Operational amplifier applications: Difference between revisions

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This article illustrates some typical '''applications''' of '''[[operational amplifier]]s'''. A simplified schematic notation is used, and the reader is reminded that many details such as device selection and power supply connections are not shown.

{{main article|Operational amplifier}}

This article illustrates some typical '''operational amplifier applications'''. Operational amplifiers are optimised for use with [[negative feedback]], and this article discusses only negative-feedback applications. When positive feedback is required, a [[comparator]] is usually more appropriate. See [[Comparator applications]] for further information.

== Practical considerations ==

__TOC__

[[File:Op-Amp_Internal.svg|right|thumb|250x250px|Fig.&nbsp;1: an equivalent circuit of an operational amplifier that models some non-ideal parameters using resistances. A real operational amplifier has a finite input impedance <math>R_\text{in}</math>, a non-zero output impedance <math>R_\text{out}</math>, and a finite gain <math>G</math>.]]

In this article, a simplified schematic notation is used that ignores details such as device selection and power supply connections. Non-ideal properties (such as those shown in Fig.&nbsp;1) are ignored.

=== Operational amplifiers parameter requirements ===

== Practical circuits==

In order for a particular device to be used in an application, it must satisfy certain requirements. The operational amplifier must

* have large open-loop signal gain (voltage gain of 200,000 is obtained in early integrated circuit exemplars), and

It is important to note that the equations shown below, pertaining to each type of circuit, assume that an [[Operational amplifier#Ideal op-amp|ideal op amp]] is used. Those interested in construction of any of these circuits for practical use should consult a more detailed reference. See the [[#External links|External links]] and [[#Further reading|Further reading]] sections.

* have input impedance large with respect to values present in the feedback network.

With these requirements satisfied, one can use the method of [[virtual ground]] to quickly and intuitively grasp the behavior of the op-amp circuits.

=== Component specification ===

Resistors used in practical solid-state op-amp circuits are typically in the kΩ range. Resistors much greater than 1 MΩ cause excessive [[thermal noise]] and make the circuit operation susceptible to significant errors due to bias or leakage currents.

=== Input bias currents and input offset ===

Practical operational amplifiers draw a small current from each of their inputs due to bias requirements and leakage. These currents flow through the resistances connected to the inputs and produce small voltage drops across those resistances. In AC signal applications this seldom matters. If high-precision DC operation is required, however, these voltage drops need to be considered. The design technique is to try to ensure that these voltage drops are equal for both inputs, and therefore cancel. If these voltage drops are equal and the [[common-mode rejection ratio]] of the operational amplifier is good, there will be considerable cancellation and improvement in DC accuracy.

Practical operational amplifiers draw a small current from each of their inputs due to bias requirements (in the case of bipolar junction transistor-based inputs) or leakage (in the case of MOSFET-based inputs).

These currents flow through the resistances connected to the inputs and produce small voltage drops across those resistances. Appropriate design of the feedback network can alleviate problems associated with input bias currents and common-mode gain, as explained below. The heuristic rule is to ensure that the impedance "looking out" of each input terminal is identical.

IfTo the input currents into the operational amplifier are equal, to reduce offset voltage the designer must ensureextent that the DC resistance looking out of each input isbias alsocurrents matched.do Innot general input currents differmatch, thethere differencewill beingbe calledan theeffective ''[[input offset current''voltage]] present, I_os.which Matchedcan externallead inputto resistancesproblems R_in willcircuit stillperformance. produceMany an input voltage error of&nbsp; R_in&middot;I_os&nbsp;.&ensp;commercial Mostop-amp manufacturersofferings provide a method for tuning the operational amplifier to balance the input currentsinputs (e.g., "offset null" or "balance" pins that can interact with an external voltage source attached to a potentiometer). OtherwiseAlternatively, a tunable external voltage can be added to one of the inputs in order to balance out the offset effect. In cases where a design calls for one input to be short-circuited to ground, that short circuit can be replaced with a variable resistance that can be tuned to mitigate the offset problem.

Note that many operationalOperational amplifiers that haveusing [[MOSFET]]-based input stages have input leakage currents that will truly be, negligiblein to mostmany designs, negligible.

===Power supply effects===

Although power supplies are not indicated in the (simplified) operational amplifier designs below, they are nonetheless present and can be critical in operational amplifier circuit design.

==== Supply noise ====

Although the power supplies are not shown in the operational amplifier designs below, they can be critical in operational amplifier design.

Power supply imperfections (e.g., power signal ripple, non-zero source impedance) may lead to noticeable deviations from ideal operational amplifier behavior. For example, operational amplifiers have a specified [[power supply rejection ratio]] that indicates how well the output can reject signals that appear on the power supply inputs. Power supply inputs are often noisy in large designs because the power supply is used by nearly every component in the design, and inductance effects prevent current from being instantaneously delivered to every component at once. As a consequence, when a component requires large injections of current (e.g., a digital component that is frequently switching from one state to another), nearby components can experience sagging at their connection to the power supply. This problem can be mitigated with appropriate use of [[decoupling capacitor|bypass capacitor]]s connected across each power supply pin and ground. When bursts of current are required by a component, the component can ''bypass'' the power supply by receiving the current directly from the nearby capacitor (which is then slowly recharged by the power supply).

==== Using power supply currents in the signal path ====

Power supply imperfections (e.g., power signal ripple, non-zero source impedance) may lead to noticeable deviations from ideal operational amplifier behavior. For example, operational amplifiers have a specified [[power supply rejection ratio]] that indicates how well the output can reject signals that appear on the power supply inputs. Power supply inputs are often noisy in large designs because the power supply is used by nearly every component in the design, and inductance effects prevent current from being instantaneously delivered to every component at once. As a consequence, when a component requires large injections of current (e.g., a digital component that is frequently switching from one state to another), nearby components can experience sagging at their connection to the power supply. This problem can be mitigated with copious use of [[decoupling capacitor|bypass capacitor]]s placed connected across each power supply pin and ground. When bursts of current are required by a component, the component can ''bypass'' the power supply by receiving the current directly from the nearby capacitor (which is then slowly charged by the power supply).

