View source for Negative feedback

{{Short description|Reuse of output to stabilize a system}}
[[Image:Ideal feedback model.svg|thumb | right | A simple negative feedback system is descriptive, for example, of some electronic amplifiers. The feedback is negative if the loop gain AB is negative.]] 

'''Negative feedback''' (or '''balancing feedback''') occurs when some function of the output of a system, process, or mechanism is [[Feedback|fed back]] in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by other disturbances. A classic example of negative feedback is a heating system thermostat — when the temperature gets high enough, the heater is turned OFF. When the temperature gets too cold, the heat is turned back ON. In each case the "feedback" generated by the thermostat "negates" the trend.

The opposite tendency — called [[Positive feedback|positive feedback]] — is when a trend is positively reinforced, creating amplification, such as the squealing "feedback" loop that can occur when a mic is brought too close to a speaker which is amplifying the very sounds the mic is picking up, or the runaway heating and ultimate [[Physics:Nuclear meltdown|meltdown]] of a nuclear reactor.

Whereas [[Positive feedback|positive feedback]] tends to lead to instability via [[Exponential growth|exponential growth]], [[Oscillation|oscillation]] or [[Chaos theory|chaotic behavior]], negative feedback generally promotes stability. Negative feedback tends to promote a settling to equilibrium, and reduces the effects of perturbations. Negative feedback loops in which just the right amount of correction is applied with optimum timing, can be very stable, accurate, and responsive.

Negative feedback is widely used in mechanical and [[Electronic engineering|electronic engineering]], and also within living organisms,<ref name=Ashby>
{{cite book |title=Introduction to cybernetics |chapter=Chapter 12: The error-controlled regulator |author=W. Ross Ashby |chapter-url=http://pcp.vub.ac.be/books/IntroCyb.pdf |pages=219–243 |year=1957 |publisher=Chapman & Hall Ltd.; Internet (1999)}} 
</ref><ref name=Ricklefs>{{cite book |title=Ecology |chapter=§6.1 Homeostasis depends upon negative feedback |chapter-url=https://books.google.com/books?id=6TMvdZQiySoC&pg=PA92 |page=92 |isbn=9780716728290 |year=2000 |publisher=Macmillan |author1=Robert E. Ricklefs |author2=Gary Leon Miller }}
</ref> and can be seen in many other fields from chemistry and economics to physical systems such as the climate. General negative feedback systems are studied in control systems engineering.

Negative feedback loops also play an integral role in maintaining the atmospheric balance in various systems on Earth. One such feedback system is the interaction between [[Astronomy:Solar irradiance|solar radiation]], [[Earth:Cloud cover|cloud cover]], and planet temperature.

[[File:Negative Feedback Gif.gif|thumb|Blood glucose levels are maintained at a constant level in the body by a negative feedback mechanism. When the blood glucose level is too high, the pancreas secretes insulin and when the level is too low, the pancreas then secretes glucagon. The flat line shown represents the homeostatic set point. The sinusoidal line represents the blood glucose level.]]

== General description ==
[[Image:105 Negative Feedback Loops.jpg|thumb|right|Feedback loops in the human body]]

In many physical and biological systems, qualitatively different influences can oppose each other. For example, in biochemistry, one set of chemicals drives the system in a given direction, whereas another set of chemicals drives it in an opposing direction. If one or both of these opposing influences are non-linear, equilibrium point(s) result.

In [[Biology:Biology|biology]], this process (in general, [[Biology:Biochemistry|biochemical]]) is often referred to as [[Homeostasis|homeostasis]]; whereas in [[Physics:Mechanics|mechanics]], the more common term is [[Physics:Mechanical equilibrium|equilibrium]].

In [[Engineering:Engineering|engineering]], [[Mathematics|mathematics]] and the physical, and biological sciences, common terms for the points around which the system gravitates include: attractors, [[Stability theory|stable]] states, eigenstates/eigenfunctions, equilibrium points, and [[Setpoint (control system)|setpoint]]s.

