 Negative feedback amplifier

A negative feedback amplifier (or more commonly simply a feedback amplifier) is an amplifier which combines a fraction of the output with the input so that a negative feedback opposes the original signal. The applied negative feedback improves performance (gain stability, linearity, frequency response, step response) and reduces sensitivity to parameter variations due to manufacturing or environment. Because of these advantages, negative feedback is used in this way in many amplifiers and control systems.^{[1]}
A negative feedback amplifier is a system of three elements (see Figure 1): an amplifier with gain A_{OL}, an attenuating feedback network with a constant β < 1 and a summing circuit acting as a subtractor (the circle in the figure). The amplifier is the only obligatory; the other elements may be omitted in some cases. For example, in a voltage (emitter, source, opamp) follower the feedback network and the summing circuit are not necessary.
Contents
 1 Overview
 2 History
 3 Classical feedback
 4 Asymptotic gain model
 5 Feedback and amplifier type
 6 Twoport analysis of feedback
 7 See also
 8 References and notes
Overview
Fundamentally, all electronic devices used to provide power gain (e.g. vacuum tubes, bipolar transistors, MOS transistors) are nonlinear. Negative feedback allows gain to be traded for higher linearity (reducing distortion), amongst other things. If not designed correctly amplifiers with negative feedback can become unstable, resulting in unwanted behavior, such as oscillation. The Nyquist stability criterion developed by Harry Nyquist of Bell Laboratories can be used to study the stability of feedback amplifiers.
Feedback amplifiers share these properties:^{[2]}
Pros:
 Can increase or decrease input impedance (depending on type of feedback)
 Can increase or decrease output impedance (depending on type of feedback)
 Reduces distortion (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
 The gain of the amplifier decreases
 The input and output impedances of the amplifier with feedback (the closedloop amplifier) become sensitive to the gain of the amplifier without feedback (the openloop amplifier); that exposes these impedances to variations in the open loop gain, for example, due to parameter variations or due to nonlinearity of the openloop gain
History
The negative feedback amplifier was invented by Harold Stephen Black (US patent 2,102,671 (issued in 1937)^{[3]} ) while a passenger on the Lackawanna Ferry (from Hoboken Terminal to Manhattan) on his way to work at Bell Laboratories (historically located in Manhattan instead of New Jersey in 1927) on August 2, 1927. Black had been toiling at reducing distortion in repeater amplifiers used for telephone transmission. On a blank space in his copy of The New York Times,^{[4]} he recorded the diagram found in Figure 1, and the equations derived below.^{[5]} Black submitted his invention to the U. S. Patent Office on August 8, 1928, and it took more than nine years for the patent to be issued. 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."^{[6]}
Classical feedback
Gain reduction
Below, the voltage gain of the amplifier with feedback, the closedloop gain A_{fb}, is derived in terms of the gain of the amplifier without feedback, the openloop gain A_{OL} and the feedback factor β, which governs how much of the output signal is applied to the input. See Figure 1, top right. The openloop gain A_{OL} 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 frequencydependent closedloop gain as in equalization/tonecontrol 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 β.V_{out} of the output to one of the subtractor inputs so that it subtracts from the circuit input voltage V_{in} 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 closedloop gain, A_{fb} is given by,
If A_{OL} >> 1, then A_{fb} ≈ 1 / β and the effective amplification (or closedloop gain) A_{fb} 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. Note also that if there are conditions where β A_{OL} = −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_{OL} 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 A_{OL} appears commonly in feedback analysis and is called the loop gain. The combination ( 1 + β A_{OL} ) also appears commonly and is variously named as the desensitivity factor or the improvement factor.