Additionally, current drawn into the operational amplifier from the power supply can be used as inputs to external circuitry that augment the capabilities of the operational amplifier. For example, an operational amplifier may not be fit for a particular high-gain application because its output would be required to generate signals outside of the safe range generated by the amplifier. In this case, an external [[push&ndash;pull output|push&ndash;pull amplifier]] can be controlled by the current into and out of the operational amplifier. Thus, the operational amplifier may itself operate within its factory specified bounds while still allowing the negative feedback path to include a large output signal well outside of those bounds.<ref>Paul Horowitz and Winfield Hill, ''[[The Art of Electronics]]''. 2nd ed. Cambridge University Press, Cambridge, 1989 {{ISBN|0-521-37095-7}}</ref>

==Amplifiers==

Additionally, current drawn into the operational amplifier from the power supply can be used as inputs to external circuitry that augment the capabilities of the operational amplifier. For example, an operational amplifier may not be fit for a particular high-gain application because its output would be required to generate signals outside of the safe range generated by the amplifier. In this case, an external [[push&ndash;pull output|push&ndash;pull amplifier]] can be controlled by the current into and out of the operational amplifier. Thus, the operational amplifier may itself operate within its factory specified bounds while still allowing the negative feedback path to include a large output signal well outside of those bounds.<ref>Paul Horowitz and Winfield Hill, ''[[The Art of Electronics]]''. 2^nd ed. Cambridge University Press, Cambridge, 1989 ISBN 0-521-37095-7</ref>

The first example is the differential amplifier, from which many of the other applications can be derived, including the [[#Inverting amplifier|inverting]], [[#Non-inverting amplifier configuration|non-inverting]], and [[summing amplifier]], the [[#Voltage follower|voltage follower]], [[#Inverting integrator|integrator]], [[#Inverting differentiator|differentiator]], and [[#Inductance gyrator|gyrator]].

=== Differential amplifier (difference amplifier) ===

== Linear circuit applications ==

{{main article|Differential amplifier}}

=== Comparator ===

[[File:Op-Amp Differential Amplifier.svg|300px||class=skin-invert]]

Amplifies the difference in voltage between its inputs.

[[Image:Op-Amp Comparator.svg|frame|Comparator]]

:The name "differential amplifier" must not be confused with the "[[#Inverting differentiator|differentiator]]", which is also shown on this page.

:The "[[#Instrumentation amplifier|instrumentation amplifier]]", which is also shown on this page, is a modification of the differential amplifier that also provides high [[input impedance]].

{{main|Comparator}}

The circuit shown computes the [[subtraction|difference]] of two voltages, multiplied by some gain factor. The output voltage

Compares two voltages and switches its output to indicate which voltage is larger.

*:<math> V_{\text{out}} = \frac{\left( R_\text{f} + R_1 \beginright) R_\text{matrixg}} V_{\left( R_\text{Sg} +} R_2 \right) R_1} &V_2 - \frac{R_\text{f}}{R_1} V_1 >= V_2\left( \frac{R_1 + R_\text{f}}{R_1} V_\right) \cdot \left( \frac{R_\text{S-g}}{R_\text{g} &+ V_1R_2} <\right) V_2 - \endfrac{matrix}R_\righttext{f}}{R_1} V_1. </math>

Or, expressed as a function of the common-mode input ''V''_com and difference input ''V''_dif:

:<math>V_\text{com} = (V_1 + V_2) / 2; V_\text{dif} = V_2 - V_1,</math>

the output voltage is

:<math>V_\text{out} \frac{R_1}{R_\text{f}} = V_\text{com} \frac{R_1 / R_\text{f} - R_2 / R_\text{g}}{1 + R_2 / R_\text{g}} + V_\text{dif} \frac{1 + (R_2 / R_\text{g} + R_1 / R_\text{f}) / 2}{1 + R_2 / R_\text{g}}.</math>

In order for this circuit to produce a signal proportional to the voltage difference of the input terminals, the coefficient of the ''V''_com term (the common-mode gain) must be zero, or

:<math>R_1 / R_\text{f} = R_2 / R_\text{g}.</math>

With this constraint<ref group="nb">If you think of the left-hand side of the relation as the closed-loop gain of the inverting input, and the right-hand side as the gain of the non-inverting input, then matching these two quantities provides an output insensitive to the common-mode voltage of <math>V_1</math> and <math>V_2</math>.</ref> in place, the [[common-mode rejection ratio]] of this circuit is infinitely large, and the output

:<math>V_\text{out} = \frac{R_\text{f}}{R_1} V_\text{dif} = \frac{R_\text{f}}{R_1} \left(V_2 - V_1\right),</math>

where the simple expression ''R''_''f'' / ''R''₁ represents the closed-loop gain of the differential amplifier.

The special case when the closed-loop gain is unity is a differential follower, with

(where <math>V_{\text{s}}</math> is the supply voltage and the opamp is powered by <math>+V_{\text{s}}</math> and <math>-V_{\text{s}}</math>.)

:<math>V_\text{out} = V_2 - V_1.</math>

{{clear}}

=== Inverting amplifier ===

[[File:Op-Amp Inverting Amplifier.svg|300px|class=skin-invert]]

An inverting amplifier is a special case of the [[differential amplifier]] in which that circuit's non-inverting input ''V''₂ is grounded, and inverting input ''V''₁ is identified with ''V''_in above. The closed-loop gain is ''R''_f / ''R''_in, hence

[[Image:Op-Amp Inverting Amplifier.svg|300px|thumb|right|Inverting amplifier]]

:<math>V_{\text{out}} = -\frac{R_{\text{f}}}{R_{\text{in}}} V_{\text{in}}\!\,</math>.

The simplified circuit above is like the differential amplifier in the limit of ''R''₂ and ''R''_g very small. In this case, though, the circuit will be susceptible to input bias current drift because of the mismatch between ''R''_f and ''R''_in.

To intuitively see the gain equation above, calculate the current in ''R''_in:

An inverting amplifier uses negative feedback to invert and [[amplifier|amplify]] a voltage. The R_f resistor allows some of the output signal to be returned to the input. Since the output is 180° out of phase, this amount is effectively subtracted from the input, thereby reducing the input into the operational amplifier. This reduces the overall gain of the amplifier and is dubbed negative feedback.<ref>Basic Electronics Theory, Delton T. Horn, 4th ed. McGraw-Hill Professional, 1994, p.342-343.</ref>

:<math> i_{\text{in}} = \frac{ V_{\text{in}} }{ R_{\text{in}} } </math>

then recall that this same current must be passing through ''R''_f, therefore (because ''V''₋ = ''V''₊ = 0):

:<math> V_{\text{out}} = -i_{\text{in}} R_{\text{f}} = - V_{\text{in}} \frac{ R_{\text{f}} }{ R_{\text{in}} }</math>

A mechanical analogy is a seesaw, with the ''V''₋ node (between ''R''_in and ''R''_f) as the fulcrum, at ground potential. ''V''_in is at a length ''R''_in from the fulcrum; ''V''_out is at a length ''R''_f. When ''V''_in descends "below ground", the output ''V''_out rises proportionately to balance the seesaw, and ''vice versa''.<ref>Basic Electronics Theory, Delton T. Horn, 4th ed. McGraw-Hill Professional, 1994, p.&nbsp;342&ndash;343.</ref>

:<math> V_{\text{out}} = -\frac{R_{\text{f}}}{R_{\text{in}}} V_{\text{in}}\!\ </math>

* <math>Z_{\text{in}} = R_{\text{in}}</math> (because <math>V_-</math> is a [[virtual ground]])

* A third resistor, of value <math>R_{\text{f}} \| R_{\text{in}} \triangleq R_{\text{f}} R_{\text{in}} / (R_{\text{f}} + R_{\text{in}}) </math>, added between the non-inverting input and ground, while not necessary, minimizes errors due to input bias currents.

