Editing Negative feedback (section)

== Detailed implementations ==

===Error-controlled regulation===

[[File:Basic error-controlled regulator.svg|thumb|Basic error-controlled regulator loop]]

{{See also|Control engineering|Homeostasis|Allostasis}}

[[File:Regulator with feedback.png|thumb|200px|A regulator ''R'' adjusts the input to a system ''T'' so the monitored essential variables ''E'' are held to set-point values ''S'' that result in the desired system output despite disturbances ''D''.<ref name=Ashby/><ref name=Bhagade>{{cite book |title=Process Dynamics and Control |author1=Sudheer S Bhagade |author2=Govind Das Nageshwar |url=https://books.google.com/books?id=rD0xPl56hZEC&pg=PA9 |isbn=9788120344051 |year=2011 |publisher=PHI Learning Pvt. Ltd |pages=6, 9}}</ref>]]

One use of feedback is to make a system (say ''T'') [[Homeostasis|self-regulating]] to minimize the effect of a disturbance (say ''D''). Using a negative feedback loop, a measurement of some variable (for example, a  [[process variable]], say ''E'') is [[Subtraction|subtracted]] from a required value (the [[Setpoint (control system)|'set point']]) to estimate an operational error in system status, which is then used by a [[Regulator (automatic control)|regulator]] (say ''R'') to reduce the gap between the measurement and the required value.<ref name=Wilts>
{{cite book |author=Charles H. Wilts |title=Principles of Feedback Control |url=https://archive.org/details/principlesoffeed00wilt |url-access=registration |publisher=Addison-Wesley Pub. Co |year=1960 |page=[https://archive.org/details/principlesoffeed00wilt/page/1 1] |quote=In a simple feedback system a specific physical quantity is being controlled, and control is brought about by making an actual comparison of this quantity with its desired value and utilizing the difference to reduce the error observed. Such a system is self-correcting in the sense that any deviations from the desired performance are used to produce corrective action.}}</ref><ref name=Singh>{{cite book |title=Process Control: Concepts Dynamics And Applications |author=SK Singh |url=https://books.google.com/books?id=CRQr3HvzN40C&pg=PA222 |isbn=9788120336780 |year=2010 |publisher=PHI Learning Pvt. Ltd |page=222}}</ref> The regulator modifies the input to the system ''T'' according to its interpretation of the error in the status of the system. This error may be introduced by a variety of possible disturbances or 'upsets', some slow and some rapid.<ref name= Svrcek>For example, input and load disturbances. See {{cite book |title=A Real-Time Approach to Process Control |author1=William Y. Svrcek |author2=Donald P. Mahoney |author3=Brent R. Young |url=https://books.google.com/books?id=WnFPAgAAQBAJ&pg=PA57 |page=57 |isbn=9781118684733 |year=2013 |edition=3rd |publisher=John Wiley & Sons}}</ref> The [[Controller (control theory)|regulation]] in such systems can range from a simple 'on-off' control to a more complex processing of the error signal.<ref name=Exeter>{{cite web |title= Types of feedback control |url=http://newton.ex.ac.uk/teaching/cdhw/Feedback/ControlTypes.html |publisher=University of Exeter: Physics and astronomy |work=Feedback and temperature control |author=Charles D H Williams|access-date=2014-06-08}}</ref>

In this framework, the physical form of a signal may undergo multiple transformations. For example, a change in weather may cause a disturbance to the ''heat'' input to a house (as an example of the system ''T'') that is monitored by a thermometer as a change in ''temperature'' (as an example of an 'essential variable' ''E''). This quantity, then, is converted by the thermostat (a 'comparator') into an ''electrical'' error in status compared to the 'set point' ''S'', and subsequently used by the [[Regulator (automatic control)|regulator]] (containing a 'controller' that commands ''gas'' control valves and an ignitor) ultimately to change the ''heat'' provided by a furnace (an 'effector') to counter the initial weather-related disturbance in heat input to the house.<ref>{{Cite journal|last1=Giannini|first1=Alessandra|last2=Biasutti|first2=Michela|last3=Verstraete|first3=Michel M.|date=2008-12-01|title=A climate model-based review of drought in the Sahel: Desertification, the re-greening and climate change|journal=Global and Planetary Change|series=Climate Change and Desertification|volume=64|issue=3|pages=119–128|doi=10.1016/j.gloplacha.2008.05.004|issn=0921-8181|bibcode=2008GPC....64..119G}}</ref>