In [[Control theory|control theory]], ''negative'' refers to the sign of the multiplier in mathematical models for feedback. In delta notation, −Δoutput is added to or mixed into the input. In multivariate systems, vectors help to illustrate how several influences can both partially complement and partially oppose each other.<ref name=mindell/>

Some authors, in particular with respect to modelling [[System dynamics|business systems]], use ''negative'' to refer to the reduction in difference between the desired and actual behavior of a system.<ref name="Ramaprasad">{{cite journal |author=Arkalgud Ramaprasad |title=On The Definition of Feedback |journal=Behavioral Science |volume=28 |issue=1 |pages=4–13 |year=1983 |doi=10.1002/bs.3830280103}}</ref><ref name=Sterman>
John D.Sterman, ''Business Dynamics: Systems Thinking and Modeling for a Complex World'' McGraw Hill/Irwin, 2000. {{ISBN|9780072389159}}
</ref> In a psychology context, on the other hand, ''negative'' refers to the valence of the feedback – attractive versus aversive, or praise versus criticism.<ref name=herold1977>{{cite journal | last1 = Herold | first1 = David M. | last2 = Greller | first2 = Martin M. | year = 1977 | title = Research Notes. Feedback: The Definition of a Construct | journal = Academy of Management Journal | volume = 20 | issue = 1| pages = 142–147 | doi=10.2307/255468| jstor = 255468 }}</ref>

In contrast, [[Positive feedback|positive feedback]] is feedback in which the system responds so as to increase the magnitude of any particular perturbation, resulting in amplification of the original signal instead of stabilization. Any system in which there is positive feedback together with a gain greater than one will result in a runaway situation. Both positive and negative feedback require a feedback loop to operate.

However, negative feedback systems can still be subject to [[Oscillation|oscillation]]s. This is caused by a phase shift around any loop. Due to these phase shifts the feedback signal of some frequencies can ultimately become in phase with the input signal and thus turn into positive feedback, creating a runaway condition. Even before the point where the phase shift becomes 180 degrees, stability of the negative feedback loop will become compromised, leading to increasing under- and overshoot following a disturbance. This problem is often dealt with by attenuating or changing the phase of the problematic frequencies in a design step called compensation. Unless the system naturally has sufficient damping, many negative feedback systems have low pass filters or [[Engineering:Dashpot|dampers]] fitted.

== Examples ==
* [[Engineering:Thermostat#History|Mercury thermostats]] (circa 1600) using expansion and contraction of columns of mercury in response to temperature changes were used in negative feedback systems to control vents in furnaces, maintaining a steady internal temperature.
* In the invisible hand of the market metaphor of economic theory (1776), reactions to price movements provide a feedback mechanism to match [[Finance:Supply (economics)|supply]] and [[Finance:Demand|demand]].
* In [[Centrifugal governor|centrifugal governor]]s (1788), negative feedback is used to maintain a near-constant speed of an engine, irrespective of the load or fuel-supply conditions.
* In a [[Engineering:Steering engine|steering engine]] (1866), power assistance is applied to the rudder with a feedback loop, to maintain the direction set by the steersman.
* In [[Servomechanism|servomechanism]]s, the [[Physics:Speed|speed]] or position of an output, as determined by a [[Engineering:Sensor|sensor]], is compared to a set value, and any error is reduced by negative feedback to the input.
* In [[Physics:Sound|audio]] amplifiers, negative feedback flattens [[Frequency response|frequency response]], reduces [[Physics:Distortion|distortion]], minimises the effect of manufacturing variations in component parameters, and compensates for changes in characteristics due to temperature change.
* In analog computing, feedback around operational amplifiers is used to generate mathematical functions such as [[Addition|addition]], [[Subtraction|subtraction]], [[Integral|integration]], [[Derivative|differentiation]], [[Logarithm|logarithm]], and antilog functions.
* In [[Delta-sigma modulation|delta-sigma]] [[Analog-to-digital converter|analog-to-digital]] and digital-to-analog converters (particularly for high [[Engineering:Sound quality|quality]] [[Engineering:Audio engineer|audio]]), a negative feedback loop is used to repeatedly correct accumulated [[Quantization (signal processing)|quantization error]] during conversion.
* In a phase locked loop (1932), feedback is used to maintain a generated alternating [[Waveform|waveform]] in a constant phase to a reference signal. In many implementations the generated waveform is the output, but when used as a demodulator in an [[Engineering:FM broadcasting|FM]] radio receiver, the error feedback voltage serves as the demodulated output signal. If there is a [[Engineering:Frequency divider|frequency divider]] between the generated waveform and the phase comparator, the device acts as a [[Engineering:Frequency multiplier|frequency multiplier]].
* In organisms, feedback enables various measures (e.g. body temperature, or blood sugar level) to be maintained within a desired range by homeostatic processes.
[[File:An example of a negative feedback loop is the process of increasing and decreasing glucose levels in out bloodstream.svg|thumb|]]