<ref>Malmstadt, Enke and Crouch, Electronics and Instrumentation for Scientists, The Benjamin/Cummings Publishing Company, Inc., 1981, ISBN 0-8053-6917-1, Chapter 5. pp 118.</ref>

The gain of the amplifier is determined by the ratio of R_f to R_in. That is:

<math>A = -\frac{R_f}{R_{in}}</math>

The presence of the negative sign is a convention indicating that the output is inverted. For example, if R_f is 10,000 Ω and R_in is 1,000 Ω, then the gain would be -10000Ω/1000Ω, which is -10.

<ref>Basic Electronics Theory, Delton T. Horn, 4th ed. McGraw-Hill Professional, 1994, p.342-343.</ref>

As the negative input of the op-amp acts as a virtual ground, the input impedance of this circuit is equal to ''R''_in.

{{clear}}

===Non-inverting amplifier{{anchor|Non-inverting amplifier configuration}}===

[[File:Op-Amp Non-Inverting Amplifier.svg|300px|class=skin-invert]]

A non-inverting amplifier is a special case of the [[differential amplifier]] in which that circuit's inverting input ''V''₁ is grounded, and non-inverting input ''V''₂ is identified with ''V''_in above, with ''R''₁ ≫ ''R''₂.

[[Image:Op-Amp Non-Inverting Amplifier.svg|thumb|300px|Non-inverting amplifier]]

Referring to the circuit immediately above,

:<math>V_{\text{out}} = \left(1 + \frac{ R_{\text{2}} }{ R_{\text{1}} } \right) V_{\text{in}}\!\,</math>.

To intuitively see this gain equation, use the virtual ground technique to calculate the current in resistor ''R''₁:

:<math> i_1 = \frac{ V_{\text{in}} }{ R_1 }\,, </math>

then recall that this same current must be passing through ''R''₂, therefore:

:<math> V_{\text{out}} = V_{\text{in}} + i_1 R_2 = V_{\text{in}} \left( 1 + \frac{ R_2 }{ R_1 } \right)</math>

Unlike the inverting amplifier, a non-inverting amplifier cannot have a gain of less than 1.

Amplifies a voltage (multiplies by a constant greater than 1)

A mechanical analogy is a [[Lever#Classes of levers|class-2 lever]], with one terminal of ''R''₁ as the fulcrum, at ground potential. ''V''_in is at a length ''R''₁ from the fulcrum; ''V''_out is at a length ''R''₂ further along. When ''V''_in ascends "above ground", the output ''V''_out rises proportionately with the lever.

:<math>V_{\text{out}} = V_{\text{in}} \left( 1 + \frac{R_2}{R_1} \right)\,</math>

* Input impedance <math>Z_{\text{in}} \approx \infin</math>

** The input impedance is ''at least'' the impedance between non-inverting (<math>+</math>) and inverting (<math>-</math>) inputs, which is typically 1 MΩ to 10 TΩ, plus the impedance of the path from the inverting (<math>-</math>) input to ground (i.e., <math>R_1</math> in parallel with <math>R_2</math>).

** Because negative feedback ensures that the non-inverting and inverting inputs match, the input impedance is actually '''much higher.'''

* Although this circuit has a large input impedance, it suffers from error of input bias current.

** The non-inverting (<math>+</math>) and inverting (<math>-</math>) inputs draw small leakage currents into the operational amplifier.

** These input currents generate voltages that act like unmodeled input offsets. These unmodeled effects can lead to noise on the output (e.g., offsets or drift).

** Assuming that the two leaking currents are '''matched,''' their effect can be mitigated by ensuring the DC impedance looking '''out''' of each input is the same.

*** The voltage produced by each bias current is equal to the product of the bias current with the equivalent DC impedance looking out of each input. Making those impedances equal makes the offset voltage at each input equal, and so the non-zero bias currents will have no impact on the '''difference''' between the two inputs.

*** A resistor of value

***:<math>R_1 \| R_2 \triangleq \left(\frac{1}{R_1} + \frac{1}{R_2}\right)^{-1} = \frac{ R_1 R_2 }{ R_1 + R_2 },\,</math>

**:which is the equivalent resistance of <math>R_1</math> in parallel with <math>R_2</math>, between the <math>V_{\text{in}}</math> source and the non-inverting (<math>+</math>) input will ensure the impedances looking '''out''' of each input will be matched.

*** The matched bias currents will then generate matched offset voltages, and their effect will be hidden to the operational amplifier (which acts on the difference between its inputs) so long as the [[common-mode rejection ratio|CMRR]] is good.

** Very often, the input currents are ''not'' matched.

*** Most operational amplifiers provide some method of balancing the two input currents (e.g., by way of an external [[potentiometer]]).

*** Alternatively, an external offset can be added to the operational amplifier input to nullify the effect.

*** Another solution is to insert a variable resistor between the <math>V_{\text{in}}</math> source and the non-inverting (<math>+</math>) input. The resistance can be tuned until the offset voltages at each input are matched.

*** Operational amplifiers with [[MOSFET]]-based input stages have input currents that are so small that they often can be neglected.

The input impedance of the simplified non-inverting amplifier is high:

{{clear}}

:<math> Z_{\text{in}} = (1+A_\text{OL}B)Z_{\text{dif}}</math>

=== Differential amplifier ===

where ''Z''_dif is the op-amp's input impedance to differential signals, and ''A''_OL is the open-loop voltage gain of the op-amp (which varies with frequency), and ''B'' is the [[Negative-feedback amplifier#Gain reduction|feedback factor]] (the fraction of the output signal that returns to the input).<ref name=":1">{{Cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Electronic/feedn.html#c3|title=Benefits of Negative Feedback|website=HyperPhysics|access-date=2018-05-07}}</ref><ref name=":0">{{Cite book|last=Simpson|first=Robert E.|title=Introductory electronics for scientists and engineers|date=1987|publisher=Allyn and Bacon|isbn=0205083773|edition=2nd|___location=Boston|pages=291|chapter=7.2 Negative Voltage Feedback|oclc=13821010|quote=input impedance of an amplifier without negative feedback is ''increased'' by adding negative feedback. .. <math>Z_\mathrm{inf}=(1+A_0 B)Z_\mathrm{ino}</math> .. output impedance .. is ''decreased'' .. <math>Z_\mathrm{outf}=Z_\mathrm{out} / (1+A_0 B)</math>}}</ref> In the case of the ideal op-amp, with ''A''_OL infinite and ''Z''_dif infinite, the input impedance is also infinite. In this case, though, the circuit will be susceptible to input bias current drift because of the mismatch between the impedances driving the ''V''₊ and ''V''₋ op-amp inputs.

