R=4\Omega} \] which corresponds to option (C).