# Negative feedback

A simple negative feedback system, and many electronic amplifiers, can be represented by this diagram. The feedback is negative if the loop gain AB is negative.

Negative feedback occurs when a result of a process influences the operation of the process itself in such a way as to reduce changes. Negative feedback can produce equilibrium and reduces the effects of perturbations. Negative feedback loops in which just the right amount of correction is applied in the most timely manner can be very stable, accurate, and responsive.

Negative feedback is widely used in mechanical and electronic engineering, but it also occurs naturally within living organisms,[1][2] and can be seen in many other fields from chemistry and economics to social behavior and physical systems such as the climate. General negative feedback systems are studied in control systems engineering. A more qualitative application of feedback is found in educational and management assessment, which is related by Roos and Hamilton to the early work on cybernetics by Norbert Wiener.[3][4][5]

## History

Negative feedback as a control technique may be seen in the refinements of the water clock introduced by 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.[6] Cornelius Drebbel had built thermostatically-controlled incubators and ovens in the early 1600s,[7] James Watt regulated the speed of the steam engine using a governor (patented in 1788), and 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.[8]

The general idea of feedback was well established by the 1920s, in reference to a means of boosting the gain of an electronic amplifier.[9] Friis and Jensen described this action as "positive feedback" and made passing mention of a contrasting "negative feed-back action" in 1924.[10] Harold Stephen Black detailed the use of negative feedback in electronic amplifiers in 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.[11][12] Nyquist and Bode built on Black’s work to develop a theory of amplifier stability, but chose to define "negative" as applying to the polarity of the loop (rather than the effect on the gain), which gave rise to some confusion over basic definitions.[9]

Early researchers in the area of cybernetics subsequently generalized the idea of negative feedback to cover any goal-seeking or purposeful behavior.[13]

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 Norbert Wiener helped to formalize the concepts of feedback control, defining feedback in general as "the chain of the transmission and return of information",[14] and negative feedback as the case when:

The information fed back to the control center tends to oppose the departure of the controlled from the controlling quantity...(p97)

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".[1] Ashby pointed out the limitations of the concept of "feedback":

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. (p54)

Further confusion arose after BF Skinner introduced the terms positive and negative reinforcement,[15] both of which can be considered negative feedback mechanisms in the sense that they try to minimize deviations from the desired behavior.[5] In a similar context, Herold and Greller used the term "negative" to refer to the valence of the feedback: that is, cases where a subject receives an evaluation with an unpleasant emotional connotation.[16]

A common theme for the 10 items [in the feedback analysis] is their valence, all representing negative feedback. Examples are being removed from a job or suffering some adverse consequence due to poor performance or receiving more or less direct indications of dissatisfaction from co-workers or the supervisor.

To reduce confusion, later authors have suggested alternative terms such as degenerative,[17] self-correcting,[18] balancing,[19] or discrepancy-reducing[20] in place of "negative".

## Overview

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, this process (in general, biochemical) is often referred to as homeostasis; whereas in mechanics, the more common term is equilibrium.

In engineering, mathematics and the physical, and biological sciences, common terms for the points around which the system gravitates include: attractors, stable states, eigenstates/eigenfunctions, equilibrium points, and setpoints.

In 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.[9]

Some authors, in particular with respect to modelling business systems, use negative to refer to the reduction in difference between the desired and actual behavior of a system.[5][21] While in a psychology context, negative refers to the valence of the feedback - how unhappy it makes the recipient.[16]

In contrast, 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.

## Examples

There are a large number of different examples of negative feedback and some are discussed below.

### Error-controlled regulation

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.[1][22]

One use of feedback is to make a system (say T) 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 subtracted from a required value (the 'set point') to estimate an operational error in system status, which is then used by a regulator (say R) to reduce the gap between the measurement and the required value.[23][24] 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.[25] The regulation in such systems can range from a simple 'on-off' control to a more complex processing of the error signal.[26]

It may be noted that the physical form of the signals in the system may change from point to point. So, for example, a disturbance (say, a change in weather) to the heat input to a house (as an example of the system T) is interpreted by a thermometer as a change in temperature (as an example of an 'essential variable' E), converted by the thermostat (a 'comparator') into an electrical error in status compared to the 'set point' S, and subsequently used by the 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.