== Detailed implementations ==

===Error-controlled regulation===

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


[[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|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 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===
The negative feedback amplifier was invented by [[Biography:Harold Stephen Black|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 |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 negative voltage feedback in amplifiers'''
# It reduces non-linear distortion, that is, it has higher fidelity.
# It increases circuit stability: that is, the gain remains stable though there are variations in ambient temperature, frequency and signal amplitude.
# It increases bandwidth slightly.
# It modifies the input and output impedances.
# Harmonic, phase, amplitude, and frequency distortions are all reduced considerably.
# Noise is reduced considerably.

Though negative feedback has many advantages, amplifiers with feedback can oscillate. See the article on [[Step response#Step response of feedback amplifiers|step response]]. They may even exhibit [[Instability|instability]]. [[Biography:Harry Nyquist|Harry Nyquist]] of Bell Laboratories proposed the [[Nyquist stability criterion]] and the [[Nyquist plot]] that identify stable feedback systems, including amplifiers and control systems.

[[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 electrical impedance and 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 |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 |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 [[Engineering:Loop gain|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 [[Engineering: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.

===Operational amplifier circuits===

[[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 [[Engineering: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 [[Engineering: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 [[Engineering:Propagation delay|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 [[Physics: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>

== Areas of application ==

=== Mechanical engineering ===
[[File:Ballcock.svg|thumb|The ballcock or float valve uses negative feedback to control the water level in a cistern.]]
An example of the use of negative feedback control is the [[Engineering:Ballcock|ballcock]] control of water level (see diagram), or a [[Engineering:Pressure regulator|pressure regulator]]. In modern engineering, negative feedback loops are found in [[Engineering:Governor (device)|engine governor]]s, [[Engineering:Fuel injection|fuel injection]] systems and carburettors. Similar control mechanisms are used in heating and cooling systems, such as those involving air conditioners, refrigerators, or freezers.

=== Biology ===
[[Image:ACTH Negative Feedback.svg|thumb|250px|Control of endocrine hormones by negative feedback.]]

Some biological systems exhibit negative feedback such as the [[Biology:Baroreflex|baroreflex]] in [[Physics:Blood pressure|blood pressure]] regulation and erythropoiesis. Many biological processes (e.g., in the human anatomy) use negative feedback. Examples of this are numerous, from the regulating of body temperature, to the regulating of blood [[Chemistry:Glucose|glucose]] levels. The disruption of feedback loops can lead to undesirable results: in the case of [[Medicine:Blood sugar level|blood glucose levels]], if negative feedback fails, the glucose levels in the blood may begin to rise dramatically, thus resulting in [[Medicine:Diabetes mellitus|diabetes]].

