Feedback amplifier - AbsoluteAstronomy.com

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

Negative feedback

Negative feedback occurs when the output of a system acts to oppose changes to the input of the system, with the result that the changes are attenuated. If the overall feedback of the system is negative, then the system will tend to be stable.- Overview :...

opposes the original signal. The applied negative feedback improves performance (gain stability, linearity, frequency response, step response

Step response

The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...

) 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.

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

Common drain

In electronics, a common-drain amplifier, also known as a source follower, is one of three basic single-stage field effect transistor amplifier topologies, typically used as a voltage buffer. In this circuit the gate terminal of the transistor serves as the input, the source is the output, and the...

, op-amp) follower the feedback network and the summing circuit are not necessary.

Overview

Fundamentally, all electronic devices used to provide power gain (e.g. vacuum tube

Vacuum tube

In electronics, a vacuum tube, electron tube , or thermionic valve , reduced to simply "tube" or "valve" in everyday parlance, is a device that relies on the flow of electric current through a vacuum...

s, bipolar transistors, MOS transistors

Fet

Fet is a municipality in Akershus county, Norway. It is part of the Romerike traditional region. The administrative centre of the municipality is the village of Fetsund.Fet was established as a municipality on 1 January 1838...

) are nonlinear. Negative feedback allows gain

Gain

In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...

to be traded for higher linearity (reducing distortion

Distortion

A distortion is the alteration of the original shape of an object, image, sound, waveform or other form of information or representation. Distortion is usually unwanted, and often many methods are employed to minimize it in practice...

), amongst other things. If not designed correctly amplifiers with negative feedback can become unstable, resulting in unwanted behavior, such as oscillation

Oscillation

Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

. The Nyquist stability criterion

Nyquist stability criterion

When designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that overcompensates -- if the car drifts in one direction, the control system...

developed by Harry Nyquist

Harry Nyquist

Harry Nyquist was an important contributor to information theory.-Personal life:...

of Bell Laboratories can be used to study the stability of feedback amplifiers.

Feedback amplifiers share these properties:

Pros:

Can increase or decrease input impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

(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
Step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...

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 closed-loop amplifier) become sensitive to the gain of the amplifier without feedback (the open-loop amplifier); that exposes these impedances to variations in the open loop gain, for example, due to parameter variations or due to nonlinearity of the open-loop gain

History

The negative feedback amplifier was invented by Harold Stephen Black

Harold Stephen Black

Harold Stephen Black was an American electrical engineer, who revolutionized the field of applied electronics by inventing the negative feedback amplifier in 1927. To some, his invention is considered the most important breakthrough of the twentieth century in the field of electronics, since it...

(US patent 2,102,671 (issued in 1937) ) 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

Distortion

in repeater

Repeater

A repeater is an electronic device that receives asignal and retransmits it at a higher level and/or higher power, or onto the other side of an obstruction, so that the signal can cover longer distances.-Description:...

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.
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."

Gain reduction

Below, the voltage gain of the amplifier with feedback, the closed-loop gain A_fb, is derived in terms of the gain of the amplifier without feedback, the open-loop 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 open-loop 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

Operational amplifier

An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...

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 capacitor

Capacitor

A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s or inductor

Inductor

An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

s 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 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 closed-loop gain, A_fb is given by,

If A_OL >> 1, then A_fb ≈ 1 / β and the effective amplification (or closed-loop 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

Nyquist plot

A Nyquist plot is a parametric plot of a transfer function used in automatic control and signal processing. The most common use of Nyquist plots is for assessing the stability of a system with feedback. In Cartesian coordinates, the real part of the transfer function is plotted on the X axis. The...

(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. 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 f_C is the cutoff

Cutoff frequency

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.Typically in electronic systems such as filters and...

or corner frequency of the amplifier: in this example f_C = 10⁴ Hz and the gain at zero frequency A₀ = 10⁵ V/V. The figure shows 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 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
Gain-bandwidth product
The gain–bandwidth product for an amplifier is the product of the amplifier's bandwidth, and the gain at which the bandwidth is measured....

. In Figure 2, ( 1 + β A₀ ) = 10³, so A_fb(0)= 10⁵ / 10³ = 100 V/V, and f_C increases to 10⁴ × 10³ = 10⁷ Hz.

Multiple poles

When the open-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

Ringing artifacts

In signal processing, particularly digital image processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near transients, particularly sounds from...

and overshoot

Overshoot (signal)

In signal processing, control theory, electronics, and mathematics, overshoot is when a signal or function exceeds its target. It arises especially in the step response of bandlimited systems such as low-pass filters...

in its step response

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.

Asymptotic gain model

In 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 open-loop 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 open-loop 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. See the table below.

Feedback amplifier type	Input connection	Output connection	Ideal feedback	Two-port feedback
Current	Shunt	Series	CCCS	g-parameter
Transresistance	Shunt	Shunt	CCVS	y-parameter
Transconductance	Series	Series	VCCS	z-parameter
Voltage	Series	Shunt	VCVS	h-parameter

The feedback can be implemented using a two-port network

Two-port network

A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...