Error controlled regulation is typically carried out using a Proportional-Integral-Derivative Controller ([[PID controller]]). The regulator signal is derived from a weighted sum of the error signal, integral of the error signal, and derivative of the error signal. The weights of the respective components depend on the application.<ref name=Bechhoefer>{{cite journal | last = Bechhoefer | first = John | title = Feedback for Physicists: A Tutorial Essay On Control | journal = Reviews of Modern Physics | volume = 77 | issue = 3 | pages = 783–835 | doi=10.1103/revmodphys.77.783| citeseerx = 10.1.1.124.7043 | year = 2005 | bibcode = 2005RvMP...77..783B }}</ref>

Mathematically, the regulator signal is given by:
:<math>\mathrm{MV(t)}=K_p\left(\,{e(t)} + \frac{1}{T_i}\int_{0}^{t}{e(\tau)}\,{d\tau} + T_d\frac{d}{dt}e(t)\right)</math>
where
:<math>T_i</math>  is the ''integral time''
:<math>T_d</math>  is the ''derivative time''

===Negative feedback amplifier===
{{main|Negative feedback amplifier}}
The negative feedback amplifier was invented by [[Harold Stephen Black]] at [[Bell Laboratories]] in 1927, and granted a patent in 1937 (US Patent 2,102,671)<ref>{{Cite web |last=Black |first=Harold |date=1937-12-21 |title=U.S. Patent 2,102,671: Wave Translation System |url=http://www.sos.siena.edu/~aweatherwax/electronics/black_patent.pdf |url-status=dead |archive-url=https://web.archive.org/web/20141006074403/http://www.sos.siena.edu/~aweatherwax/electronics/black_patent.pdf |archive-date=2014-10-06 |access-date= |website=www.eepatents.com}}</ref> "a continuation of application Serial No. 298,155, filed August 8, 1928 ...").<ref name=Brittain>{{cite journal |author=James E Brittain |title=Electrical engineering hall of fame: Harold S Black |journal=Proceedings of the IEEE |date=February 2011 |issue=2 |volume=99 |pages=351–353 |url=http://www.ieee.org/documents/proc_scanpast0211.pdf |archive-url=https://web.archive.org/web/20141129085736/http://www.ieee.org/documents/proc_scanpast0211.pdf |url-status=dead |archive-date=November 29, 2014 |doi=10.1109/jproc.2010.2090997}}</ref><ref name=Desoer>{{cite journal |author=CA Desoer |title=In Memoriam: Harold Stephen Black |journal=IEEE Transactions on Automatic Control |volume=AC-29 |pages=673–674 |number=8 |date=August 1984 |doi=10.1109/tac.1984.1103645 }}</ref>
:"The patent is 52 pages long plus 35 pages of figures. The first 43 pages amount to a small treatise on feedback amplifiers!"<ref name=Desoer/>

There are many advantages to feedback in amplifiers.<ref name=Kal1>{{cite book |author=Santiram Kal |title=Basic electronics: Devices, circuits and its fundamentals |chapter=§6.3 Advantages of negative feedback amplifiers |pages=193 ''ff'' |chapter-url=https://books.google.com/books?id=_Bw_-ZyGL6YC&pg=PA193 |isbn=9788120319523 |year=2009 |publisher=PHI Learning Pvt. Ltd}}</ref>  In design, the type of feedback and amount of feedback are carefully selected to weigh and optimize these various benefits.

