Negative feedback

Negative feedback occurs when some function of the output of a system, process, or mechanism is 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.
Whereas positive feedback tends to instability via exponential growth, oscillation or 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, and it is observed in many other fields including biology, chemistry and economics. 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 climate systems on Earth. One such feedback system is the interaction between solar radiation, cloud cover, and planet temperature.

General description

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 result.
In biology, this process 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.
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. In a psychology context, on the other hand, negative refers to the valence of the feedback – attractive versus aversive, or praise versus criticism.
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.
However, negative feedback systems can still be subject to oscillations. 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 dampers fitted.

Examples

Mercury thermostats 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, reactions to price movements provide a feedback mechanism to match supply and demand.
In centrifugal governors, negative feedback is used to maintain a near-constant speed of an engine, irrespective of the load or fuel-supply conditions.
In a steering engine, power assistance is applied to the rudder with a feedback loop, to maintain the direction set by the steersman.
In servomechanisms, the speed or position of an output, as determined by a sensor, is compared to a set value, and any error is reduced by negative feedback to the input.
In audio amplifiers, negative feedback flattens frequency response, reduces 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, subtraction, integration, differentiation, logarithm, and antilog functions.
In delta-sigma analog-to-digital and digital-to-analog converters, a negative feedback loop is used to repeatedly correct accumulated quantization error during conversion.
In a phase locked loop, feedback is used to maintain a generated alternating 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 FM radio receiver, the error feedback voltage serves as the demodulated output signal. If there is a frequency divider between the generated waveform and the phase comparator, the device acts as a frequency multiplier.
In organisms, feedback enables various measures to be maintained within a desired range by homeostatic processes.
Detailed implementations

Error-controlled regulation

One use of feedback is to make a system self-regulating to minimize the effect of a disturbance. Using a negative feedback loop, a measurement of some variable is subtracted from a required value to estimate an operational error in system status, which is then used by a regulator to reduce the gap between the measurement and the required value. 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. The regulation in such systems can range from a simple 'on-off' control to a more complex processing of the error signal.
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 that is monitored by a thermometer as a change in temperature. This quantity, then, is converted by the thermostat into an electrical error in status compared to the 'set point' S, and subsequently used by the regulator ultimately to change the heat provided by a furnace to counter the initial weather-related disturbance in heat input to the house.
Error controlled regulation is typically carried out using a Proportional-Integral-Derivative 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.
Mathematically, the regulator signal is given by:
where

Negative feedback amplifier

The negative feedback amplifier was invented by Harold Stephen Black at Bell Laboratories in 1927, and granted a patent in 1937.
There are many advantages to feedback in amplifiers. 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

Negative voltage feedback in amplifiers has the following advantages; it

reduces non-linear distortion, i.e., produces higher fidelity;
increases circuit stability: i.e., gains remain stable over variations in ambient temperature, frequency, and signal amplitude;
slightly increases bandwidth;
modifies input and output impedances;
considerably reduces harmonic, phase, amplitude, and frequency distortions; and
considerably reduces noise.

Though negative feedback has many advantages, amplifiers with feedback can oscillate, and they may exhibit instability. Harry Nyquist of Bell Laboratories proposed the a stability criterion and a plot to identify stable feedback systems, including amplifiers and control systems.
The figure shows a simplified block diagram of a negative feedback amplifier.
The feedback sets the overall amplifier gain at a value:
where the approximate value assumes βA >> 1. This expression shows that a gain greater than one requires β < 1. Because the approximate gain 1/β is independent of the open-loop gain A, the feedback is said to 'desensitize' the closed-loop gain to variations in A , provided only that the gain A is sufficiently large. In this context, the factor is often called the 'desensitivity factor', and in the broader context of feedback effects that include other matters like electrical impedance and bandwidth, the 'improvement factor'.
If the disturbance D is included, the amplifier output becomes:
which shows that the feedback reduces the effect of the disturbance by the 'improvement factor'. The disturbance D might arise from fluctuations in the amplifier output due to noise and nonlinearity within this amplifier, or from other noise sources such as power supplies.
The difference signal I–βO at the amplifier input is sometimes called the "error signal". According to the diagram, the error signal is:
From this expression, it can be seen that a large 'improvement factor' tends to keep this error signal small.
Although the diagram illustrates the principles of the negative feedback amplifier, modeling a real amplifier as a unilateral forward amplification block and a unilateral feedback block has significant limitations. For methods of analysis that do not make these idealizations, see the article Negative feedback amplifier.