CR · Feedback

Negative Feedback and Gain Desensitization

Why does an op-amp’s virtual short fit so well? The answer is negative feedback. Trim a huge raw gain A with feedback, and the closed-loop gain is set not by the amplifier but by the feedback network β. Learn this gain desensitization, where the output holds steady even as A wobbles and drifts.

The larger the raw gain, the more the gain locks onto 1/β

Slide the raw gain (open-loop gain) A. Under negative feedback the closed-loop gain is G = A/(1+Aβ), where β is the fraction of the output fed back to the input. When A is small G is small too, but once the loop gain Aβ grows far past 1, G approaches and locks onto 1/β, independent of A. Since β = 0.1 the ideal gain is 10. Raise A to bring the closed-loop gain to the ideal value of 10.

Raw gain A (open-loop)A = 5

Closed-loop gain G = A/(1+Aβ)

G = 3.34  ·  Aβ = 0.5

Far from 1/β

Feedback trims the gain

An amplifier’s raw gain A is enormous, but it varies from part to part and drifts with temperature and frequency. Used as is, it cannot be trusted. Negative feedback subtracts a fraction β of the output back at the input, so the input the amplifier actually sees shrinks, and the closed-loop gain G = A/(1+Aβ) becomes smaller than the raw gain. The Aβ in the denominator is called the loop gain, the total factor a signal is multiplied by going once around through the amplifier and the feedback network.

With large loop gain, the amplifier drops out

When the loop gain Aβ is far greater than 1, the denominator 1+Aβ ≈ Aβ, and G = A/(Aβ) = 1/β. The A has cancelled out of the closed-loop gain. So the gain is set not by the amplifier’s fickle A but by the reciprocal of the feedback fraction β, fixed precisely by a few resistors. That the non-inverting amplifier’s gain was 1/β = 1 + Rf/Rg is exactly this result, and the virtual short is the same story told in the language of the circuit.

Desensitization buys precision

Even if A doubles, the closed-loop gain barely moves. Precisely, the relative change in G is reduced to 1/(1+Aβ) of the relative change in A. The larger the loop gain, the more desensitized. So we trade a cheap, enormous but imprecise raw gain, through negative feedback, for a smaller but precise and stable gain. This is also why an ideal op-amp’s virtual short fits so well: A is effectively infinite, the loop gain overwhelming, so the closed loop is fully locked onto 1/β. Negative feedback is the bargain that converts surplus gain into precision and stability.

ObserveG = A/(1+Aβ)

The definition of closed-loop gain.

ChooseAβ >> 1 → G ≈ ?

With large loop gain, G is 1/β.

Fill inβ = 0.1 → 1/β = ?

The reciprocal of β is the ideal gain.

On your own(dG/G)/(dA/A) = ?

Large loop gain desensitizes to changes in A.

Back to the first screen

As you slid the raw gain A, the closed-loop gain G grew with it at first, but once the loop gain Aβ passed 1, G stopped growing and approached and locked onto the ideal value 1/β = 10 (β = 0.1). Raise A further and G would not budge from around 10. The A had cancelled out of the closed-loop gain, leaving only the precise β. A raw gain that was enormous but untrustworthy became, through negative feedback, a smaller but trustworthy gain. This desensitization is exactly why every op-amp circuit solved so cleanly with the virtual short.

The closed-loop gain of an amplifier under negative feedback is G = A/(1+Aβ). A is the raw (open-loop) gain and β the fraction of the output fed back to the input. When the loop gain Aβ is far greater than 1, the A cancels and G ≈ 1/β, so the gain is set by the precise feedback network β, not the fickle amplifier. Moreover, when A changes, G’s relative change shrinks to 1/(1+Aβ), becoming desensitized. This gain desensitization converts an enormous but imprecise raw gain into a smaller but precise and stable one, and is the basis on which the virtual short holds for an ideal op-amp (A→∞).