Bandwidth extension
Feedback can be used to extend the bandwidth of an amplifier (speed it up) at the cost of lowering the amplifier gain.^{[7]} Figure 2 shows such a comparison. The figure is understood as follows. Without feedback the socalled openloop gain in this example has a single time constant frequency response given by
where f_{C} is the cutoff or corner frequency of the amplifier: in this example f_{C} = 10^{4} Hz and the gain at zero frequency A_{0} = 10^{5} V/V. The figure shows the gain is flat out to the corner frequency and then drops. When feedback is present the socalled closedloop gain, as shown in the formula of the previous section, becomes,
The last expression shows the feedback amplifier still has a single time constant behavior, but the corner frequency is now increased by the improvement factor ( 1 + β A_{0} ), and the gain at zero frequency has dropped by exactly the same factor. This behavior is called the gainbandwidth tradeoff. In Figure 2, ( 1 + β A_{0} ) = 10^{3}, so A_{fb}(0)= 10^{5} / 10^{3} = 100 V/V, and f_{C} increases to 10^{4} × 10^{3} = 10^{7} Hz.
Multiple poles
When the openloop 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 twopole 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.^{[8]}
Asymptotic gain model
Main article: asymptotic gain modelIn the above analysis the feedback network is unilateral. However, real feedback networks often exhibit feed forward as well, that is, they feed a small portion of the input to the output, degrading performance of the feedback amplifier. A more general way to model negative feedback amplifiers including this effect is with the asymptotic gain model.
Feedback and amplifier type
Amplifiers use current or voltage as input and output, so four types of amplifier are possible. See classification of amplifiers. Any of these four choices may be the openloop amplifier used to construct the feedback amplifier. The objective for the feedback amplifier also may be any one of the four types of amplifier, not necessarily the same type as the openloop amplifier. For example, an op amp (voltage amplifier) can be arranged to make a current amplifier instead. The conversion from one type to another is implemented using different feedback connections, usually referred to as series or shunt (parallel) connections.^{[9]}^{[10]} See the table below.
Feedback amplifier type Input connection Output connection Ideal feedback Twoport feedback Current Shunt Series CCCS gparameter Transresistance Shunt Shunt CCVS yparameter Transconductance Series Series VCCS zparameter Voltage Series Shunt VCVS hparameter The feedback can be implemented using a twoport network. There are four types of twoport network, and the selection depends upon the type of feedback. For example, for a current feedback amplifier, current at the output is sampled and combined with current at the input. Therefore, the feedback ideally is performed using an (output) currentcontrolled current source (CCCS), and its imperfect realization using a twoport network also must incorporate a CCCS, that is, the appropriate choice for feedback network is a gparameter twoport.
Twoport analysis of feedback
One approach to feedback is the use of return ratio. Here an alternative method used in most textbooks^{[11]}^{[12]}^{[13]} is presented by means of an example treated in the article on asymptotic gain model.
Figure 3 shows a twotransistor amplifier with a feedback resistor R_{f}. The aim is to analyze this circuit to find three items: the gain, the output impedance looking into the amplifier from the load, and the input impedance looking into the amplifier from the source.
Replacement of the feedback network with a twoport
The first step is replacement of the feedback network by a twoport. Just what components go into the twoport?
On the input side of the twoport we have R_{f}. If the voltage at the right side of R_{f} changes, it changes the current in R_{f} that is subtracted from the current entering the base of the input transistor. That is, the input side of the twoport is a dependent current source controlled by the voltage at the top of resistor R_{2}.
One might say the second stage of the amplifier is just a voltage follower, transmitting the voltage at the collector of the input transistor to the top of R_{2}. That is, the monitored output signal is really the voltage at the collector of the input transistor. That view is legitimate, but then the voltage follower stage becomes part of the feedback network. That makes analysis of feedback more complicated.
An alternative view is that the voltage at the top of R_{2} is set by the emitter current of the output transistor. That view leads to an entirely passive feedback network made up of R_{2} and R_{f}. The variable controlling the feedback is the emitter current, so the feedback is a currentcontrolled current source (CCCS). We search through the four available twoport networks and find the only one with a CCCS is the gparameter twoport, shown in Figure 4. The next task is to select the gparameters so that the twoport of Figure 4 is electrically equivalent to the Lsection made up of R_{2} and R_{f}. That selection is an algebraic procedure made most simply by looking at two individual cases: the case with V_{1} = 0, which makes the VCVS on the right side of the twoport a shortcircuit; and the case with I_{2} = 0. which makes the CCCS on the left side an open circuit. The algebra in these two cases is simple, much easier than solving for all variables at once. The choice of gparameters that make the twoport and the Lsection behave the same way are shown in the table below.
g_{11} g_{12} g_{21} g_{22} ' Smallsignal circuit
The next step is to draw the smallsignal schematic for the amplifier with the twoport in place using the hybridpi model for the transistors. Figure 5 shows the schematic with notation R_{3} = R_{C2} // R_{L} and R_{11} = 1 / g_{11}, R_{22} = g_{22} .