====Advantages of amplifier negative voltage feedback====
{{refimprove section|date=March 2025}}
Negative voltage feedback in amplifiers has the following advantages; it
# reduces non-linear distortion, i.e., produces higher fidelity;{{cn|date=March 2025}}
# increases circuit stability: i.e., gains remain stable over variations in ambient temperature, frequency, and signal amplitude;{{cn|date=March 2025}}
# slightly increases bandwidth;{{cn|date=March 2025}}
# modifies input and output impedances;{{cn|date=March 2025}}
# considerably reduces harmonic, phase, amplitude, and frequency distortions;{{cn|date=March 2025}} and
# considerably reduces noise.{{cn|date=March 2025}}

<!--NOTE: WIKILINKS DO NOT REPLACE REQUIREMENT FOR CITATIONS IN THIS ARTICLE, SEE [[WP:VERIFY]].-->
Though negative feedback has many advantages, amplifiers with feedback can [[oscillate]] (see  [[Step response#Step response of feedback amplifiers|Step response of feedback amplifiers]]),{{cn|date=March 2025}} and they may exhibit [[instability]].{{cn|date=March 2025}} [[Harry Nyquist]] of [[Bell Laboratories]] proposed the [[Nyquist stability criterion|a stability criterion]] and a [[Nyquist plot|plot]] to identify stable feedback systems, including amplifiers and control systems.{{cn|date=March 2025}}

<!--TEXT CHECKS/VERIFICATIONS MADE THROUGH THIS POINT OF THIS SECTION. REST OF SECTION NOT YET CHECKED.-->
[[File:Negative feedback amplifier with disturbance.png|200px|thumb|Negative feedback amplifier with external disturbance.<ref name=Thompson>
{{cite book |author=Marc Thomson |title=Intuitive Analog Circuit Design |chapter=Figure 11-4: Classical single input, single output control loop |chapter-url=https://books.google.com/books?id=d8EJP8qQQcwC&q=%22Classical+single+input,+single+output+control+loop%22&pg=PA308 |isbn=9780080478753 |year=2006 |publisher=Newnes}}
</ref> The feedback is negative if &beta;''A''&thinsp;>0.]]
The figure shows a simplified block diagram of a [[negative feedback amplifier]].

The feedback sets the overall (closed-loop) amplifier gain at a value:

:<math>\frac{O}{I} =\frac {A}  { 1+\beta A }  \approx \frac {1}{\beta} \ ,</math>

where the approximate value assumes &beta;''A '' >> 1. This expression shows that a gain greater than one requires &beta; < 1. Because the approximate gain 1/&beta; is independent of the open-loop gain ''A'', the feedback is said to 'desensitize' the closed-loop gain to variations in ''A '' (for example, due to manufacturing variations between units, or temperature effects upon components), provided only that the gain ''A'' is sufficiently large.<ref name=Kal3>
{{cite book |chapter-url=https://books.google.com/books?id=_Bw_-ZyGL6YC&q=%22the+percentage+change+in%22,+%22is+smaller+than+the+percentage+change+in%22,+%22by+an+amount+of+feedback+factor%22&pg=PA194 |chapter=§6.3.1 Gain stability |author=Santiram Kal |title=Basic Electronics: Devices, Circuits, and IT Fundamentals |isbn=9788120319523 |year=2009 |publisher= PHI Learning Pvt. Ltd |pages=193–194}}
</ref> In this context, the factor (1+&beta;''A'') is often called the 'desensitivity factor',<ref name=Thompson2>
[https://books.google.com/books?id=d8EJP8qQQcwC&dq=%22is+called+the+desensitivity+of+the+system%22&pg=PA309 Marc T Thompson, p. 309]</ref><ref name=Lee>
{{cite book |title=The Design of CMOS Radio Frequency Circuits |author1=Thomas H Lee |page=447 |edition=2nd |publisher=Cambridge University Press |year=2004 |url=https://books.google.com/books?id=io1hL48OqBsC&q=%22of+a+feedback+system%22+%22is+often+called%22&pg=PA447 |isbn=9780521835398}}
</ref> and in the broader context of feedback effects that include other matters like [[Negative feedback amplifier#Input and output resistances|electrical impedance]] and [[Negative feedback amplifier#Bandwidth extension|bandwidth]], the 'improvement factor'.<ref name=Malik>
{{cite book |title=Electronic Circuits: Analysis simulation and design |author=Norbert A Malik |page=671 |chapter=Improvement Factor |chapter-url=https://books.google.com/books?id=7AJTAAAAMAAJ&q=improvement+factor |isbn=9780023749100 |year=1995 |publisher=Prentice Hall}}
</ref>