[[Image:Op-Amp Differential Amplifier.svg|thumb|300px|right|Differential amplifier]]

The feedback loop similarly decreases the output impedance:

{{main|Differential amplifier}}

:<math> Z_{\text{out}} = \frac{Z_{\text{OL}}}{1+A_\text{OL}B}</math>

The circuit shown is used for finding the [[subtraction|difference]] of two voltages each multiplied by some constant (determined by the resistors).

where ''Z''_out is the output impedance with feedback, and ''Z''_OL is the open-loop output impedance.<ref name=":0" />

''The name "differential amplifier" should not be confused with the "[[#Differentiator|differentiator]]", also shown on this page.''

{{clear}}

=== Voltage follower (unity buffer amplifier){{anchor|Voltage follower}} ===

:<math> V_{\text{out}} = \frac{ \left( R_{\text{f}} + R_1 \right) R_{\text{g}} }{\left( R_{\text{g}} + R_2 \right) R_1} V_2 - \frac{R_{\text{f}}}{R_1} V_1 </math>

[[File:Op-Amp Unity-Gain Buffer.svg|200px|class=skin-invert]]

Used as a [[buffer amplifier]] to eliminate loading effects (e.g., connecting a device with a high [[source impedance]] to a device with a low [[input impedance]]).

* Differential <math>Z_{\text{in}}</math> (between the two input pins) = <math>R_1 + R_2</math> (Note: this is approximate)

:<math> V_{\text{out}} = V_{\text{in}} \! </math>

For common-mode rejection, anything done to one input must be done to the other. The addition of a compensation capacitor in parallel with Rf, for instance, must be balanced by an equivalent capacitor in parallel with Rg.

:<math>Z_{\text{in}} = \infty</math> (realistically, the differential input impedance of the op-amp itself (1 MΩ to 1 TΩ), multiplied by the open-loop gain of the op-amp)

Due to the strong (i.e., [[unity (mathematics)|unity]] gain) feedback and certain non-ideal characteristics of real operational amplifiers, this feedback system is prone to have poor [[phase margin|stability margin]]s. Consequently, the system may be [[BIBO stability|unstable]] when connected to sufficiently capacitive loads. In these cases, a [[frequency compensation|lag compensation]] network (e.g., connecting the load to the voltage follower through a resistor) can be used to restore stability. The manufacturer [[data sheet]] for the operational amplifier may provide guidance for the selection of components in external compensation networks. Alternatively, another operational amplifier can be chosen that has more appropriate internal compensation.

''The "[[#Instrumentation amplifier|instrumentation amplifier]]", which is also shown on this page, is another form of differential amplifier that also provides high [[input impedance]].''

The input and output impedance are affected by the feedback loop in the same way as the non-inverting amplifier, with ''B''=1.<ref name=":1" /><ref name=":0" />

==== Amplified difference ====

Whenever <math>R_1 = R_2\,</math> and <math>R_{\text{f}} = R_{\text{g}}\,</math>,

:<math>V_{\text{out}} = A (V_{\text{2}} - V_{\text{1}})\,</math> &nbsp; and &nbsp; <math>A \triangleq \frac{ R_{\text{f}} }{ R_1 }</math>

==== Difference amplifier ====

When <math>R_1 = R_\mathrm{f}\,</math> and <math>R_2 = R_\mathrm{g}\,</math>:

:<math> V_\mathrm{out} = V_2 - V_1 \,\!</math>

{{clear}}

=== Voltage follower ===

[[Image:Op-Amp Unity-Gain Buffer.svg|frame|Voltage follower]]

Used as a [[buffer amplifier]], to eliminate loading effects or to interface impedances (connecting a device with a high [[source impedance]] to a device with a low [[input impedance]]). Due to the strong feedback, this circuit

tends to get unstable when driving a high capacity load. This can be avoided by connecting the load through a

resistor.

:<math> V_{\text{out}} = V_{\text{in}} \!\ </math>

* <math>Z_{\text{in}} = \infin</math> (realistically, the differential input impedance of the op-amp itself, 1 MΩ to 1 TΩ)

{{clear}}

=== Summing amplifier ===

[[ImageFile:Op-Amp Summing Amplifier.svg|thumb200px|300px|Summing amplifierclass=skin-invert]]

A summing ampliferamplifier sumsproduces the negative of the sum of several (weighted) voltages:

:<math> V_{\text{out}} = -R_{\text{f}} \left( \frac{V_1}{R_1} + \frac{V_2}{R_2} + \cdots + \frac{V_n}{R_n} \right) </math>

Line 137 ⟶ 130:

* When <math>R_1 = R_2 = \cdots = R_n</math>, and <math>R_{\text{f}}</math> independent

:<math> V_{\text{out}} = -\frac{R_{\text{f}}}{R_1} ( V_1 + V_2 + \cdots + V_n ) \!\ </math>

* When <math>R_1 = R_2 = \cdots = R_n = R_{\text{f}}</math>

:<math> V_{\text{out}} = -( V_1 + V_2 + \cdots + V_n ) \!\ </math>

* Input impedance of the '''''n''th''' input is <math>Z_n = R_n</math> (<math>V_-</math> is a [[virtual ground]])

* Output is inverted

* Input impedance of the ''n''th input is <math>Z_n = R_n</math> (<math>V_-</math> is a [[virtual ground]])

=== Instrumentation amplifier ===

{{clear}}

{{main article|Instrumentation amplifier}}

[[File:Op-Amp Instrumentation Amplifier.svg|300px|class=skin-invert]]

Combines very high [[input impedance]], high [[common-mode rejection]], low [[DC offset]], and other properties used in making very accurate, low-noise measurements

=== Integrator ===

* Is made by adding a [[#Non-inverting amplifier|non-inverting]] [[buffer amplifier|buffer]] to each input of the [[#Differential_amplifier_.28difference_amplifier.29|differential amplifier]] to increase the input impedance.

[[Image:Op-Amp Integrating Amplifier.svg|thumb|300px|Integrating amplifier]]

[[Integral|Integrates]] the (inverted) signal over time

:<math>V_{\text{out}} = -\int_0^t \frac{ V_{\text{in}} }{RC} \, \operatorname{d}t + V_{\text{initial}}\,</math>

(where <math>V_{\text{in}}</math> and <math>V_{\text{out}}</math> are functions of time, <math>V_{\text{initial}}</math> is the output voltage of the integrator at time ''t'' = 0.)

* Note that this can also be viewed as a [[Low-pass filter|low-pass]] [[electronic filter]]. It is a filter with a single [[complex pole|pole]] at DC (i.e., where <math>\omega = 0</math>) and gain.

* There are several potential problems with this circuit.

** It is usually assumed that the input <math>V_{\text{in}}</math> has zero DC component (i.e., has a zero average value). Otherwise, unless the capacitor is periodically discharged, the output will drift outside of the operational amplifier's operating range.

** Even when <math>V_{\text{in}}</math> has no offset, the leakage or bias currents into the operational amplifier inputs can add an unexpected offset voltage to <math>V_{\text{in}}</math> that causes the output to drift. Balancing input currents '''and''' replacing the non-inverting (<math>+</math>) short-circuit to ground with a resistor with resistance <math>R</math> can reduce the severity of this problem.

