# Operational amplifier

An operational amplifier (op-amp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output.[1] In this configuration, an op-amp produces an output potential (relative to circuit ground) that is typically hundreds of thousands of times larger than the potential difference between its input terminals.[2]

Operational amplifiers had their origins in analog computers, where they were used to do mathematical operations in many linear, non-linear and frequency-dependent circuits. Characteristics of a circuit using an op-amp are set by external components with little dependence on temperature changes or manufacturing variations in the op-amp itself, which makes op-amps popular building blocks for circuit design.

### Negative feedback applications

#### Non-inverting amplifier

An op-amp connected in the non-inverting amplifier configuration

In a non-inverting amplifier, the output voltage changes in the same direction as the input voltage.

The gain equation for the op-amp is:

${\displaystyle V_{\text{out}}=A_{OL}\,(V_{\!+}-V_{\!-})}$

However, in this circuit V is a function of Vout because of the negative feedback through the R1 R2 network. R1 and R2 form a voltage divider, and as V is a high-impedance input, it does not load it appreciably. Consequently:

${\displaystyle V_{\!-}\,\,=\beta \cdot V_{\text{out}}}$

where

${\displaystyle \beta ={\frac {R_{1}}{R_{1}+R_{2}}}}$

Substituting this into the gain equation, we obtain:

${\displaystyle V_{\text{out}}=A_{OL}(V_{\text{in}}-\beta \cdot V_{\text{out}})}$
${\displaystyle V_{\text{out}}=V_{\text{in}}\left({\frac {1}{\beta +1/A_{OL}}}\right)}$

If ${\displaystyle A_{OL}}$ is very large, this simplifies to

${\displaystyle V_{\text{out}}\approx {\frac {V_{\text{in}}}{\beta }}={\frac {V_{\text{in}}}{\frac {R_{\text{1}}}{R_{\text{1}}+R_{\text{2}}}}}=V_{\text{in}}\left(1+{\frac {R_{2}}{R_{1}}}\right)}$.

The non-inverting input of the operational amplifier needs a path for DC to ground; if the signal source does not supply a DC path, or if that source requires a given load impedance, then the circuit will require another resistor from the non-inverting input to ground. When the operational amplifier's input bias currents are significant, then the DC source resistances driving the inputs should be balanced.[12] The ideal value for the feedback resistors (to give minimum offset voltage) will be such that the two resistances in parallel roughly equal the resistance to ground at the non-inverting input pin. That ideal value assumes the bias currents are well-matched, which may not be true for all op-amps.[13]

#### Inverting amplifier

An op-amp connected in the inverting amplifier configuration

In an inverting amplifier, the output voltage changes in an opposite direction to the input voltage.

As with the non-inverting amplifier, we start with the gain equation of the op-amp:

${\displaystyle V_{\text{out}}=A_{OL}\,(V_{\!+}-V_{\!-})}$

This time, V is a function of both Vout and Vin due to the voltage divider formed by Rf and Rin. Again, the op-amp input does not apply an appreciable load, so:

${\displaystyle V_{\!-}\,\,={\frac {1}{R_{\text{f}}+R_{\text{in}}}}\left(R_{\text{f}}V_{\text{in}}+R_{\text{in}}V_{\text{out}}\right)}$

Substituting this into the gain equation and solving for ${\displaystyle V_{\text{out}}}$:

${\displaystyle V_{\text{out}}=-V_{\text{in}}\cdot {\frac {A_{OL}R_{\text{f}}}{R_{\text{f}}+R_{\text{in}}+A_{OL}R_{\text{in}}}}}$

If ${\displaystyle A_{OL}}$ is very large, this simplifies to

${\displaystyle V_{\text{out}}\approx -V_{\text{in}}{\frac {R_{\text{f}}}{R_{\text{in}}}}}$

A resistor is often inserted between the non-inverting input and ground (so both inputs "see" similar resistances), reducing the input offset voltage due to different voltage drops due to bias current, and may reduce distortion in some op-amps.

A DC-blocking capacitor may be inserted in series with the input resistor when a frequency response down to DC is not needed and any DC voltage on the input is unwanted. That is, the capacitive component of the input impedance inserts a DC zero and a low-frequency pole that gives the circuit a bandpass or high-pass characteristic.

The potentials at the operational amplifier inputs remain virtually constant (near ground) in the inverting configuration. The constant operating potential typically results in distortion levels that are lower than those attainable with the non-inverting topology.