### Negative feedback amplifier

Negative feedback amplifier using ideal unilateral elements.[27] The feedback is negative if βAOL> 0.

The figure shows a simplified block diagram of a negative feedback amplifier in which the feedback sets the overall ('closed-loop') amplifier gain at a value:

$\frac{O}{I} =\frac {A_{OL}} { 1+\beta A_{OL}} \approx \frac {1}{\beta}$

where the approximate value assumes βAOL >> 1, and 1/β as set by the feedback branch is independent of undesirable variations in the 'open-loop' gain AOL (for example, due to manufacturing variations between units, or temperature effects upon components) provided only that this gain is sufficiently large.

The difference signal I–βO that is applied to the open-loop amplifier often is called the "error signal".[28] This difference is given by:

$I - \beta O = I\left ( 1- \beta \frac {O}{I}\right ) =\frac {I}{1+ \beta A_{OL}} .$

Even though the negative feedback amplifier does not attempt to correct variations in the open-loop gain by opposing such changes, the feedback circuit does reduce their impact on the overall closed-loop system behavior.[29]

The negative feedback amplifier was invented by Harold Stephen Black at Bell Laboratories in 1927, and granted a patent in 1937.[30]

There are many advantages to feedback in amplifiers.[31] For example, all electronic devices (among them, vacuum tubes, bipolar transistors, MOS transistors) exhibit some nonlinear behavior. Negative feedback corrects this failure by trading unused gain for higher linearity (lower distortion), substituting the smaller, but constant gain 1/β for the nonlinear but large gain AOL. An amplifier with too large an open-loop gain, possibly in a specific frequency range, will result in a large feedback signal in that same range. This feedback signal, when subtracted from the original input, will act to reduce the input to the open-loop amplifier, thereby lowering the overall closed-loop gain. That is, the reduced input, although amplified again by the "too large" open-loop gain AOL, results in an output signal that is amplified by only 1/β. Because the feedback network provides a value of β that is not amplitude dependent (that is, it provides a linear output, directly proportional to its input), the net result is a flattening (desensitizing) of the amplifier's gain over those frequencies such that, of course, βAOL >> 1. Though feedback renders the gain much more predictable, amplifiers with negative feedback can oscillate. See the article on step response. They may even exhibit instability. Harry Nyquist of Bell Laboratories proposed a criterion to insure that behavior remains stable.

Negative feedback is used in this way in many types of amplification systems to stabilize and improve their operating characteristics (see e.g., operational amplifiers).

### Operational amplifier circuits

A voltage amplifier using an operational amplifier to set the voltage across R1 to Vin.

Almost all operational amplifier circuits employ negative feedback. Since the open-loop gain of an op-amp is extremely large, the tiniest of input signals 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.

Ideally the operational amplifier draws no current from the resistor divider because its input impedance is infinite.[32] The infinite gain of the ideal op-amp means this feedback circuit drives the voltage difference between the two op-amp inputs to zero.[33] Consequently, the ideal voltage gain of this circuit is derived as:

$V_{\text{out}} = \left(1 + \frac{ R_{\text{2}} }{ R_{\text{1}} } \right) V_{\text{in}}\!\,$.

Practical op-amps are not ideal, but their input impedance is extremely high and they have very high gain. So the model of an ideal op-amp often suffices to understand circuit operation, putting aside assessment of non-ideal behaviors.

If the ideal op-amp is replaced by a realistic op-amp with a finite gain and other distortions, the unwanted disturbances in the amplifier properties that would appear in an open-loop operation of the op-amp are very well suppressed by the external circuit when the op-amp gain is large.[34]

### Mechanical engineering

The fly-ball governor is an early example of negative feedback.