If the disturbance ''D'' is included, the amplifier output becomes:
:<math>O =\frac {AI}  { 1+\beta A }  +\frac {D}{1+ \beta A} \ , </math>

which shows that the feedback reduces the effect of the disturbance by the 'improvement factor' (1+&beta; ''A''). The disturbance ''D'' might arise from fluctuations in the amplifier output due to noise and nonlinearity (distortion) within this amplifier, or from other noise sources such as power supplies.<ref name=Kal4>
{{cite book |title=Basic Electronics: Devices, Circuits and ''IT'' fundamentals |author=Santiram Kal |chapter-url=https://books.google.com/books?id=_Bw_-ZyGL6YC&q=%22The+sources+of+noise+in+an+amplifier%22&pg=PA194 |page=194 |chapter=§6.3.2 Noise Reduction|date=14 January 2009 |publisher=PHI Learning Pvt. |isbn=9788120319523 }}
</ref><ref name=Bhattacharya3>
{{cite book |title=Linear Control Systems: For Punjab Technical University |author=SK Bhattacharya |chapter=§5.3.3 Effect of feedback on disturbance signal |publisher=Pearson Education India |chapter-url=https://books.google.com/books?id=e5Z1A_6jxAUC&q=%22Effect+of+feedback+on+disturbance+signal%22&pg=PA137 |isbn=9788131759523}}
</ref>

The difference signal ''I''–&beta;''O'' at the amplifier input is sometimes called the "error signal".<ref name=Rashid>
{{cite book |url=https://www.google.com/search?q=%22the+difference+between+the+input+and+the+feedback+signals,+called+the+error+signal%22 |page=642 |author=Muhammad Rashid |title=Microelectronic Circuits: Analysis & Design |isbn=9780495667728 |publisher=Cengage Learning |edition=2nd  |year=2010}}
</ref> According to the diagram, the error signal is:
:<math> \text{Error signal} = I - \beta O = I \left ( 1-\beta \frac{O}{I} \right ) =\frac {I} {1 + \beta A} - \frac { \beta D} {1+\beta A} \ . </math>

From this expression, it can be seen that a large 'improvement factor' (or a large [[loop gain]] &beta;''A'') tends to keep this error signal small.

Although the diagram illustrates the principles of the negative feedback amplifier, modeling a real amplifier as a [[Amplifier#Unilateral or bilateral|unilateral forward amplification block]] and a unilateral feedback block has significant limitations.<ref name=Chen>
{{cite book |title=Circuit Analysis and Feedback Amplifier Theory |author=Wai-Kai Chen |chapter=Chapter 13: General feedback theory |chapter-url=https://books.google.com/books?id=ZlJM1OLDQx0C&q=%22THe+ideal+feedback+model+is+not+an+adequate+representation+of+a+practical+amplifier%22&pg=SA13-PA1 |quote=[In a practical amplifier] the forward path may not be strictly unilateral, the feedback path is usually bilateral, and the input and output coupling networks are often complicated. |pages=13–1 |isbn=9781420037272 |year=2005 |publisher=CRC Press}}
</ref> For methods of analysis that do not make these idealizations, see the article [[Negative feedback amplifier#Signal flow analysis|Negative feedback amplifier]].

