Negative feedback

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Negative feedback is the process of feeding back to the input a part of a system's output, so as to reverse the direction of change of the output. This tends to keep the output from changing, so it is stabilizing and attempts to maintain homeostasis. When a change of variable occurs within a stable range, the system will attempt to establish equilibrium. Negative feedback is also used in many types of amplification systems to stabilise and improve their amplification characteristics (see e.g. Operational amplifiers).

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. The same applies to a cooling system, such as an air conditioner, a refrigerator, or a freezer.

'Positive' and 'negative' do not refer to desirability, but rather to the sign of the multiplier in the mathematical feedback equation. The negative feedback loop tends to bring a process to equilibrium, while the positive feedback loop tends to accelerate it away from equilibrium.

While it has some advantages,such as increased stability of the system, it also has disadvantages,like loss of gain.

Some biological systems exhibit negative feedback such as the baroreflex in blood pressure regulation and erythropoiesis.

In amplifiers

Consider a voltage amplifier (other systems are similar).

Without feedback, the output voltage V_out = A_O.V_in, where the amplification A_O (also known as the open-loop gain) may in general be a function of both frequency and voltage.

The open-loop gain A_O is given as

$A_{O}={\frac {V_{out}}{V_{in}}}$ .....(1)

Suppose we have a feedback loop so that a fraction β.V_out of the output is added to the input. β is known as the feedback factor and is determined by the feedback network that is connected around the amplifier. For an operational amplifier just two resistors are required for the feedback network to set the closed-loop gain. This network may be modified using reactive elements like capacitors or inductors to (a) give frequency dependent closed-loop gain as in equalisation/tone-control circuits or (b) construct oscillators.

The input to the amplifier is now V'_in, where

$V'_{in}=V_{in}+\beta .V_{out}$ ..... (2)

The closed-loop gain A_C is given by,

$A_{C}={\frac {V_{out}}{V'_{in}}}$ ..... (3)

Substituting for V'_in from (2),

$A_{C}={\frac {V_{out}}{V_{in}+\beta .V_{out}}}$ ..... (4)

Rearranging,

$V_{in}+\beta .V_{out}={\frac {V_{out}}{A_{C}}}$ ..... (5)

Dividing both sides of (5) by V_in,

$1+\beta .{\frac {V_{out}}{V_{in}}}={\frac {V_{out}}{V_{in}.A_{c}}}$ ..... (6)

Substituting for V_out/V_in from (1),

$1+\beta .A_{O}={\frac {A_{O}}{A_{C}}}$ ..... (7)

And hence

$A_{C}={\frac {A_{O}}{1+\beta .A_{O}}}$ ..... (8)

If A_O >> 1, then A_C ≈ 1/β and the effective amplification (or closed-loop gain) A_C is set by the characteristics of the feedback constant β, thus making linearising and stabilising the amplification characteristics straightforward.

Note also that if there are conditions where β.A_O = -1, the amplifier has infinite amplification - it has become an oscillator, and the system is unstable.

The stability characteristics of the gain feedback product (β.A_O) are often displayed and investigated on a Nyquist plot (a polar plot of the gain/phase shift as a parametric function of frequency).

These are the advantages of NFB, but there is the primary disadvantage, of loss of gain:

1. Improves stability of gain
2. Increases input impedence
3. Decreases output impedence
4. Reduced distortion and noise
5. Increases the bandwidth

Disadvantage: The gain of the amplifier decreases.

See also

• Stability criterion
• Nyquist
• Closed loop
• Positive feedback
• Cybernetics

External links

• http://www.biology-online.org/4/1_physiological_homeostasis.htm

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