Question

For the inverting operational amplifier circuit shown below, determine the closed-loop voltage gain (\(A_{cl} = V_{out}/V_{in}\)). The op-amp has an open-loop gain \(A_{OL} = 10^5\).

• A. -20
• B. 20
• C. -19.996
• D. -20.042

Hint:

When an op-amp's open-loop gain (\(A_{OL}\)) is finite, the actual closed-loop gain will always have a smaller magnitude than the ideal gain. For an inverting amplifier, this means the gain will be slightly closer to zero (e.g., -19.996 instead of -20).

Answer 1

The Correct Option is C

Step 1: Ideal vs. Real Gain
Ideally, for an inverting amplifier with \(R_f = 100k\) and \(R_{in} = 5k\), the gain is \(-R_f / R_{in} = -20\). However, the finite open-loop gain (\(A_{OL}\)) introduces an error.
Step 2: Using the Finite Gain Formula
The precise gain formula for an inverting op-amp is: \[ A_{cl} = \frac{-R_f / R_{in}}{1 + \frac{1 + R_f / R_{in}}{A_{OL}}} \]
Step 3: Calculation
Substitute the values: \(R_f / R_{in} = 20\) and \(A_{OL} = 100,000\). \[ A_{cl} = \frac{-20}{1 + \frac{1 + 20}{10^5}} = \frac{-20}{1 + \frac{21}{100,000}} \] \[ A_{cl} = \frac{-20}{1 + 0.00021} = \frac{-20}{1.00021} \approx -19.9958 \] Rounding to the nearest provided option gives \(-19.996\).