. There are four types of two-port 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) current-controlled current source (CCCS), and its imperfect realization using a two-port network also must incorporate a CCCS, that is, the appropriate choice for feedback network is a g-parameter two-port.

Two-port analysis of feedback

One approach to feedback is the use of return ratio

Return ratio

The return ratio of a dependent source in a linear electrical circuit is the negative of the ratio of the current returned to the site of the dependent source to the current of a replacement independent source...

. Here an alternative method used in most textbooks is presented by means of an example treated in the article on asymptotic gain model.

Figure 3 shows a two-transistor 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 two-port

The first step is replacement of the feedback network by a two-port

Two-port network

A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...

. Just what components go into the two-port?

On the input side of the two-port 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 two-port is a dependent current source controlled by the voltage at the top of resistor R₂.

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 R₂. 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 R₂ is set by the emitter current of the output transistor. That view leads to an entirely passive feedback network made up of R₂ and R_f. The variable controlling the feedback is the emitter current, so the feedback is a current-controlled current source (CCCS). We search through the four available two-port network

Two-port network

A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...

s and find the only one with a CCCS is the g-parameter two-port, shown in Figure 4. The next task is to select the g-parameters so that the two-port of Figure 4 is electrically equivalent to the L-section made up of R₂ and R_f. That selection is an algebraic procedure made most simply by looking at two individual cases: the case with V₁ = 0, which makes the VCVS on the right side of the two-port a short-circuit; and the case with I₂ = 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 g-parameters that make the two-port and the L-section behave the same way are shown in the table below.

g₁₁	g₁₂	g₂₁	g₂₂

Small-signal circuit

The next step is to draw the small-signal schematic for the amplifier with the two-port in place using the hybrid-pi model

Hybrid-pi model

The hybrid-pi model is a popular circuit model used for analyzing the small signal behavior of bipolar junction and field effect transistors. The model can be quite accurate for low-frequency circuits and can easily be adapted for higher frequency circuits with the addition of appropriate...

for the transistors. Figure 5 shows the schematic with notation R₃ = R_C2 // R_L and R₁₁ = 1 / g₁₁, R₂₂ = g₂₂ .

Loaded open-loop gain

Figure 3 indicates the output node, but not the choice of output variable. A useful choice is the short-circuit current output of the amplifier (leading to the short-circuit current gain). Because this variable leads simply to any of the other choices (for example, load voltage or load current), the short-circuit current gain is found below.

First the loaded open-loop gain is found. The feedback is turned off by setting g₁₂ = g₂₁ = 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 R₁₁, R_B, and r_π1 all are in parallel and v₁ = v_π. Let R₁ = R₁₁ // R_B // r_π1. In addition, i₂ = −(β+1) i_B. The result for the open-loop current gain A_OL is:

Gain with feedback

In the classical approach to feedback, the feedforward represented by the VCVS (that is, g₂₁ v₁) is neglected. 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 = −g₁₂. 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 two-port 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

Kirchhoff's circuit laws

Kirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff...

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

Norton's theorem

Norton's theorem for linear electrical networks, known in Europe as the Mayer–Norton theorem, states that any collection of voltage sources, current sources, and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R...

or Thévenin

Thévenin's theorem

In circuit theory, Thévenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem...

drives, arbitrary loads, and general two-port feedback networks

Two-port network

A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...

. However, the results do depend upon the main amplifier having a representation as a two-port – 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 = R₁ = R₁₁ // 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 two-port 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.

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).

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

Ohm's law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

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 small-signal schematic with the main amplifier and the feedback two-port in shaded boxes. The two-port satisfies the port conditions

Two-port network

A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...

: 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 G₁ must be connected, as well as emitter lead G_E1. Likewise, on the output side, all ground connections G₂ must be connected and also ground connection G_E2. Then, at the bottom of the schematic, underneath the feedback two-port and outside the amplifier blocks, G₁ is connected to G₂. 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 base-side and a collector-side – 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. 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 two-port 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

SPICE

SPICE is a general-purpose, open source analog electronic circuit simulator.It is a powerful program that is used in integrated circuit and board-level design to check the integrity of circuit designs and to predict circuit behavior.- Introduction :Unlike board-level designs composed of discrete...

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 two-port approach altogether, and use return ratio

Return ratio

s. That choice might be advisable if small-signal device models are complex, or are not available (for example, the devices are known only numerically, perhaps from measurement or from SPICE

SPICE

simulations).

Overview

History

Gain reduction

Bandwidth extension

Multiple poles

Asymptotic gain model

Feedback and amplifier type

Two-port analysis of feedback

Replacement of the feedback network with a two-port

Small-signal circuit

Loaded open-loop gain

Gain with feedback

Input and output resistances

Background on resistance determination

Application to the example amplifier

Load voltage and load current

Is the main amplifier block a two port?

See also