For hormone secretion regulated by the negative feedback loop: when gland X releases hormone X, this stimulates target cells to release hormone Y. When there is an excess of hormone Y, gland X "senses" this and inhibits its release of hormone X. As shown in the figure, most endocrine [[Biology:Hormone|hormone]]s are controlled by a [[Biology:Physiology|physiologic]] negative feedback inhibition loop, such as the [[Chemistry:Glucocorticoid|glucocorticoid]]s secreted by the adrenal cortex.  The [[Biology:Hypothalamus|hypothalamus]] secretes [[Biology:Corticotropin-releasing hormone|corticotropin-releasing hormone (CRH)]], which directs the [[Medicine:Pituitary gland|anterior pituitary gland]] to secrete adrenocorticotropic hormone (ACTH).  In turn, ACTH directs the adrenal cortex to secrete glucocorticoids, such as [[Chemistry:Cortisol|cortisol]].  Glucocorticoids not only perform their respective functions throughout the body but also negatively affect the release of further stimulating secretions of both the hypothalamus and the pituitary gland, effectively reducing the output of glucocorticoids once a sufficient amount has been released.<ref>Raven, PH; Johnson, GB. ''Biology'', Fifth Edition, Boston: Hill Companies, Inc. 1999. page 1058.</ref>

=== Chemistry ===

Closed systems containing substances undergoing a [[Physics:Reversible reaction|reversible chemical reaction]] can also exhibit negative feedback in accordance with [[Biology:Le Chatelier's principle|Le Chatelier's principle]] which shift the [[Chemistry:Chemical equilibrium|chemical equilibrium]] to the opposite side of the reaction in order to reduce a stress. For example, in the reaction 
: N<sub>2</sub> + 3 H<sub>2</sub> ⇌ 2 NH<sub>3</sub> + 92 kJ/mol
If a mixture of the reactants and products exists at equilibrium in a sealed container and nitrogen gas is added to this system, then the equilibrium will shift toward the product side in response. If the temperature is raised, then the equilibrium will shift toward the reactant side which, since the reverse reaction is endothermic, will partially reduce the temperature.

===Self-organization===
{{main|Physics:Self-organization|Philosophy:Emergence}}
Self-organization is the capability of certain systems "of organizing their own behavior or structure".<ref name=Uttal>
{{cite book |url=https://books.google.com/books?id=vil6AgAAQBAJ&pg=PA95 |author= William R. Uttal |title=Psychomythics: Sources of Artifacts and Misconceptions in Scientific Psychology |isbn=9781135623722 |publisher=Psychology Press |year=2014 |pages=95 ''ff''}}
</ref> There are many possible factors contributing to this capacity, and most often [[Positive feedback|positive feedback]] is identified as a possible contributor. However, negative feedback also can play a role.<ref name=Camazine>
{{cite book |title=Self-organization in biological systems |author1=Scott Camazine |author2=Jean-Louis Deneubourg |author3=Nigel R Franks |author4=James Sneyd |author5=Guy Theraulaz |author6=Eric Bonabeau |chapter=Chapter 2: How self-organization works |pages=15 ''ff'' |isbn= 9780691116242 |year=2003 |publisher=Princeton University Press |chapter-url=https://books.google.com/books?id=zMgyNN6Ufj0C&pg=PA15}}
</ref>

=== Economics ===

In economics, automatic stabilisers are government programs that are intended to work as negative feedback to dampen fluctuations in real GDP.

