Question

In the circuit shown in the figure, both OPAMPs are ideal. The output for the circuit \(V_{out}\) is: 

• A. \(20V_1 + 10V_2\)
• B. \(-20V_1 + 10V_2\)
• C. \(10V_1 - 20V_2\)
• D. \(20V_1 - 10V_2\)

Hint:

Always track the sign of gains carefully across multiple inverting stages; two inversions restore the original polarity, while an odd number reverses it.

Answer 1

The Correct Option is D

Step 1: Analyze the first OPAMP (inverting amplifier). 
For the first OPAMP: - Input \(V_1\) passes through resistor \(R\). - Feedback resistor = \(10R\). Gain = \(-\dfrac{10R}{R} = -10\). Thus, the output of the first OPAMP is \[ V_{A} = -10V_1 \]

Step 2: Input to the second OPAMP. 
The second OPAMP is also in inverting configuration: - Input \(V_A\) enters through \(5R\), and feedback resistor = \(10R\). Thus, gain = \(-\dfrac{10R}{5R} = -2\). This gives contribution from the first stage as \(+20V_1\).

Step 3: Add contribution from \(V_2\). 
The second OPAMP also receives \(V_2\) through resistor \(R\). For this branch: gain = \(-\dfrac{10R}{R} = -10\). Therefore, the total output is \[ V_{out} = (20V_1) + (-10V_2) = 20V_1 - 10V_2 \] 

Step 4: Conclusion. 
Hence, the output voltage is \(20V_1 - 10V_2\).