Negative feedback was first implemented in the 16th Century with the invention of the centrifugal governor. Its operation is most easily seen in its use by James Watt to control the speed of his steam engine. Two heavy balls on an upright frame rotate at the same speed as the engine. As their speed increases they swing up and outwards due to centrifugal force. This causes them to lift a mechanism that closes the steam inlet valve, and the engine slows. When the speed of the engine falls too far, the balls will fall by gravity and open the steam valve.

### Control systems

The ballcock or float valve uses negative feedback to control the water level in a cistern.

Examples of the use of negative feedback to control its system are: thermostat control, the phase-locked loop, control of water level (see diagram at left), and temperature regulation in animals.

A simple and practical example is a thermostat. When the temperature in a heated room reaches a certain upper limit, the room heating is switched off so that the temperature begins to fall. When the temperature drops to a lower limit, the heating is switched on again. Provided the limits are close to each other, a steady room temperature is maintained. Similar control mechanisms are used in cooling systems, such as an air conditioner, a refrigerator, or a freezer.

### Biology and chemistry

Control of endocrine hormones by negative feedback.

Some biological systems exhibit negative feedback such as the baroreflex in blood pressure regulation and erythropoiesis. Many biological process (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 glucose levels. The disruption of feedback loops can lead to undesirable results: in the case of blood glucose levels, if negative feedback fails, the glucose levels in the blood may begin to rise dramatically, thus resulting in 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 hormones are controlled by a physiologic negative feedback inhibition loop, such as the glucocorticoids secreted by the adrenal cortex. The hypothalamus secretes corticotropin-releasing hormone (CRH), which directs the anterior pituitary gland to secrete adrenocorticotropic hormone (ACTH). In turn, ACTH directs the adrenal cortex to secrete glucocorticoids, such as 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.[35]

