Finite op-amp open-loop gain limit closed-loop accuracy and introduce error terms

• A. Finite Aol improves accuracy at high frequencies
• B. Finite Aol has no effect on closed-loop accuracy
• C. The closed-loop error is proportional to β Aol
• D. Finite Aol causes the actual closed-loop gain to deviate from the ideal by roughly 1/(1+βAol); accuracy worsens with frequency

Answer: (D)

Finite open-loop gain limits how closely the amplifier can follow its intended closed-loop value. If A is the open-loop gain and β is the feedback factor, the actual closed-loop gain is G = A / (1 + Aβ). The ideal closed-loop gain, when A would be infinite, is G_ideal = 1/β. The difference between the actual and ideal gains is a fractional error on the order of 1/(Aβ+1), so the actual gain is about G_ideal × [Aβ/(Aβ+1)] or G_ideal × [1/(1 + 1/(Aβ))]. This shows that finite A reduces the closed-loop gain away from the ideal value by roughly 1/(1 + βA).

As frequency increases, the op-amp’s open-loop gain A typically decreases (finite bandwidth). That makes βA smaller, increasing the fractional error and thus worsening accuracy. This is why closed-loop accuracy degrades with frequency.

The other statements don’t fit because finite A does not improve accuracy at high frequencies, it does affect accuracy, the error isn’t simply proportional to βA, and the commonly cited magnitude of deviation is about 1/(1 + βA).