===Operational amplifier circuits===
{{main|Operational amplifier applications}}

[[File:Feedback op-amp voltage amplifier.png|thumb|200px|A feedback voltage amplifier using an op amp with finite gain but infinite input impedances and zero output impedance.<ref name=Franco>See, for example, Figure 1.4, p. 7 ''Ideal op amp model'' in {{cite book |title=Design with operational amplifiers and analog integrated circuits |edition=3rd |author=Sergio Franco |url=https://books.google.com/books?id=em1BnAEACAAJ |publisher=McGraw-Hill |year=2002 |isbn=978-0078028168}} or {{cite book |title=Fundamentals of Circuits and Filters |editor=Wai-Kai Chen |author1=David G Nair |author2=Sergio B Franco |chapter=Figure 16.2: ''The four possible op-amp configurations'' |pages=16–2 |isbn=9781420058888 |year=2009 |publisher=CRC Press |edition=The Circuits and Filters Handbook, 3rd |chapter-url=https://books.google.com/books?id=_UVb4cxL0c0C&pg=SA16-PA2}}</ref>]]

The operational amplifier was originally developed as a building block for the construction of [[analog computers]], but is now used almost universally in all kinds of applications including [[audio signal|audio]] equipment and [[control systems]].

Operational amplifier circuits typically employ negative feedback to get a predictable transfer function. Since the open-loop gain of an [[Operational amplifier|op-amp]] is extremely large, a small differential input signal would drive the output of the amplifier to one rail or the other in the absence of negative feedback. A simple example of the use of feedback is the op-amp voltage amplifier shown in the figure.

The idealized model of an operational amplifier assumes that the gain is infinite, the input impedance is infinite, output resistance is zero, and input offset currents and voltages are zero. Such an ideal amplifier draws no current from the resistor divider.<ref name=Schitter>
{{cite book |title=The Design of High Performance Mechatronics |author1=G. Schitter |author2=A. Rankers |page=499 |chapter-url=https://books.google.com/books?id=3WvnAgAAQBAJ&pg=PA499 |chapter=§6.3.4 Linear amplifiers with operational amplifiers |isbn=9781614993681 |year=2014 |publisher=IOS Press}}
</ref>  
Ignoring dynamics (transient effects and [[propagation delay]]), the infinite gain of the ideal op-amp means this feedback circuit drives the voltage difference between the two op-amp inputs to zero.<ref name=Schitter/> Consequently, the voltage gain of the circuit in the diagram, assuming an ideal op amp, is the reciprocal of feedback [[Voltage divider|voltage division]] ratio &beta;:

:<math>V_{\text{out}} = \frac{ R_{\text{1}} + R_{\text{2}} }{ R_{\text{1}} } V_{\text{in}}\!  = \frac{1}{\beta} V_{\text{in}} \,</math>.

A real op-amp has a high but finite gain ''A'' at low frequencies, decreasing gradually at higher frequencies. In addition, it exhibits a finite input impedance and a non-zero output impedance. Although practical op-amps are not ideal, the model of an ideal op-amp often suffices to understand circuit operation at low enough frequencies. 
As discussed in the previous section, the feedback circuit stabilizes the closed-loop gain and desensitizes the output to fluctuations generated inside the amplifier itself.<ref name= Jung>
{{cite book |title=Op Amp Applications Handbook |author=Walter G Jung |chapter=Noise gain (NG) |pages=12 ''ff'' |isbn=9780750678445 |year=2005 |publisher=Newnes |chapter-url=https://books.google.com/books?id=dunqt1rt4sAC&q=%22Including+the+effects+of+finite+op+amp+gain,%22&pg=PA12}} 
</ref>