Editing Negative-feedback amplifier (section)

==Classical feedback==
Using the model of two unilateral blocks, several consequences of feedback are simply derived.

=== 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>

=== Summary of terms ===

*[[Open-loop gain]] = <math>A_\text{OL}</math><ref>{{Cite web|url=http://cc.ee.ntu.edu.tw/~lhlu/eecourses/Electronics2/Electronics_Ch9.pdf#page=2|title=The General Feedback Structure|last=Lu|first=L. H.|archive-url=https://web.archive.org/web/20160605063422/http://cc.ee.ntu.edu.tw/~lhlu/eecourses/Electronics2/Electronics_Ch9.pdf#page=2|archive-date=2016-06-05|url-status=dead}}</ref><ref>{{Cite book|title=Audio Power Amplifier Design|last=Self|first=Douglas|date=2013-06-18|publisher=Focal Press|isbn=9780240526133|edition=6|location=New York|page=54}}</ref><ref>{{Cite book|title=The Art of Electronics|last1=Horowitz|first1=Paul|last2=Hill|first2=Winfield|date=1989-07-28|publisher=Cambridge University Press|isbn=9780521370950|edition=2|page=[https://archive.org/details/artofelectronics00horo/page/23 23]|url-access=registration|url=https://archive.org/details/artofelectronics00horo/page/23}}</ref><ref>{{Cite web|url=http://www.analog.com/media/en/training-seminars/tutorials/MT-044.pdf|title=MT-044 Op Amp Open Loop Gain and Open Loop Gain Nonlinearity|publisher=[[Analog Devices]]|quote=β is the feedback loop attenuation, or feedback factor ... noise gain is equal to 1/β}}</ref>
* Closed-loop gain = <math>\frac{A_\text{OL}}{1 + \beta \cdot A_\text{OL}}</math>
* Feedback factor = <math>\beta</math>
* Noise gain = <math>1 / \beta</math>{{Dubious|date=January 2017|reason="the noise gain and the 1/β curve are the same—until they intercept with the gain-magnitude curve. After that, the noise gain rolls off with the amplifier open-loop response but the 1/β curve continues on its path." - Jerald Graeme|One over beta}}
*[[Loop gain]] = <math>-\beta \cdot A_\text{OL}</math>
* Desensitivity factor = <math>1 + \beta \cdot A_\text{OL}</math>

===Bandwidth extension===
[[Image:Bandwidth comparison.JPG|thumb|380px|Figure 2: Gain vs. frequency for a single-pole amplifier with and without feedback; corner frequencies are labeled]]
Feedback can be used to extend the bandwidth of an amplifier at the cost of lowering the amplifier gain.<ref>R. W. Brodersen. [http://bwrc.eecs.berkeley.edu/classes/ee140/Lectures/10_stability.pdf ''Analog circuit design: lectures on stability''].</ref> Figure 2 shows such a comparison. The figure is understood as follows. Without feedback the so-called '''open-loop''' gain in this example has a single-time-constant frequency response given by

:<math> A_\text{OL}(f) = \frac{A_0}{1 + j f / f_\text{C}},</math>

where ''f''<sub>C</sub> is the [[cutoff frequency|cutoff]] or [[corner frequency]] of the amplifier: in this example ''f''<sub>C</sub> = 10<sup>4</sup> Hz, and the gain at zero frequency ''A''<sub>0</sub> = 10<sup>5</sup> V/V. The figure shows that the gain is flat out to the corner frequency and then drops. When feedback is present, the so-called '''closed-loop''' gain, as shown in the formula of the previous section, becomes

:<math>\begin{align}
 A_\text{FB}(f) &= \frac{A_\text{OL}}{1 + \beta A_\text{OL}} \\
  &= \frac{A_0 / (1 + jf/f_\text{C})}{1 + \beta A_0 / (1 + jf/f_\text{C})} \\
  &= \frac{A_0}{1 + jf/f_\text{C} + \beta A_0} \\
  &= \frac{A_0}{(1 + \beta A_0) \left(1 + j \frac{f}{(1 + \beta A_0) f_\text{C}}\right)}.
\end{align}</math>

The last expression shows that the feedback amplifier still has a single-time-constant behavior, but the corner frequency is now increased by the improvement factor (1 + β ''A''<sub>0</sub>), and the gain at zero frequency has dropped by exactly the same factor. This behavior is called the '''[[Gain–bandwidth product|gain–bandwidth tradeoff]]'''. In Figure 2, (1 + β ''A''<sub>0</sub>) = 10<sup>3</sup>, so ''A''<sub>FB</sub>(0) = 10<sup>5</sup> / 10<sup>3</sup> = 100 V/V, and ''f''<sub>C</sub> increases to 10<sup>4</sup> × 10<sup>3</sup> = 10<sup>7</sup> Hz.

===Multiple poles===
When the close-loop gain has several poles, rather than the single pole of the above example, feedback can result in complex poles (real and imaginary parts). In a two-pole case, the result is peaking in the frequency response of the feedback amplifier near its corner frequency and [[ringing artifacts|ringing]] and [[overshoot (signal)|overshoot]] in its [[step response]]. In the case of more than two poles, the feedback amplifier can become unstable and oscillate. See the discussion of [[Bode plot#Gain margin and phase margin|gain margin and phase margin]]. For a complete discussion, see Sansen.<ref name=Sansen>
{{cite book 
|author=Willy M. C. Sansen
|title=Analog design essentials
|year= 2006 
|pages=§0513-§0533, p. 155–165
|publisher=Springer 
|location=New York; Berlin
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>