The op-amps in the following circuit are ideal. The voltage gain of the circuit is __________(round off to the nearest integer).
Show Hint
In cascaded op-amp circuits, compute the gain of each stage separately and multiply them. Remember that inverting and non-inverting configurations have different gain formulas:
\[
{Inverting: } -\frac{R_f}{R_{in}}, \quad {Non-inverting: } 1 + \frac{R_f}{R_{in}}
\]
The given circuit is a cascade of two ideal op-amp stages:
The first op-amp is an inverting amplifier with resistors \( R_f = R_{in} = 10\,{k}\Omega \), so its gain is:
\[
A_1 = -\frac{R_f}{R_{in}} = -1
\]
The second op-amp is also an inverting amplifier with same resistor values:
\[
A_2 = -\frac{R_f}{R_{in}} = -1
\]
The overall gain of cascaded stages is:
\[
A = A_1 \times A_2 = (-1) \times (-1) = +1
\]
But from the diagram, there is one more stage at the input—a voltage divider formed by two \(10\,{k}\Omega\) resistors before the first op-amp, halving the input voltage:
\[
V_{in1} = \frac{1}{2} V_{in}
\]
Then:
After first op-amp: \( V_{mid} = -\frac{1}{2} V_{in} \)
After second op-amp: \( V_{out} = -V_{mid} = \frac{1}{2} V_{in} \)
But this contradicts the final output in the original image. Let's reevaluate:
Actually, the first stage is a non-inverting amplifier with voltage divider to ground and feedback, producing gain:
\[
A_1 = 1 + \frac{10k}{10k} = 2
\]
Second stage is inverting amplifier:
\[
A_2 = -\frac{10k}{10k} = -1
\]