** Because this circuit provides no DC feedback (i.e., the capacitor appears like an open circuit to signals with <math>\omega = 0</math>), the offset of the output may not agree with expectations (i.e., <math>V_{\text{initial}}</math> may be out of the designer's control with the present circuit).

: Many of these problems can be made less severe by adding a ''large'' resistor <math>R_F</math> in parallel with the feedback capacitor. At significantly high frequencies, this resistor will have negligible effect. However, at low frequencies where there are drift and offset problems, the resistor provides the necessary feedback to hold the output steady at the correct value. In effect, this resistor reduces the DC gain of the "integrator"&nbsp;&ndash; it goes from infinite to some finite value <math>R_F/R</math>.

{{clear}}

==Oscillators==

=== Differentiator ===

=== Wien bridge oscillator ===

[[Image:Op-Amp Differentiating Amplifier.svg|thumb|300px|Differentiating amplifier]]

{{main article|Wien bridge oscillator}}

[[File:Wien bridge classic osc.svg|300px|class=skin-invert]]

Produces a very low distortion [[sine wave]]. Uses negative temperature compensation in the form of a light bulb or diode.

[[Derivative|Differentiates]] the (inverted) signal over time.

:''The name "differentiator" should not be confused with the "[[#Differential amplifier|differential amplifier,]]" which is also shown on this page. The former takes a [[derivative]] and the latter takes a difference (i.e., does subtraction).'' This is a circuit that could be used in an [[analog computer]], but in practice these circuits are difficult to keep stable and noise-free, so often the problem can be rearranged to use an integrator instead.

:<math>V_{\text{out}} = -RC \,\frac{\operatorname{d}V_{\text{in}} }{ \operatorname{d}t} \, \qquad \text{where } V_{\text{in}}\text{ and } V_{\text{out}} \text{ are functions of time.}</math>

* Note that this can also be viewed as a [[High-pass filter|high-pass]] [[electronic filter]]. It is a filter with a single [[complex zero|zero]] at DC (i.e., where <math>\omega = 0</math>) and gain.

{{clear}}

==Filters==

=== Instrumentation amplifier ===

{{main article|Active filter}}

Operational amplifiers can be used in construction of [[active filter]]s, providing high-pass, low-pass, band-pass, reject and delay functions. The high input impedance and gain of an op-amp allow straightforward calculation of element values, allowing accurate implementation of any desired filter topology with little concern for the loading effects of stages in the filter or of subsequent stages. However, the frequencies at which active filters can be implemented is limited; when the behavior of the amplifiers departs significantly from the ideal behavior assumed in elementary design of the filters, filter performance is degraded.

==Comparator==

[[Image:Op-Amp Instrumentation Amplifier.svg|300px|thumb|Instrumentation amplifier]]

{{main article|Comparator|Comparator applications}} [[File:Op-Amp Comparator.svg|200px|class=skin-invert]]

An operational amplifier can, if necessary, be forced to act as a comparator. The smallest difference between the input voltages will be amplified enormously, causing the output to swing to nearly the supply voltage. However, it is usually better to use a dedicated comparator for this purpose, as its output has a higher slew rate and can reach either power supply rail. Some op-amps have clamping diodes on the input that prevent use as a comparator.<ref>{{cite web | url=http://e2e.ti.com/blogs_/archives/b/thesignal/archive/2012/03/14/op-amps-used-as-comparators-is-it-okay.aspx | title=Op Amps used as Comparators—is it okay? - the Signal - Archives - TI E2E support forums }}</ref>

==Integration and differentiation==

{{main|Instrumentation amplifier}}

=== Inverting integrator ===

{{main article|Op amp integrator}}

The integrator is mostly used in [[analog computer]]s, [[analog-to-digital converter]]s and wave-shaping circuits. A simple version is:

Combines very high [[input impedance]], high [[common-mode rejection]], low [[DC offset]], and other properties used in making very accurate, low-noise measurements

[[File:Op-Amp Integrating Amplifier.svg|300px|class=skin-invert]]

* Is made by adding a [[#non-inverting amplifier|non-inverting]] [[buffer amplifier|buffer]] to each input of the [[#Differential amplifier|differential amplifier]] to increase the input impedance.

Assuming ideal elements, it [[Integral|integrates]] the input signal (multiplied by <math>-\tfrac{1}{RC}</math>) over a time interval from ''t''₀ to ''t''₁, yielding an output voltage at time ''t'' = ''t''₁ of:

{{clear}}

:<math>V_\text{out}(t_1) = V_\text{out}(t_0) - \frac{1}{RC} \int_{t_0}^{t_1} V_\text{in}(t) \,dt,</math>

where ''V''_out(''t''₀) is the capacitor's initial voltage at time ''t'' = ''t''₀. In other words, the circuit's output voltage changes over the time interval by an amount proportional to the time integral of the input voltage:

:<math>-\frac{1}{RC} \int_{t_0}^{t_1} V_\text{in}(t) \,dt.</math>

This circuit can be viewed as an active [[Low-pass filter|low-pass]] [[electronic filter]] with a single [[complex pole|pole]] at DC (i.e., where <math>\omega = 0</math>).

=== Schmitt trigger ===

Its practicality is limited by a significant problem: unless the capacitor is periodically discharged, the output voltage will eventually drift outside of the operational amplifier's operating range. This can be due to any combination of:

{{main|Schmitt trigger}}

* a non-zero [[DC component]] in the input ''V''_in,

* a non-zero opamp input bias current,

* a non-zero opamp input offset voltage.<ref name="microchip-opa-dc">{{cite web

|title = AN1177 Op-Amp Precision Design: DC Errors

|publisher = Microchip

|date = 2 January 2008

|url = http://ww1.microchip.com/downloads/en/AppNotes/01177a.pdf

|archive-url = https://web.archive.org/web/20190709005741/http://ww1.microchip.com/downloads/en/AppNotes/01177a.pdf

|archive-date = 2019-07-09

|url-status = live

|access-date = 26 December 2012

}}</ref>

The following slightly more complex circuit can ameliorate the second two problems, and in some cases, the first as well, but has a limited bandwidth of integration:

A [[bistable multivibrator]] implemented as a [[comparator]] with [[hysteresis]].

[[File:Op-Amp Integrating Amplifier balanced.svg|100pxl|class=skin-invert]]

{{clear}}

Here, the feedback resistor R_f provides a discharge path for capacitor C_f. The series resistor R_n at the non-inverting input alleviates input bias current and common-mode problems, provided it is set to the parallel resistance of R_i [[Parallel (operator)#Circuit analysis|||]] R_f:

[[Image:Op-Amp Schmitt Trigger.svg|frame|Schmitt trigger with non-inverting hysteretic switching characteristic]]

:<math>R_\text{n} = R_\text{i} || R_\text{f} = \frac{1}{ \frac{1}{R_\text{i}} + \frac{1}{R_\text{f}} } \, .</math>

{{Section link|Op amp integrator|Practical circuit}} explains the output drift adds a small finite DC error voltage:

: <math>V_\text{error} = \left( \frac{R_\text{f}}{R_\text{i}} + 1 \right) \left( V_\text{OS} + I_\text{B-} \left( R_\text{f} \parallel R_\text{i} \right) \right) .</math>

In this configuration, the hysteresis curve is non-inverting (i.e., very negative inputs correspond to a negative output and very positive inputs correspond to a positive output), and the switching thresholds are <math>\pm \frac{R_1}{R_2}V_{\text{sat}}</math> where <math>V_{\text{sat}}</math> is the greatest output magnitude of the operational amplifier.