### Other applications

Most single, dual and quad op-amps available have a standardized pin-out which permits one type to be substituted for another without wiring changes. A specific op-amp may be chosen for its open loop gain, bandwidth, noise performance, input impedance, power consumption, or a compromise between any of these factors.

## Historical timeline

1941: A vacuum tube op-amp. An op-amp, defined as a general-purpose, DC-coupled, high gain, inverting feedback amplifier, is first found in U.S. Patent 2,401,779 "Summing Amplifier" filed by Karl D. Swartzel Jr. of Bell Labs in 1941. This design used three vacuum tubes to achieve a gain of 90 dB and operated on voltage rails of ±350 V. It had a single inverting input rather than differential inverting and non-inverting inputs, as are common in today's op-amps. Throughout World War II, Swartzel's design proved its value by being liberally used in the M9 artillery director designed at Bell Labs. This artillery director worked with the SCR584 radar system to achieve extraordinary hit rates (near 90%) that would not have been possible otherwise.[14]

GAP/R's K2-W: a vacuum-tube op-amp (1953)

1947: An op-amp with an explicit non-inverting input. In 1947, the operational amplifier was first formally defined and named in a paper by Professor John R. Ragazzini of Columbia University. In this same paper a footnote mentioned an op-amp design by a student that would turn out to be quite significant. This op-amp, designed by Loebe Julie, was superior in a variety of ways. It had two major innovations. Its input stage used a long-tailed triode pair with loads matched to reduce drift in the output and, far more importantly, it was the first op-amp design to have two inputs (one inverting, the other non-inverting). The differential input made a whole range of new functionality possible, but it would not be used for a long time due to the rise of the chopper-stabilized amplifier.[14]

1949: A chopper-stabilized op-amp. In 1949, Edwin A. Goldberg designed a chopper-stabilized op-amp.[15] This set-up uses a normal op-amp with an additional AC amplifier that goes alongside the op-amp. The chopper gets an AC signal from DC by switching between the DC voltage and ground at a fast rate (60 Hz or 400 Hz). This signal is then amplified, rectified, filtered and fed into the op-amp's non-inverting input. This vastly improved the gain of the op-amp while significantly reducing the output drift and DC offset. Unfortunately, any design that used a chopper couldn't use their non-inverting input for any other purpose. Nevertheless, the much improved characteristics of the chopper-stabilized op-amp made it the dominant way to use op-amps. Techniques that used the non-inverting input regularly would not be very popular until the 1960s when op-amp ICs started to show up in the field.

1953: A commercially available op-amp. In 1953, vacuum tube op-amps became commercially available with the release of the model K2-W from George A. Philbrick Researches, Incorporated. The designation on the devices shown, GAP/R, is an acronym for the complete company name. Two nine-pin 12AX7 vacuum tubes were mounted in an octal package and had a model K2-P chopper add-on available that would effectively "use up" the non-inverting input. This op-amp was based on a descendant of Loebe Julie's 1947 design and, along with its successors, would start the widespread use of op-amps in industry.

GAP/R's model P45: a solid-state, discrete op-amp (1961).

1961: A discrete IC op-amp. With the birth of the transistor in 1947, and the silicon transistor in 1954, the concept of ICs became a reality. The introduction of the planar process in 1959 made transistors and ICs stable enough to be commercially useful. By 1961, solid-state, discrete op-amps were being produced. These op-amps were effectively small circuit boards with packages such as edge connectors. They usually had hand-selected resistors in order to improve things such as voltage offset and drift. The P45 (1961) had a gain of 94 dB and ran on ±15 V rails. It was intended to deal with signals in the range of ±10 V.

1961: A varactor bridge op-amp. There have been many different directions taken in op-amp design. Varactor bridge op-amps started to be produced in the early 1960s.[16][17] They were designed to have extremely small input current and are still amongst the best op-amps available in terms of common-mode rejection with the ability to correctly deal with hundreds of volts at their inputs.

GAP/R's model PP65: a solid-state op-amp in a potted module (1962)

1962: An op-amp in a potted module. By 1962, several companies were producing modular potted packages that could be plugged into printed circuit boards.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} These packages were crucially important as they made the operational amplifier into a single black box which could be easily treated as a component in a larger circuit.