Mainstream economics asserts that the market pricing mechanism operates to match [[Finance:Supply and demand|supply and demand]], because mismatches between them feed back into the decision-making of suppliers and demanders of goods, altering prices and thereby reducing any discrepancy. However [[Biography:Norbert Wiener|Norbert Wiener]] wrote in 1948:

:''"There is a belief current in many countries and elevated to the rank of an official article of faith in the United States that free competition is itself a homeostatic process... Unfortunately the evidence, such as it is, is against this simple-minded theory."''<ref>Cybernetics: Or Control and Communication in the Animal and the Machine p.158</ref>

The notion of economic equilibrium being maintained in this fashion by market forces has also been questioned by numerous [[Finance:Heterodox economics|heterodox]] economists such as financier George Soros<ref>Goeroge Soros, ''[https://books.google.com/books?id=qxkiYul2wgoC The Alchemy of Finance]''</ref> and leading [[Social:Ecological economics|ecological economist]] and [[Social:Steady-state economy#Herman Daly's concept of a steady-state economy|steady-state theorist]]  Herman Daly, who was with the [[Organization:World Bank|World Bank]] in 1988–1994.<ref>Herman Daly, ''[https://books.google.com/books?id=DwC8BwAAQBAJ Steady State Economics]''</ref>

=== Environmental Science ===
[[File:20220726 Feedbacks affecting global warming and climate change - block diagram.svg |thumb|right|upright=1.5| Some [[Biology:Effects of climate change|effects of climate change]] can either enhance ([[Positive feedback|positive feedback]]s) or weaken (negative feedbacks) global warming.<ref name=NASA_IntegratedSystem>{{cite web |title=The Study of Earth as an Integrated System |url=https://climate.nasa.gov/nasa_science/science/ |website=nasa.gov |publisher=NASA |date=2016 |archive-url=https://web.archive.org/web/20161102022200/https://climate.nasa.gov/nasa_science/science/ |archive-date=November 2, 2016 |url-status=live }}</ref><ref name=IPCC_AR6_SGI_FigTS.17>Fig. TS.17, ''[https://www.ipcc.ch/report/ar6/wg1/downloads/report/IPCC_AR6_WGI_TS.pdf Technical Summary],'' Sixth Assessment Report (AR6), Working Group I, IPCC, 2021, p. 96. [https://web.archive.org/web/20220721021347/https://www.ipcc.ch/report/ar6/wg1/downloads/report/IPCC_AR6_WGI_TS.pdf Archived] from the original on July 21, 2022.</ref> Observations and modeling studies indicate that there is a net positive feedback to Earth's current global warming.<ref name=WG1AR5_TS_FINAL>{{Cite book|url=https://www.ipcc.ch/site/assets/uploads/2018/02/WG1AR5_TS_FINAL.pdf |title=IPCC AR5 WG1. Technical Summary |last1=Stocker |first1=Thomas F. |last2=Dahe |first2=Qin |last3=Plattner |first3=Gian-Kaksper |year=2013 |archive-url=https://web.archive.org/web/20230716041216/https://www.ipcc.ch/site/assets/uploads/2018/02/WG1AR5_TS_FINAL.pdf |archive-date=16 July 2023 |url-status=live }} See esp. TFE.6: Climate Sensitivity and Feedbacks at p. 82.</ref>]]
A basic and common example of a negative feedback system in the environment is the interaction among [[Earth:Cloud cover|cloud cover]], plant growth, [[Astronomy:Solar irradiance|solar radiation]], and planet temperature.<ref>{{Cite journal|last1=Charlson|first1=Robert J.|last2=Lovelock|first2=James E.|last3=Andreae|first3=Meinrat O.|last4=Warren|first4=Stephen G.|date=1987|title=Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate|journal=Nature|language=en|volume=326|issue=6114|pages=655–661|doi=10.1038/326655a0|issn=1476-4687|bibcode=1987Natur.326..655C|s2cid=4321239}}</ref> As incoming solar radiation increases, planet temperature increases. As the temperature increases, the amount of plant life that can grow increases. This plant life can then make products such as sulfur which produce more cloud cover. An increase in cloud cover leads to higher [[Earth:Albedo|albedo]], or surface reflectivity, of the Earth. As albedo increases, however, the amount of solar radiation decreases.<ref>{{Cite journal|last=Winton|first=Michael|date=2006|title=Amplified Arctic climate change: What does surface albedo feedback have to do with it?|journal=Geophysical Research Letters|language=en|volume=33|issue=3|pages=L03701|doi=10.1029/2005GL025244|issn=1944-8007|bibcode=2006GeoRL..33.3701W|doi-access=free}}</ref> This, in turn, affects the rest of the cycle.