Because the circuit is a first-order low-pass filter with a flat response up to its cutoff frequency <math>\tfrac{1}{2 \pi C_f R_f}</math>, it only functions as an integrator for frequencies significantly higher than that cutoff.{{clear}}

{{clear}}

=== Inverting differentiator ===

[[File:Op-Amp Inverting Schmitt Trigger.svg|frame|Schmitt trigger with inverting hysteretic switching characteristic]]

{{main article|Differentiator#Active differentiator}}

Alternatively, the input and the [[electrical ground|ground]] may be swapped. In this configuration, the hysteresis curve is inverting (i.e., very negative inputs correspond to a positive output and vice versa), and the switching thresholds are <math>\pm \frac{R_1}{R_1 + R_2} V_{\text{sat}}</math>. Such a configuration is used in the [[#Relaxation_oscillator|relaxation oscillator]] shown below.

[[File:Op-Amp Differentiating Amplifier.svg|300px|class=skin-invert]]

Assuming ideal elements, this circuit [[Derivative|differentiates]] the signal (multiplied by <math>-RC</math>) over time:

{{clear}}

:<math>V_\text{out} = -RC \frac{dV_\text{in}}{dt},</math>

where <math>V_\text{in}</math> and <math>V_\text{out}</math> are functions of time.

The transfer function of the inverting differentiator has a single [[complex zero|zero]] in the origin (i.e., where [[angular frequency]] <math>\omega = 0</math>). The high-pass characteristics of a differentiating amplifier can lead to stability challenges when the circuit is used in an analog servo loop (e.g., in a [[PID controller]] with a significant derivative gain). In particular, as a [[root locus|root locus analysis]] would show, increasing feedback gain will drive a closed-loop pole toward marginal stability at the DC zero introduced by the differentiator.

=== Relaxation oscillator ===

==Synthetic elements==

{{main|Relaxation oscillator}}

[[File:Op-Amp Relaxation Oscillator.svg|frame|Relaxation oscillator implemented with inverting Schmitt trigger and RC network]]

By using an [[RC network]] to add slow [[negative feedback]] to the inverting [[#Schmitt_trigger|Schmitt trigger]], a [[relaxation oscillator]] is formed. The feedback through the RC network causes the Schmitt trigger output to [[oscillate]] in an endless symmetric [[square wave]] (i.e., the Schmitt trigger in this configuration is an [[astable multivibrator]]).

{{clear}}

=== Inductance gyrator ===

{{main article|Gyrator}}

[[File:Op-Amp Gyrator.svg|300px|class=skin-invert]]

Simulates an [[inductor]] (i.e., provides [[inductance]] without the use of a possibly costly inductor). The circuit exploits the fact that the current flowing through a capacitor behaves through time as the voltage across an inductor. The capacitor used in this circuit is geometrically smaller than the inductor it simulates, and its capacitance is less subject to changes in value due to environmental changes. Applications where this circuit may be superior to a physical inductor are simulating a variable inductance or simulating a very large inductance.

[[Image:Op-Amp Gyrator.svg|300px|thumb|Inductance gyrator]]

{{main|Gyrator}}

Simulates an [[inductor]] (i.e., provides [[inductance]] without the use of an possibly costly inductor).

This circuit is of limited use in applications relying on the [[back EMF]] property of an inductor, as this effect will be limited in a gyrator circuit to the voltage supplies of the op-amp.

{{clear}}

=== Zero level detector ===

<!-- fig needed here? -->

Voltage divider reference

* Zener sets reference voltage

=== Negative impedance converter (NIC) ===

{{main article|Negative impedance converter}}

[[File:Op-Amp Negative Impedance Converter.svg|300px|class=skin-invert]]

Creates a [[resistor]] having a negative value for any signal generator.

[[Image:Op-Amp Negative Impedance Converter.svg|300px|thumb|Negative impedance converter]]

In this case, the ratio between the input voltage and the input current (thus the input resistance) is given by

{{main|Negative impedance converter}}

: <math>R_\text{in} = -R_3 \frac{R_1}{R_2}.</math>

Creates a [[resistor]] having a negative value for any signal generator

* In this case, the ratio between the input voltage and the input current (thus the input resistance) is given by:

:<math>R_{\text{in}} = -R_3 \frac{R_1}{R_2}</math>

In general, the components <math>R_1</math>, <math>R_2</math>, and <math>R_3</math> need not be resistors; they can be any component that can be described with an [[impedance]].

In general, the components <math>R_1</math>, <math>R_2</math>, and <math>R_3</math> need not be resistors; they can be any component that can be described with an [[Electrical impedance|impedance]].

{{clear}}

==Non-linear==

=== Wien bridge oscillator ===

{{Unreferenced section|date=February 2022}}

[[File:Wien bridge classic osc.png|300px|thumb|Wien bridge oscillator]]

{{main|Wien bridge oscillator}}

Produces a pure [[sine wave]].

{{clear}}

== Non-linear configurations ==

=== Precision rectifier ===

{{main article|Precision rectifier}}

[[File:Op-Amp Precision Rectifier.svg|250px|class=skin-invert]]

The voltage drop ''V''_F across the forward-biased diode in the circuit of a passive rectifier is undesired. In this active version, the problem is solved by connecting the diode in the negative feedback loop. The op-amp compares the output voltage across the load with the input voltage and increases its own output voltage with the value of ''V''_F. As a result, the voltage drop ''V''_F is compensated, and the circuit behaves very nearly as an ideal (''super'') [[diode]] with ''V''_F = 0&nbsp;V.

[[Image:Op-Amp Precision Rectifier.svg|thumb|250px|Precision rectifier]]

The circuit has speed limitations at high frequency because of the slow negative feedback and due to the low slew rate of many non-ideal op-amps.

{{main|Precision rectifier}}

Although there are some limitations, this '''super diode''' circuit behaves like an ideal [[diode]] for the load <math>R_{\text{L}}</math>.