Loaded openloop gain
Figure 3 indicates the output node, but not the choice of output variable. A useful choice is the shortcircuit current output of the amplifier (leading to the shortcircuit current gain). Because this variable leads simply to any of the other choices (for example, load voltage or load current), the shortcircuit current gain is found below.
First the loaded openloop gain is found. The feedback is turned off by setting g_{12} = g_{21} = 0. The idea is to find how much the amplifier gain is changed because of the resistors in the feedback network by themselves, with the feedback turned off. This calculation is pretty easy because R_{11}, R_{B}, and r_{π1} all are in parallel and v_{1} = v_{π}. Let R_{1} = R_{11} // R_{B} // r_{π1}. In addition, i_{2} = −(β+1) i_{B}. The result for the openloop current gain A_{OL} is:
Gain with feedback
In the classical approach to feedback, the feedforward represented by the VCVS (that is, g_{21} v_{1}) is neglected.^{[14]} That makes the circuit of Figure 5 resemble the block diagram of Figure 1, and the gain with feedback is then:
where the feedback factor β_{FB} = −g_{12}. Notation β_{FB} is introduced for the feedback factor to distinguish it from the transistor β.
Input and output resistances
First, a digression on how twoport theory approaches resistance determination, and then its application to the amplifier at hand.
Background on resistance determination
Figure 6 shows an equivalent circuit for finding the input resistance of a feedback voltage amplifier (left) and for a feedback current amplifier (right). These arrangements are typical Miller theorem applications.
In the case of the voltage amplifier, the output voltage βV_{out} of the feedback network is applied in series and with an opposite polarity to the input voltage V_{x} travelling over the loop (but in respect to ground, the polarities are the same). As a result, the effective voltage across and the current through the amplifier input resistance R_{in} decrease so that the circuit input resistance increases (one might say that R_{in} apparently increases). Its new value can be calculated by applying Miller theorem (for voltages) or the basic circuit laws. Thus Kirchhoff's voltage law provides:
where v_{out} = A_{v} v_{in} = A_{v} I_{x} R_{in}. Substituting this result in the above equation and solving for the input resistance of the feedback amplifier, the result is:
The general conclusion to be drawn from this example and a similar example for the output resistance case is:
A series feedback connection at the input (output) increases the input (output) resistance by a factor ( 1 + β A_{OL} ), where A_{OL} = open loop gain.
On the other hand, for the current amplifier, the output current βI_{out} of the feedback network is applied in parallel and with an opposite direction to the input current I_{x}. As a result, the total current flowing through the circuit input (not only through the input resistance R_{in}) increases and the voltage across it decreases so that the circuit input resistance decreases (R_{in} apparently decreases). Its new value can be calculated by applying the dual Miller theorem (for currents) or the basic Kirchhoff's laws:
where i_{out} = A_{i} i_{in} = A_{i} V_{x} / R_{in}. Substituting this result in the above equation and solving for the input resistance of the feedback amplifier, the result is:
The general conclusion to be drawn from this example and a similar example for the output resistance case is:
A parallel feedback connection at the input (output) decreases the input (output) resistance by a factor ( 1 + β A_{OL} ), where A_{OL} = open loop gain.
These conclusions can be generalized to treat cases with arbitrary Norton or Thévenin drives, arbitrary loads, and general twoport feedback networks. However, the results do depend upon the main amplifier having a representation as a twoport – that is, the results depend on the same current entering and leaving the input terminals, and likewise, the same current that leaves one output terminal must enter the other output terminal.
A broader conclusion to be drawn, independent of the quantitative details, is that feedback can be used to increase or to decrease the input and output impedances.