Cloud cover, and in turn planet albedo and temperature, is also influenced by the [[Physics:Water cycle|hydrological cycle]].<ref>{{Cite journal|last=Stephens|first=Graeme L.|date=2005|title=Cloud Feedbacks in the Climate System: A Critical Review|journal=Journal of Climate|volume=18|issue=2|pages=237–273|doi=10.1175/JCLI-3243.1|issn=0894-8755|bibcode=2005JCli...18..237S|s2cid=16122908 |doi-access=free}}</ref> As planet temperature increases, more water vapor is produced, creating more clouds.<ref>{{Cite journal|last1=Jickells|first1=T. D.|last2=An|first2=Z. S.|last3=Andersen|first3=K. K.|last4=Baker|first4=A. R.|last5=Bergametti|first5=G.|last6=Brooks|first6=N.|last7=Cao|first7=J. J.|last8=Boyd|first8=P. W.|last9=Duce|first9=R. A.|last10=Hunter|first10=K. A.|last11=Kawahata|first11=H.|date=2005|title=Global Iron Connections Between Desert Dust, Ocean Biogeochemistry, and Climate|journal=Science|language=en|volume=308|issue=5718|pages=67–71|doi=10.1126/science.1105959|issn=0036-8075|pmid=15802595|bibcode=2005Sci...308...67J|s2cid=16985005}}</ref> The clouds then block incoming solar radiation, lowering the temperature of the planet. This interaction produces less [[Physics:Water vapor|water vapor]] and therefore less cloud cover. The cycle then repeats in a negative feedback loop. In this way, negative feedback loops in the environment have a stabilizing effect.<ref>{{Cite journal|last1=Giannini|first1=Alessandra|last2=Biasutti|first2=Michela|last3=Verstraete|first3=Michel M.|date=2008|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>

== History ==
Negative feedback as a control technique may be seen in the refinements of the [[Physics:Water clock|water clock]] introduced by [[Biography:Ctesibius|Ktesibios]] of Alexandria in the 3rd century BCE. Self-regulating mechanisms have existed since antiquity, and were used to maintain a constant level in the reservoirs of water clocks as early as 200 BCE.<ref>{{cite journal | last1 = Breedveld | first1 = Peter C | year = 2004 | title = Port-based modeling of mechatronic systems | journal = Mathematics and Computers in Simulation | volume = 66 | issue = 2| pages = 99–128 | doi=10.1016/j.matcom.2003.11.002| citeseerx = 10.1.1.108.9830 }}</ref>

[[File:Centrifugal governor.png|thumb|The [[Centrifugal governor|fly-ball governor]] is an early example of negative feedback.]]
Negative feedback was implemented in the 17th century. Cornelius Drebbel had built [[Engineering:Thermostat#History|thermostatically controlled]] incubators and ovens in the early 1600s,<ref>{{cite web|url=http://www.drebbel.net/Tierie.pdf|title=Tierie, Gerrit. Cornelis Drebbel. Amsterdam: HJ Paris, 1932.|access-date=2013-05-03}}</ref> and [[Centrifugal governor|centrifugal governor]]s were used to regulate the distance and pressure between [[Engineering:Millstone|millstone]]s in windmills.<ref>{{cite book| last=Hills|first=Richard L|title=Power From the Wind |url=https://books.google.com/books?id=FoVkfkBV1_8C |publisher=Cambridge University Press|year=1996|isbn=9780521566865}}</ref> James Watt patented a form of governor in 1788 to control the speed of his [[Physics:Steam engine|steam engine]], and [[Biography:James Clerk Maxwell|James Clerk Maxwell]] in 1868 described "component motions" associated with these governors that lead to a decrease in a disturbance or the amplitude of an oscillation.<ref name=maxwell>{{cite journal|last=Maxwell|first=James Clerk|title=On Governors|url=https://commons.wikimedia.org/wiki/File:On_Governors.pdf|journal=Proceedings of the Royal Society of London|volume= 16|year= 1868 |pages= 270–283|via=Wikimedia|doi=10.1098/rspl.1867.0055|s2cid=51751195|doi-access=free}}</ref>

