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,Currently on display at Bell Laboratories in Mountainside, New Jersey. 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
Suppose now that an attenuating feedback loop applies a fraction 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
Substituting for V′in in the first expression,
Rearranging:
Then the gain of the amplifier with feedback, called the closed-loop gain, AFB is given by
If AOL ≫ 1, then AFB ≈ 1 / β, and the effective amplification (or closed-loop gain) AFB is set by the feedback constant β, and hence set by the feedback network, usually a simple reproducible network, thus making linearizing and stabilizing the amplification characteristics straightforward. If there are conditions where β AOL = −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 AOL are often displayed and investigated on a Nyquist plot (a polar plot of the gain/phase shift as a parametric function of frequency). A simpler, but less general technique, uses Bode plots.
The combination L = −β AOL appears commonly in feedback analysis and is called the loop gain. The combination (1 + β AOL) also appears commonly and is variously named as the desensitivity factor, return difference, or improvement factor.
Summary of terms
Open-loop gain = Closed-loop gain = Feedback factor = Noise gain = Loop gain = Desensitivity factor =
Bandwidth extension
Feedback can be used to extend the bandwidth of an amplifier at the cost of lowering the amplifier gain.R. W. Brodersen. Analog circuit design: lectures on stability. Figure 2 shows such a comparison. The figure is understood as follows. Without feedback the so-called open-loop gain in this example has a single-time-constant frequency response given by
where fC is the cutoff or corner frequency of the amplifier: in this example fC = 104 Hz, and the gain at zero frequency A0 = 105 V/V. The figure shows that the gain is flat out to the corner frequency and then drops. When feedback is present, the so-called closed-loop gain, as shown in the formula of the previous section, becomes
The last expression shows that the feedback amplifier still has a single-time-constant behavior, but the corner frequency is now increased by the improvement factor (1 + β A0), and the gain at zero frequency has dropped by exactly the same factor. This behavior is called the gain–bandwidth tradeoff. In Figure 2, (1 + β A0) = 103, so AFB(0) = 105 / 103 = 100 V/V, and fC increases to 104 × 103 = 107 Hz.
Multiple poles When the close-loop gain has several poles, rather than the single pole of the above example, feedback can result in complex poles (real and imaginary parts). In a two-pole case, the result is peaking in the frequency response of the feedback amplifier near its corner frequency and ringing and overshoot in its step response. In the case of more than two poles, the feedback amplifier can become unstable and oscillate. See the discussion of gain margin and phase margin. For a complete discussion, see Sansen.
Signal-flow analysis
A principal idealization behind the formulation of the Introduction is the network's division into two autonomous blocks (that is, with their own individually determined transfer functions), a simple example of what often is called "circuit partitioning", which refers in this instance to the division into a forward amplification block and a feedback block. In practical amplifiers, the information flow is not unidirectional as shown here. Frequently these blocks are taken to be two-port networks to allow inclusion of bilateral information transfer. Casting an amplifier into this form is a non-trivial task, however, especially when the feedback involved is not global (that is directly from the output to the input) but local (that is, feedback within the network, involving nodes that do not coincide with input and/or output terminals).
In these more general cases, the amplifier is analyzed more directly without the partitioning into blocks like those in the diagram, using instead some analysis based upon signal-flow analysis, such as the return-ratio method or the asymptotic gain model.For an introduction, see Commenting upon the signal-flow approach, Choma says: "In contrast to block diagram and two-port approaches to the feedback network analysis problem, signal flow methods mandate no a priori assumptions as to the unilateral or bilateral properties of the open loop and feedback subcircuits. Moreover, they are not predicated on mutually independent open loop and feedback subcircuit transfer functions, and they do not require that feedback be implemented only globally. Indeed signal flow techniques do not even require explicit identification of the open loop and feedback subcircuits. Signal flow thus removes the detriments pervasive of conventional feedback network analyses but additionally, it proves to be computationally efficient as well."
Following up on this suggestion, a signal-flow graph for a negative-feedback amplifier is shown in the figure, which is patterned after one by D'Amico et al.. Following these authors, the notation is as follows: "Variables xS, xO represent the input and output signals, moreover, two other generic variables, xi, xj linked together through the control (or critical) parameter P are explicitly shown. Parameters aij are the weight branches. Variables xi, xj and the control parameter, P, model a controlled generator, or the relation between voltage and current across two nodes of the circuit.
The term a11 is the transfer function between the input and the output [after] setting the control parameter, P, to zero; term a12 is the transfer function between the output and the controlled variable xj [after] setting the input source, xS, to zero; term a21 represents the transfer function between the source variable and the inner variable, xi when the controlled variable xj is set to zero (i.e., when the control parameter, P is set to zero); term a22 gives the relation between the independent and the controlled inner variables setting control parameter, P and input variable, xS, to zero."
Using this graph, these authors derive the generalized gain expression in terms of the control parameter P that defines the controlled source relationship xj = Pxi:
Combining these results, the gain is given by
To employ this formula, one has to identify a critical controlled source for the particular amplifier circuit in hand. For example, P could be the control parameter of one of the controlled sources in a two-port network, as shown for a particular case in D'Amico et al. As a different example, if we take a12 = a21 = 1, P = A, a22 = –β (negative feedback) and a11 = 0 (no feedforward), we regain the simple result with two unidirectional blocks.
Two-port analysis of feedback
Although, as mentioned in the section Signal-flow analysis, some form of signal-flow analysis is the most general way to treat the negative-feedback amplifier, representation as two two-ports is the approach most often presented in textbooks and is presented here. It retains a two-block circuit partition of the amplifier, but allows the blocks to be bilateral. Some drawbacks of this method are described at the end.