### Economics

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

## References

1. ^ a b c W. Ross Ashby (1957). "Chapter 12: The error-controlled regulator". Introduction to cybernetics. Chapman & Hall Ltd.; Internet (1999). pp. 219–243.
2. ^ Robert E. Ricklefs, Gary Leon Miller (2000). "§6.1 Homeostasis depends upon negative feedback". Ecology. Macmillan. p. 92. ISBN 9780716728290.
3. ^ Bertil Roos, David Hamilton (March 2005). "Formative and Summative Assessment: A cybernetic viewpoint". Assessment in Education Principles Policy and Practice 12: 7–20.
4. ^ Gary E Davis, Merecedes A McGowan (2007). "Formative feedback and the mindful teaching of mathematics". Australian Senior Mathematics Journal 21 (1): 19.
5. ^ a b c Arkalgud Ramaprasad (1983). "On The Definition of Feedback". Behavioral Science 28 (1). doi:10.1002/bs.3830280103.
6. ^ Breedveld, Peter C. "Port-based modeling of mechatronic systems." Mathematics and Computers in Simulation 66.2 (2004): 99-128.
7. ^ "Tierie, Gerrit. Cornelis Drebbel. Amsterdam: HJ Paris, 1932.". Retrieved 2013-05-03.
8. ^ Maxwell, James Clerk (1868). On Governors 16. Proceedings of the Royal Society of London. pp. 270–283.
9. ^ a b c David A. Mindell (2002). Between Human and Machine : Feedback, Control, and Computing before Cybernetics.. Baltimore, MD, USA: Johns Hopkins University Press.
10. ^ Friis,H.T., and A.G.Jensen. "High Frequency Amplifiers" Bell System Technical Journal 3 (April 1924):181-205.
11. ^ Black, H.S. (January 1934). "Stabilized Feedback Amplifiers". Bell System Tech. J. (American Telephone & Telegraph) 13 (1): 1–18. Retrieved January 2, 2013.
12. ^ Bennett, Stuart (1993). A History of Control Engineering: 1930-1955. IET. p. 70. ISBN 0863412998.
13. ^ Rosenblueth, Arturo, Norbert Wiener, and Julian Bigelow. "Behavior, purpose and teleology." Philosophy of science 10.1 (1943): 18-24.
14. ^ 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.
15. ^ BF Skinner, The Experimental Analysis of Behavior, American Scientist, Vol. 45, No. 4 (SEPTEMBER 1957), pp. 343-371
16. ^ a b Herold, David M., and Martin M. Greller. "Research Notes. Feedback: The definition of a construct." Academy of management Journal 20.1 (1977): 142-147.
17. ^ Hermann A Haus and Richard B. Adler, Circuit Theory of Linear Noisy Networks, MIT Press, 1959
18. ^ Peter M. Senge (1990). The Fifth Discipline: The Art and Practice of the Learning Organization. New York: Doubleday. p. 424. ISBN 0-385-26094-6.
19. ^ John D.Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World McGraw Hill/Irwin, 2000. ISBN 978-0-07-238915-9
20. ^ Charles S. Carver, Michael F. Scheier: On the Self-Regulation of Behavior Cambridge University Press, 2001
21. ^ John D.Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World McGraw Hill/Irwin, 2000. ISBN 9780072389159
22. ^ Sudheer S Bhagade, Govind Das Nageshwar (2011). Process Dynamics and Control. PHI Learning Pvt. Ltd. pp. 6, 9. ISBN 9788120344051.
23. ^ Charles H. Wilts (1960). Principles of Feedback Control. Addison-Wesley Pub. Co. p. 1. "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."
24. ^ SK Singh (2010). Process Control: Concepts Dynamics And Applications. PHI Learning Pvt. Ltd. p. 222. ISBN 9788120336780.
25. ^ For example, input and load disturbances. See William Y. Svrcek, Donald P. Mahoney, Brent R. Young (2013). A Real-Time Approach to Process Control (3rd ed.). John Wiley & Sons. p. 57. ISBN 9781118684733.
26. ^ Charles D H Williams. "Types of feedback control". Feedback and temperature control. University of Exeter: Physics and astronomy. Retrieved 2014-06-08.
27. ^ Wai-Kai Chen (2005). "Chapter 13: General feedback theory". Circuit Analysis and Feedback Amplifier Theory. CRC Press. p. 13-1. ISBN 9781420037272. "[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."
28. ^ Muhammad Rashid (2010). Microelectronic Circuits: Analysis & Design (2nd ed.). Cengage Learning. p. 642. ISBN 9780495667728.
29. ^ Santiram Kal (2009). "§6.3.1 Gain stability". Basic Electronics: Devices, Circuits, and IT Fundamentals. PHI Learning Pvt. Ltd. pp. 193–194. ISBN 9788120319523.
30. ^ James E Brittain (February 2011). "Electrical engineering hall of fame: Harold S Black". Proceedings IEEE 99 (2): 351–353.
31. ^ Santiram Kal (2009). "§6.3 Advantages of negative feedback amplifiers". Basic electronics: Devices, circuits and its fundamentals. PHI Learning Pvt. Ltd. pp. 193 ff. ISBN 9788120319523.
32. ^ Ideal op-amp characteristics are described by, for example, G. Schitter, A. Rankers (2014). "§6.3.4 Linear amplifiers with operational amplifiers". The Design of High Performance Mechatronics. IOS Press. p. 499. ISBN 9781614993681.
33. ^ G. Schitter, A. Rankers (2014). "§6.3.4.2 Non-inverting amplifier". The Design of High Performance Mechatronics. IOS Press. p. 500. ISBN 9781614993681.
34. ^ Walter G Jung (2005). "Noise gain (NG)". Op Amp Applications Handbook. Newnes. pp. 12 ff. ISBN 9780750678445.
35. ^ Raven, PH; Johnson, GB. Biology, Fifth Edition, Boston: Hill Companies, Inc. 1999. page 1058.