Editing Negative-feedback amplifier (section)

=== Gain reduction ===
Below, the voltage gain of the amplifier with feedback, the '''closed-loop gain''' ''A''<sub>FB</sub>, is derived in terms of the gain of the amplifier without feedback, the '''open-loop gain''' ''A''<sub>OL</sub> and the '''feedback factor''' β, which governs how much of the output signal is applied to the input (see Figure 1). The open-loop gain ''A''<sub>OL</sub> in general may be a function of both frequency and voltage; the feedback parameter β is determined by the feedback network that is connected around the amplifier. For an [[operational amplifier]], two resistors forming a voltage divider may be used for the feedback network to set β between 0 and 1. This network may be modified using reactive elements like [[capacitor]]s or [[inductor]]s to (a) give frequency-dependent closed-loop gain as in equalization/tone-control circuits or (b) construct oscillators. The gain of the amplifier with feedback is derived below in the case of a voltage amplifier with voltage feedback.

Without feedback, the input voltage ''V′''<sub>in</sub> is applied directly to the amplifier input. The according output voltage is

:<math>V_\text{out} =  A_\text{OL}\cdot V'_\text{in}.</math>

Suppose now that an attenuating feedback loop applies a fraction <math>\beta \cdot V_\text{out}</math> of the output to one of the subtractor inputs so that it subtracts from the circuit input voltage  ''V''<sub>in</sub> applied to the other subtractor input. The result of subtraction applied to the amplifier input is

:<math>V'_\text{in} = V_\text{in} - \beta \cdot V_\text{out}.</math>

Substituting for ''V′''<sub>in</sub> in the first expression,

:<math>V_\text{out} =  A_\text{OL} (V_\text{in} - \beta \cdot V_\text{out}).</math>

Rearranging:

:<math>V_\text{out} (1 + \beta \cdot A_\text{OL}) =  V_\text{in} \cdot A_\text{OL}.</math>

Then the gain of the amplifier with feedback, called the closed-loop gain, ''A''<sub>FB</sub> is given by

:<math>A_\text{FB} = \frac{V_\text{out}}{V_\text{in}} = \frac{A_\text{OL}}{1 + \beta \cdot A_\text{OL}}.</math>

If ''A''<sub>OL</sub> ≫ 1, then ''A''<sub>FB</sub> ≈ 1 / β, and the effective amplification (or closed-loop gain) ''A''<sub>FB</sub> is set by the feedback constant β, and hence set by the feedback network, usually a simple reproducible network, thus making linearizing and stabilizing the amplification characteristics straightforward. If there are conditions where β ''A''<sub>OL</sub> = −1, the amplifier has infinite amplification – it has become an oscillator, and the system is unstable. The stability characteristics of the gain feedback product β ''A''<sub>OL</sub> are often displayed and investigated on a [[Nyquist plot]] (a polar plot of the gain/phase shift as a parametric function of frequency). A simpler, but less general technique, uses [[Bode plot#Gain margin and phase margin|Bode plots]].

The combination ''L'' = −β ''A''<sub>OL</sub> appears commonly in feedback analysis and is called the '''[[loop gain]]'''. The combination (1 + β ''A''<sub>OL</sub>) also appears commonly and is variously named as the '''desensitivity factor''', '''return difference''', or '''improvement factor'''.<ref>{{Cite book|url=https://books.google.com/books?id=7AJTAAAAMAAJ&q=improvement+factor|title=Electronic Circuits: Analysis, Simulation, and Design|last=Malik|first=Norbert R.|date=January 1995|publisher=Prentice Hall|isbn=9780023749100|language=en}}</ref>