The term "[[Feedback|feedback]]" was well established by the 1920s, in reference to a means of [[Engineering:Regenerative circuit|boosting the gain]] of an electronic amplifier.<ref name=mindell>{{Cite book
|author=David A. Mindell
|title=Between Human and Machine : Feedback, Control, and Computing before Cybernetics
|year= 2002
|publisher=Johns Hopkins University Press
|location=Baltimore, MD, USA
|url=https://books.google.com/books?id=sExvSbe9MSsC|isbn=9780801868955
}}</ref> Friis and Jensen described this action as "positive feedback" and made passing mention of a contrasting "negative feed-back action" in 1924.<ref name=friis>{{Cite journal | doi=10.1002/j.1538-7305.1924.tb01354.x|title = High Frequency Amplifiers| journal=Bell System Technical Journal| volume=3| issue=2| pages=181–205|year = 1924|last1 = Friis|first1 = H. T.| last2=Jensen| first2=A. G.}}</ref> [[Biography:Harold Stephen Black|Harold Stephen Black]] came up with the idea of using negative feedback in electronic amplifiers in 1927, submitted a patent application in 1928,<ref name=Brittain/> and detailed its use in his paper of 1934, where he defined negative feedback as a type of coupling that ''reduced'' the gain of the amplifier, in the process greatly increasing its stability and bandwidth.<ref name="Black">{{cite journal
  | last = Black
  | first = H.S.
  | title = Stabilized Feedback Amplifiers
  | journal = Bell System Tech. J.
  | volume = 13
  | issue = 1
  | pages = 1–18
  | date = January 1934
  | url = http://www3.alcatel-lucent.com/bstj/vol13-1934/articles/bstj13-1-1.pdf
  | doi = 10.1002/j.1538-7305.1934.tb00652.x
  | access-date = January 2, 2013}}
</ref><ref name=BennettS>{{cite book | title=A history of control engineering 1930-1955 |chapter=Chapter 3: The electronic negative feedback amplifier |pages=70 ''ff'' |author=Stuart Bennett |isbn=9780863412806 |year=1993 |publisher=Institution of Electrical Engineers |chapter-url=https://books.google.com/books?id=JMUPAjK490gC&pg=PA70}}</ref>

[[Biography:Karl Küpfmüller|Karl Küpfmüller]] published papers on a negative-feedback-based [[Physics:Automatic gain control|automatic gain control]] system and a feedback system stability criterion in 1928.<ref>{{cite journal 
 | author = C. Bissell
 | title = Karl Kupfmuller, 1928: an early time-domain, closed-loop, stability criterion
 | journal = IEEE Control Systems Magazine
 | year = 2006
 | pages = 115–116, 126
 | url = http://oro.open.ac.uk/5575/1/01636314.pdf }}</ref>

Nyquist and Bode built on Black's work to develop a theory of amplifier stability.<ref name=BennettS/>