Application to the example amplifier
These resistance results now are applied to the amplifier of Figure 3 and Figure 5. The improvement factor that reduces the gain, namely ( 1 + β_{FB} A_{OL}), directly decides the effect of feedback upon the input and output resistances of the amplifier. In the case of a shunt connection, the input impedance is reduced by this factor; and in the case of series connection, the impedance is multiplied by this factor. However, the impedance that is modified by feedback is the impedance of the amplifier in Figure 5 with the feedback turned off, and does include the modifications to impedance caused by the resistors of the feedback network.
Therefore, the input impedance seen by the source with feedback turned off is R_{in} = R_{1} = R_{11} // R_{B} // r_{π1}, and with the feedback turned on (but no feedforward)
where division is used because the input connection is shunt: the feedback twoport is in parallel with the signal source at the input side of the amplifier. A reminder: A_{OL} is the loaded open loop gain found above, as modified by the resistors of the feedback network.
The impedance seen by the load needs further discussion. The load in Figure 5 is connected to the collector of the output transistor, and therefore is separated from the body of the amplifier by the infinite impedance of the output current source. Therefore, feedback has no effect on the output impedance, which remains simply R_{C2} as seen by the load resistor R_{L} in Figure 3.^{[15]}^{[16]}
If instead we wanted to find the impedance presented at the emitter of the output transistor (instead of its collector), which is series connected to the feedback network, feedback would increase this resistance by the improvement factor ( 1 + β_{FB} A_{OL}).^{[17]}
Load voltage and load current
The gain derived above is the current gain at the collector of the output transistor. To relate this gain to the gain when voltage is the output of the amplifier, notice that the output voltage at the load R_{L} is related to the collector current by Ohm's law as v_{L} = i_{C} (R_{C2} // R_{L}). Consequently, the transresistance gain v_{L} / i_{S} is found by multiplying the current gain by R_{C2} // R_{L}:
Similarly, if the output of the amplifier is taken to be the current in the load resistor R_{L}, current division determines the load current, and the gain is then:
Is the main amplifier block a two port?
Some complications follow, intended for the attentive reader.
Figure 7 shows the smallsignal schematic with the main amplifier and the feedback twoport in shaded boxes. The twoport satisfies the port conditions: at the input port, I_{in} enters and leaves the port, and likewise at the output, I_{out} enters and leaves. The main amplifier is shown in the upper shaded box. The ground connections are labeled.
Figure 7 shows the interesting fact that the main amplifier does not satisfy the port conditions at its input and output unless the ground connections are chosen to make that happen. For example, on the input side, the current entering the main amplifier is I_{S}. This current is divided three ways: to the feedback network, to the bias resistor R_{B} and to the base resistance of the input transistor r_{π}. To satisfy the port condition for the main amplifier, all three components must be returned to the input side of the main amplifier, which means all the ground leads labeled G_{1} must be connected, as well as emitter lead G_{E1}. Likewise, on the output side, all ground connections G_{2} must be connected and also ground connection G_{E2}. Then, at the bottom of the schematic, underneath the feedback twoport and outside the amplifier blocks, G_{1} is connected to G_{2}. That forces the ground currents to divide between the input and output sides as planned. Notice that this connection arrangement splits the emitter of the input transistor into a baseside and a collectorside – a physically impossible thing to do, but electrically the circuit sees all the ground connections as one node, so this fiction is permitted.
Of course, the way the ground leads are connected makes no difference to the amplifier (they are all one node), but it makes a difference to the port conditions. That is a weakness of this approach: the port conditions are needed to justify the method, but the circuit really is unaffected by how currents are traded among ground connections.
However, if there is no possible arrangement of ground conditions that will lead to the port conditions, the circuit might not behave the same way.^{[18]} The improvement factors ( 1 + β_{FB} A_{OL}) for determining input and output impedance might not work. This situation is awkward, because a failure to make a twoport may reflect a real problem (it just is not possible), or reflect a lack of imagination (for example, just did not think of splitting the emitter node in two). As a consequence, when the port conditions are in doubt, at least two approaches are possible to establish whether improvement factors are accurate: either simulate an example using Spice and compare results with use of an improvement factor, or calculate the impedance using a test source and compare results.
