Negative-feedback amplifier

A negative-feedback amplifier (or feedback amplifier) is an electronic amplifier that subtracts a fraction of its output from its input, so that negative feedback opposes the original signal. The applied negative feedback can improve its performance (gain stability, linearity, frequency response, step response) and reduces sensitivity to parameter variations due to manufacturing or environment. Because of these advantages, many amplifiers and control systems use negative feedback. An idealized negative-feedback amplifier as shown in the diagram is a system of three elements (see Figure 1):

an amplifier with gain AOL, a feedback network β, which senses the output signal and possibly transforms it in some way (for example by attenuating or filtering it), a summing circuit that acts as a subtractor (the circle in the figure), which combines the input and the transformed output.

Overview Fundamentally, all electronic devices that provide power gain (e.g., vacuum tubes, bipolar transistors, MOS transistors) are nonlinear. Negative feedback trades gain for higher linearity (reducing distortion) and can provide other benefits. If not designed correctly, amplifiers with negative feedback can under some circumstances become unstable due to the feedback becoming positive, resulting in unwanted behavior such as oscillation. The Nyquist stability criterion developed by Harry Nyquist of Bell Laboratories is used to study the stability of feedback amplifiers. Feedback amplifiers share these properties: Pros:

Can increase or decrease input impedance (depending on type of feedback). Can increase or decrease output impedance (depending on type of feedback). Reduces total distortion if sufficiently applied (increases linearity). Increases the bandwidth. Desensitizes gain to component variations. Can control step response of amplifier. Cons:

May lead to instability if not designed carefully. Amplifier gain decreases. Input and output impedances of a negative-feedback amplifier (closed-loop amplifier) become sensitive to the gain of an amplifier without feedback (open-loop amplifier)—that exposes these impedances to variations in the open-loop gain, for example, due to parameter variations or nonlinearity of the open-loop gain. Changes the composition of the distortion (increasing audibility) if insufficiently applied.

History Paul Voigt patented a negative feedback amplifier in January 1924, though his theory lacked detail. Harold Stephen Black independently invented the negative-feedback amplifier while he was a passenger on the Lackawanna Ferry (from Hoboken Terminal to Manhattan) on his way to work at Bell Laboratories (located in Manhattan instead of New Jersey in 1927) on August 6th, 1927 (US Patent 2,102,671, issued in 1937). Black was working on reducing distortion in repeater amplifiers used for telephone transmission. On a blank space in his copy of The New York Times, he recorded the diagram found in Figure 1 and the equations derived below. On August 8, 1928, Black submitted his invention to the U. S. Patent Office, which took more than 9 years to issue the patent. Black later wrote: "One reason for the delay was that the concept was so contrary to established beliefs that the Patent Office initially did not believe it would work."

Classical feedback Using the model of two unilateral blocks, several consequences of feedback are simply derived.

Gain reduction Below, the voltage gain of the amplifier with feedback, the closed-loop gain AFB, is derived in terms of the gain of the amplifier without feedback, the open-loop gain AOL and the feedback factor β, which governs how much of the output signal is applied to the input (see Figure 1). The open-loop gain AOL in general may be a function of both frequency and voltage; the feedback parameter β is determined by the feedback network that is connected around the amplifier. For an operational amplifier, two resistors forming a voltage divider may be used for the feedback network to set β between 0 and 1. This network may be modified using reactive elements like capacitors or inductors to (a) give frequency-dependent closed-loop gain as in equalization/tone-control circuits or (b) construct oscillators. The gain of the amplifier with feedback is derived below in the case of a voltage amplifier with voltage feedback. Without feedback, the input voltage V′in is applied directly to the amplifier input. The according output voltage is

V out = A OL ⋅ V in ′ . {\displaystyle V_{\text{out}}=A_{\text{OL}}\cdot V'_{\text{in}}.}

Suppose now that an attenuating feedback loop applies a fraction β ⋅ V out {\displaystyle \beta \cdot V_{\text{out}}} of the output to one of the subtractor inputs so that it subtracts from the circuit input voltage Vin applied to the other subtractor input. The result of subtraction applied to the amplifier input is

V in ′ = V in − β ⋅ V out . {\displaystyle V'_{\text{in}}=V_{\text{in}}-\beta \cdot V_{\text{out}}.}

Substituting for V′in in the first expression,

V out = A OL ( V in − β ⋅ V out ) . {\displaystyle V_{\text{out}}=A_{\text{OL}}(V_{\text{in}}-\beta \cdot V_{\text{out}}).}

Rearranging:

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