Early researchers in the area of [[Cybernetics|cybernetics]] subsequently generalized the idea of negative feedback to cover any goal-seeking or purposeful behavior.<ref name=Rosenblueth>Rosenblueth, Arturo, Norbert Wiener, and Julian Bigelow. "[http://courses.media.mit.edu/2004spring/mas966/rosenblueth_1943.pdf Behavior, purpose and teleology]." Philosophy of science 10.1 (1943): 18-24.</ref>
{{quote|text=All purposeful behavior may be considered to require negative feed-back. If a goal is to be attained, some signals from the goal are necessary at some time to direct the behavior.}}
Cybernetics pioneer [[Biography:Norbert Wiener|Norbert Wiener]] helped to formalize the concepts of feedback control, defining feedback in general as "the chain of the transmission and return of information",<ref name=wiener>Norbert Wiener ''Cybernetics: Or Control and Communication in the Animal and the Machine''. Cambridge, Massachusetts: The Technology Press; New York: John Wiley & Sons, Inc., 1948.</ref> and negative feedback as the case when:
{{quote|text=The information fed back to the control center tends to oppose the departure of the controlled from the controlling quantity...{{rp|page=97}}}}
While the view of feedback as any "circularity of action" helped to keep the theory simple and consistent, Ashby pointed out that, while it may clash with definitions that require a "materially evident" connection, "the exact definition of feedback is nowhere important".<ref name=Ashby/> Ashby pointed out the limitations of the concept of "feedback":
{{quote|text=The concept of 'feedback', so simple and natural in certain elementary cases, becomes artificial and of little use when the interconnections between the parts become more complex...Such complex systems cannot be treated as an interlaced set of more or less independent feedback circuits, but only as a whole. For understanding the general principles of dynamic systems, therefore, the concept of feedback is inadequate in itself. What is important is that complex systems, richly cross-connected internally, have complex behaviors, and that these behaviors can be goal-seeking in complex patterns.{{rp|page=54}} }}

To reduce confusion, later authors have suggested alternative terms such as ''degenerative'',<ref>
Hermann A Haus and Richard B. Adler, ''Circuit Theory of Linear Noisy Networks'', MIT Press, 1959
</ref> ''self-correcting'',<ref name="senge">{{Cite book
|author=Peter M. Senge
|title=The Fifth Discipline: The Art and Practice of the Learning Organization
|year=1990
|publisher=Doubleday
|location=New York
|isbn=978-0-385-26094-7
|pages=424
|url-access=registration
|url=https://archive.org/details/fifthdisciplineasen00seng
}}
</ref> ''balancing'',<ref name=Hobbs>{{cite book |title=Science and Policy in Natural Resource Management: Understanding System Complexity |isbn= 9781139458603 |year=2006 |author1=Helen E. Allison |author2=Richard J. Hobbs |url=https://books.google.com/books?id=bgHVGj_DOxAC |publisher=Cambridge University Press|quote=Balancing or negative feedback counteracts and opposes change |page=205}}
</ref> or ''discrepancy-reducing''<ref name="carver">{{cite book |url=https://books.google.com/books?id=U9xi8wlfWccC |title = On the Self-Regulation of Behavior|isbn = 9780521000994|last1 = Carver|first1 = Charles S.|last2 = Scheier|first2 = Michael F.|date = 2001-05-07}}</ref> in place of "negative".

==See also==

{{colbegin}}
*{{annotated link|Asymptotic gain model}}
*{{annotated link|Biology:Biofeedback}}
*{{annotated link|Control theory}}
*{{annotated link|Cybernetics}}
*{{annotated link|Earth:Climate change feedback}}
*{{annotated link|Nyquist stability criterion}}
*{{annotated link|Open-loop controller}}
*{{annotated link|Perceptual control theory}}
*{{annotated link|Positive feedback}}
*{{annotated link|Stability criterion}}
*{{annotated link|Step response}}
{{colend}}

== References ==

{{Reflist|30em}}

== External links ==
*{{cite web |title=Physiological Homeostasis  |url=http://www.biology-online.org/4/1_physiological_homeostasis.htm |work=biology online: answers to your biology questions |date=30 January 2020 |publisher=Biology-Online.org}}


[[Category:Control theory]]
[[Category:Cybernetics]]
[[Category:Signal processing]]
[[Category:Analog circuits]]

{{Sourceattribution|Negative feedback}}