{{clear}}

=== LogarithmicExponential output ===

[[File:Op-Amp Exponential Amplifier.svg|300px|class=skin-invert]]

{{See also|Log amplifier}}

[[Image:Op-Amp Logarithmic Amplifier.svg|thumb|300px|Logarithmic configuration]]

The [[Shockley diode equation]] gives the [[current–voltage relationship]] for an ideal semiconductor [[diode]]:

* The relationship between the input voltage <math>v_{\text{in}}</math> and the output voltage <math>v_{\text{out}}</math> is given by:

:: <math>v_{I_\text{out}D} = -V_{I_\gammatext{S} \ln \left( e^{\frac{v_{V_\text{in}D}}{I_{V_\text{ST}}} \, R}- 1\right),</math>

where <math>I_\text{S}</math> is the [[saturation current]], <math>V_\text{D}</math> is the forward voltage across the diode, and <math>V_\text{T}</math> is the [[thermal voltage]] (approximately 26&nbsp;mV at room temperature). When <math>V_\text{D} \gg V_\text{T},</math> the diode's current is approximately proportional to an [[exponential function]]:

:where <math>I_{\text{S}}</math> is the ''saturation current''.

: <math>I_\text{D} \simeq I_\text{S} e^{\frac{V_\text{D}}{V_\text{T}}}.</math>

* If the operational amplifier is considered ideal, the negative pin is virtually grounded, so the current flowing into the resistor from the source (and thus through the diode to the output, since the op-amp inputs draw no current) is:

The opamp's inverting input is virtually grounded and ideally draws no current. Thus, the output voltage will be:

::<math>\frac{v_{\text{in}}}{R} = I_{\text{R}} = I_{\text{D}}</math>

: <math>V_\text{out} = -R I_\text{D}.</math>

The output voltage <math>V_\text{out}</math> is thus approximately an exponential function of the input voltage <math>V_\text{in}</math>:

: <math>V_\text{out} \simeq -R I_\text{S} e^{\frac{V_\text{in}}{V_\text{T}}} \, .</math>

:where <math>I_{\text{D}}</math> is the current through the diode. As known, [[Diode#Shockley diode equation|the relationship between the current and the voltage]] for a [[diode]] is:

This implementation does not consider temperature stability and other non-ideal effects.{{clear}}

=== Logarithmic output ===

::<math>I_{\text{D}} = I_{\text{S}} \left( e^{\frac{V_{\text{D}}}{V_{\gamma}}} - 1 \right).</math>

{{See also|Log amplifier}}Since the [[logarithm]] is the [[inverse function]] of exponentiation, the exponential output circuit described above can be rearranged by swapping the diode into the feedback path of the opamp to form a [[log amplifier]]:

[[File:Op-Amp Logarithmic Amplifier.svg|300px|class=skin-invert]]

:This, when the voltage is greater than zero, can be approximated by:

Since the opamp's inverting input is virtually grounded and ideally draws no current, <math>V_\text{out} {=} V_\text{D}</math> and the current flowing from the source through the resistor and diode is:

::<math>I_{\text{D}} \simeq I_{\text{S}} e^{\frac{V_{\text{D}}}{V_{\gamma}}}. </math>

: <math>\frac{V_\text{in}}{R} = I_\text{R} = I_\text{D},</math>

:Putting these two formulae together and considering that the output voltage is the negative of the voltage across the diode (<math>V_{\text{out}} = -V_{\text{D}}</math>), the relationship is proven.

where <math>I_\text{D}</math> is the current through the diode, which as described earlier is approximately:

Note that this implementation does not consider temperature stability and other non-ideal effects.

: <math>I_\text{D} \simeq I_\text{S} e^{\frac{V_\text{D}}{V_\text{T}}}.</math>

{{clear}}

Solving for <math>V_\text{out}</math> gives an approximately logarithmic relationship between the input voltage <math>V_\text{in}</math> and the output voltage <math>V_\text{out}</math>:

=== Exponential output===

: <math>V_\text{out} \simeq -V_\text{T} \ln \left(\frac{V_\text{in}}{I_\text{S} R}\right) \, .</math>

[[Image:Op-Amp Exponential Amplifier.svg|thumb|300px|Exponential configuration]]

This implementation does not consider temperature stability and other non-ideal effects.

* The relationship between the input voltage <math>v_{\text{in}}</math> and the output voltage <math>v_{\text{out}}</math> is given by:

{{clear}}

=== Piecewise linear output ===

:<math>v_{\text{out}} = -R I_{\text{S}} e^{\frac{v_{\text{in}}}{V_{\gamma}}}</math>

[[Piecewise linear function|Piecewise linear functions]] can approximate [[Nonlinear system|non-linear functions]] as a series of connected [[line segments]]. [[Gain compression]] circuits (like [[Sine and cosine|sine]] or [[square root]]) use diodes or transistors to switch between line segments with slopes determined by resistive [[voltage divider]] networks. Expansion circuits may be built using a compression circuit as feedback of an opamp.<ref name=":2">{{Cite web |last=Kuhn |first=Kenneth A. |date=2004-03-24 |title=Piecewise Linear Circuits |url=https://www.kennethkuhn.com/students/ee431/piecewise_linear_circuits.pdf}}</ref>

==== Temperature-compensated compression ====

where <math>I_{\text{S}}</math> is the ''saturation current''.

[[File:Piecewise linear temperature compensated amplifier.gif|thumb|Temperature-compensated three-segment compression function]]The schematic shown for a "temperature-compensated three-segment compression function"<ref>{{Cite web |date=April 1968 |title=AN-4 Monolithic Op Amp—the Universal Linear Component |url=https://www.ti.com/lit/an/snoa650b/snoa650b.pdf |website=[[Texas Instruments]]}}</ref><ref>{{Cite web |date=September 2002 |title=National Semiconductor Application Note 31 |url=http://www.frankshospitalworkshop.com/electronics/documents/OP%20Amp%20Circuit%20Collection%20-%20National.pdf}}</ref> produces a gain compression transfer function where each subsequent line segment reduces the steepness of the transfer function. For small signals, transistors Q2 and Q3 produce very little base current, and so the circuit's gain is determined just by the feedback resistance of R2 divided by the input resistance of R1. Once the output voltage exceeds around 2&nbsp;V (whose exact voltage depends on R3 and R4 and the -15&nbsp;V supply), then Q3 saturates, so the circuit's feedback resistance is determined by R4 in parallel with R2, reducing the gain. As the output voltage increases more, Q2 will saturate, so the circuit's gain is again reduced by the additional inclusion of R6 into the parallel feedback resistance. Temperature-compensation transistors Q4 and Q1 cancel out the temperature-dependent [[p–n junction]] base-emitter forward voltage drop of Q3 and Q2. Additional linear segments can be added using additional copies of the resistor-transistor-resistor chains (like the chain R5, Q2, R6 or the chain R3, Q3, R4 but with different values) in a similar manner to further compress the input. This circuit's compression function only works for negative inputs. Diode D1 forces the output to zero if a positive input is applied.