A more radical choice is to drop the twoport approach altogether, and use return ratios. That choice might be advisable if smallsignal device models are complex, or are not available (for example, the devices are known only numerically, perhaps from measurement or from SPICE simulations).
See also
 Asymptotic gain model
 Bode plot
 Buffer amplifier considers the basic opamp amplifying stage with negative feedback
 Common collector (emitter follower) is dedicated to the basic transistor amplifying stage with negative feedback
 Frequency compensation
 Miller theorem is a powerful tool for determining the input/output impedances of negative feedback circuits
 Operational amplifier presents the basic opamp noninverting amplifier and inverting amplifier
 Operational amplifier applications shows the most typical opamp circuits with negative feedback
 Phase margin
 Pole splitting
 Return ratio
References and notes
 ^ Kuo, Benjamin C & Farid Golnaraghi (2003). Automatic control systems (Eighth edition ed.). NY: Wiley. p. 46. ISBN 0471134767. http://worldcat.org/isbn/0471134767.
 ^ Palumbo, Gaetano & Salvatore Pennisi (2002). Feedback amplifiers: theory and design. Boston/Dordrecht/London: Kluwer Academic. p. 64. ISBN 0792376439. http://worldcat.org/isbn/0792376439.
 ^ ".". http://eepatents.com/patents/2102671.pdf. Retrieved 20051024.^{[dead link]}
 ^ Currently on display at Bell Laboratories in Mountainside, New Jersey
 ^ Waldhauer, Fred (1982). Feedback. NY: Wiley. p. 3. ISBN 0471053198. http://worldcat.org/isbn/0471053198.
 ^ Black, Harold (Dec. 1977). "Inventing the negative feedback amplifier". IEEE Spectrum.
 ^ RW Brodersen Analog circuit design: lectures on stability
 ^ Willy M. C. Sansen (2006). Analog design essentials. New York; Berlin: Springer. pp. §0513§0533, p. 155–165. ISBN 0387257462. http://worldcat.org/isbn/0387257462.
 ^ Ashok K. Goel Feedback topologies
 ^ Zimmer T & Geoffreoy D: Feedback amplifier
 ^ Vivek Subramanian: Lectures on feedback
 ^ P R Gray, P J Hurst, S H Lewis, and R G Meyer (2001). Analysis and Design of Analog Integrated Circuits (Fourth Edition ed.). New York: Wiley. pp. 586–587. ISBN 0471321680. http://worldcat.org/isbn/0471321680.
 ^ A. S. Sedra and K.C. Smith (2004). Microelectronic Circuits (Fifth Edition ed.). New York: Oxford. Example 8.4, pp. 825–829 and PSpice simulation pp. 855–859. ISBN 0195142519. http://worldcat.org/isbn/0195142519.
 ^ If the feedforward is included, its effect is to cause a modification of the openloop gain, normally so small compared to the openloop gain itself that it can be dropped. Notice also that the main amplifier block is unilateral.
 ^ The use of the improvement factor ( 1 + β_{FB} A_{OL}) requires care, particularly for the case of output impedance using series feedback. See Jaeger, note below.
 ^ R.C. Jaeger and T.N. Blalock (2006). Microelectronic Circuit Design (Third Edition ed.). McGrawHill Professional. Example 17.3 pp. 1092–1096. ISBN 9780073191638. http://worldcat.org/isbn/9780073191638.
 ^ That is, the impedance found by turning off the signal source I_{S} = 0, inserting a test current in the emitter lead I_{x}, finding the voltage across the test source V_{x}, and finding R_{out} = V_{x} / I_{x}.
 ^ The equivalence of the main amplifier block to a twoport network guarantees that performance factors work, but without that equivalence they may work anyway. For example, in some cases the circuit can be shown to be equivalent to another circuit that is a two port, by "cooking up" different circuit parameters that are functions of the original ones. There is no end to creativity!
Categories: Electronic feedback
 Electronic amplifiers
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