* Considering the operational amplifier ideal, then the negative pin is virtually grounded, so the current through the diode is given by:

:<math>I_{\text{D}} = I_{\text{S}} \left( e^{\frac{V_{\text{D}}}{V_{\gamma}}} - 1 \right)</math>

when the voltage is greater than zero, it can be approximated by:

:<math>I_{\text{D}} \simeq I_{\text{S}} e^{\frac{V_{\text{D}}}{V_{\gamma}}}. </math>

The output voltage is given by:

:<math>v_{\text{out}} = -R I_{\text{D}}.\,</math>

{{clear}}

== Other applications ==

{{div col|colwidth=30em}}

<div class="references" style="-moz-column-count:3; column-count:3;">

* audio and video [[preamplifier|pre]]-[[amplifier]]s and [[buffer amplifier|buffer]]s

* voltage [[comparator]]s

* [[differential amplifier]]s

* [[Derivative|differentiator]]s and [[integrator]]s

* [[Filter (signal processing)|filter]]s

* precision [[rectifier]]s

* [[voltage regulator]] and [[current regulator]]

* [[analog-to-digital converter]]

* [[digital-to-analog converter]]

* [[Clamper (electronics)|voltage clamp]]s

* [[electronic oscillator|oscillator]]s and [[waveform generator]]s

* [[Schmitt trigger]]

* [[Gyrator]]

* [[Comparator]]

* [[Active filter]]

* [[Analog computer]]

* [[Capacitance multiplier]]

* [[Charge amplifier]]

</div>

{{div col end}}

== See also ==

{{div col|colwidth=30em}}

* [[Current-feedback operational amplifier]]

* [[Frequency compensation]]

* [[Operational amplifier]]

* [[Current-feedback operational amplifier]]

* [[Operational transconductance amplifier]]

* [[FrequencyTransimpedance compensationamplifier]]

{{div col end}}

* [[George A. Philbrick]]

== Notes ==

{{Reflist|group=nb}}

== References ==

{{reflistReflist}}

== Further reading ==

{{Main|Operational amplifier#Further reading|l1=Operational amplifier further reading}}

{{refbegin}}

* Paul Horowitz and Winfield Hill, ''[[The Art of Electronics]]''. 2^nd ed. Cambridge University Press, Cambridge, 1989 ISBN 0-521-37095-7

* Sergio Franco, ''Design with Operational Amplifiers and Analog Integrated Circuits'', 3^rd ed., McGraw-Hill, New York, 2002 ISBN 0-07-232084-2

{{refend}}

== External links ==

{{Wikibooks|Electronics|Op-Amps}}

* {{cite web|url= https://web.archive.org/web/20060108060032/http://instruct1.cit.cornell.edu/courses/bionb440/datasheets/SingleSupply.pdf |title=Single supply op-amp circuit collection }}&nbsp;{{small|(163&nbsp;[[Kibibyte|KiB]])}}

{{Wikibooks|Circuit Idea|Voltage Compensation}}

* {{cite web|url= https://www.ti.com/lit/an/snla140d/snla140d.pdf |title=Op-amp circuit collection }}&nbsp;{{small|(2980&nbsp;[[Kibibyte|KiB]])}}

* [http://ourworld.compuserve.com/homepages/Bill_Bowden/opamp.htm Introduction to op-amp circuit stages, second order filters, single op-amp bandpass filters, and a simple intercom]

* {{PDFlinkcite web|[url= http://focuswww.tianalog.com/litstatic/animported-files/slod006bapplication_notes/slod006b28080533AN106.pdf Op|title=A AmpsCollection forof Everyone]Amp Applications }}&nbsp;{{small|(1.9606&nbsp;[[Mebibyte|MiB]]<!--)}}&nbsp;&ndash; application/pdf,[[Analog 2061665Devices]] bytes[[Application -->}}note]]

* {{PDFlinkcite web|[url= http://instruct1www.citligo.cornellcaltech.edu/courses~vsanni/bionb440ph5/datasheetspdf/SingleSupplyBasicOpAmpApplications.pdf Single|title=Basic supplyOpAmp op-ampApplications circuit collection]}}&nbsp;{{small|163(173&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 167039 bytes -->)}}

* {{cite web|url= http://focus.ti.com/lit/an/sboa092a/sboa092a.pdf |title=Handbook of operational amplifier applications }}&nbsp;{{small|(2.00&nbsp;[[Mebibyte|MiB]])}}&nbsp;&ndash; [[Texas Instruments]] [[Application note]]

* {{PDFlink|[http://www.national.com/an/AN/AN-31.pdf Op-amp circuit collection]|962&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 985824 bytes -->}}

* [http://focus.ti.com/analog/docs/gencontent.tsp?familyId=57&genContentId=28017 Low Side Current Sensing Using Operational Amplifiers] {{Webarchive|url=https://web.archive.org/web/20090408192711/http://focus.ti.com/analog/docs/gencontent.tsp?familyId=57&genContentId=28017 |date=2009-04-08 }}

* {{PDFlink|[http://www.analog.com/static/imported-files/application_notes/28080533AN106.pdf A Collection of Amp Applications]|1.06&nbsp;[[Mebibyte|MiB]]<!-- application/pdf, 1120990 bytes -->}}&nbsp;&ndash; [[Analog Devices]] [[Application note]]

* {{cite web |url= http://www.national.com/an/AN/AN-30.pdf |title= Log/anti-log generators, cube generator, multiply/divide amp |url-status= dead |archive-url= https://web.archive.org/web/20080509065104/http://www.national.com/an/AN/AN-30.pdf |archive-date= 2008-05-09 }}&nbsp;{{small|(165&nbsp;[[Kibibyte|KiB]])}}

* {{PDFlink|[http://www.ligo.caltech.edu/~vsanni/ph5/BasicOpAmpApplications.pdf Basic OpAmp Applications]|173&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 177205 bytes -->}}

* [https://web.archive.org/web/20060215074343/http://www.edn.com/archives/1994/030394/05di7.htm Logarithmically variable gain from a linear variable component]

* {{PDFlink|[http://focus.ti.com/lit/an/sboa092a/sboa092a.pdf Handbook of operational amplifier applications]|2.00&nbsp;[[Mebibyte|MiB]]<!-- application/pdf, 2103557 bytes -->}}&nbsp;&ndash; [[Texas Instruments]] [[Application note]]

* [http://focus.ti.com/analog/docs/gencontent.tsp?familyId=57&genContentId=28017 Low Side Current Sensing Using Operational Amplifiers]

* {{PDFlink|[http://www.national.com/an/AN/AN-30.pdf Log/anti-log generators, cube generator, multiply/divide amp]|165&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 169719 bytes -->}}

* [http://www.edn.com/archives/1994/030394/05di7.htm Logarithmically variable gain from a linear variable component]

* [http://www.philbrickarchive.org/1964-1_v12_no1_the_lightning_empiricist.htm Impedance and admittance transformations using operational amplifiers] by D. H. Sheingold

* [http://www.linear.com/docs/4138 ''High Speed Amplifier Techniques ''] very practical and readable{{spaced ndash}}with photos and real waveforms

* [http://www.electronicproducts.com/Analog_Mixed_Signal_ICs/Amplifiers/Properly_terminating_an_unused_op_amp.aspx Properly terminating an unused op-amp]

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