1963: A monolithic IC op-amp. In 1963, the first monolithic IC op-amp, the μA702 designed by Bob Widlar at Fairchild Semiconductor, was released. Monolithic ICs consist of a single chip as opposed to a chip and discrete parts (a discrete IC) or multiple chips bonded and connected on a circuit board (a hybrid IC). Almost all modern op-amps are monolithic ICs; however, this first IC did not meet with much success. Issues such as an uneven supply voltage, low gain and a small dynamic range held off the dominance of monolithic op-amps until 1965 when the μA709[18] (also designed by Bob Widlar) was released.

1968: Release of the μA741. The popularity of monolithic op-amps was further improved upon the release of the LM101 in 1967, which solved a variety of issues, and the subsequent release of the μA741 in 1968. The μA741 was extremely similar to the LM101 except that Fairchild's facilities allowed them to include a 30 pF compensation capacitor inside the chip instead of requiring external compensation. This simple difference has made the 741 the canonical op-amp and many modern amps base their pinout on the 741s. The μA741 is still in production, and has become ubiquitous in electronics—many manufacturers produce a version of this classic chip, recognizable by part numbers containing 741. The same part is manufactured by several companies.

1970: First high-speed, low-input current FET design. In the 1970s high speed, low-input current designs started to be made by using FETs. These would be largely replaced by op-amps made with MOSFETs in the 1980s. During the 1970s single sided supply op-amps also became available.

ADI's HOS-050: a high speed hybrid IC op-amp (1979)

1972: Single sided supply op-amps being produced. A single sided supply op-amp is one where the input and output voltages can be as low as the negative power supply voltage instead of needing to be at least two volts above it. The result is that it can operate in many applications with the negative supply pin on the op-amp being connected to the signal ground, thus eliminating the need for a separate negative power supply.

The LM324 (released in 1972) was one such op-amp that came in a quad package (four separate op-amps in one package) and became an industry standard. In addition to packaging multiple op-amps in a single package, the 1970s also saw the birth of op-amps in hybrid packages. These op-amps were generally improved versions of existing monolithic op-amps. As the properties of monolithic op-amps improved, the more complex hybrid ICs were quickly relegated to systems that are required to have extremely long service lives or other specialty systems.

An op-amp in a modern mini DIP

Recent trends. Recently supply voltages in analog circuits have decreased (as they have in digital logic) and low-voltage op-amps have been introduced reflecting this. Supplies of ±5 V and increasingly 3.3 V (sometimes as low as 1.8 V) are common. To maximize the signal range modern op-amps commonly have rail-to-rail output (the output signal can range from the lowest supply voltage to the highest) and sometimes rail-to-rail inputs.

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## Notes

1. This definition hews to the convention of measuring op-amp parameters with respect to the zero voltage point in the circuit, which is usually half the total voltage between the amplifier's positive and negative power rails.
2. Many older designs of operational amplifiers have offset null inputs to allow the offset to be manually adjusted away. Modern precision op-amps can have internal circuits that automatically cancel this offset using choppers or other circuits that measure the offset voltage periodically and subtract it from the input voltage.
3. That the output cannot reach the power supply voltages is usually the result of limitations of the amplifier's output stage transistors. See Output stage.
4. Widlar used this same trick in μA702 and μA709

## References

1. Maxim Application Note 1108: Understanding Single-Ended, Pseudo-Differential and Template:SicDifferential ADC Inputs – Retrieved November 10, 2007
2. Analog devices MT-044 Tutorial
3. Template:Cite web
4. Jacob Millman, Microelectronics: Digital and Analog Circuits and Systems, McGraw-Hill, 1979, ISBN 0-07-042327-X, pp. 523-527
5. {{#invoke:citation/CS1|citation |CitationClass=book }}
6. D.F. Stout Handbook of Operational Amplifier Circuit Design (McGraw-Hill, 1976, ISBN 0-07-061797-X ) pp. 1–11.
7. Template:Cite web
8. Template:Cite web
9. Template:Cite web
10. Template:Cite web
11. The μA741 Operational Amplifier
12. An input bias current of 1 µA through a DC source resistance of 10 kΩ produces a 10 mV offset voltage. If the other input bias current is the same and sees the same source resistance, then the two input offset voltages will cancel out. Balancing the DC source resistances may not be necessary if the input bias current and source resistance product is small.
13. http://www.analog.com/static/imported-files/tutorials/MT-038.pdf
14. {{#invoke:citation/CS1|citation |CitationClass=book }}
15. http://www.analog.com/library/analogDialogue/archives/39-05/Web_ChH_final.pdf
16. http://www.philbrickarchive.org/
18. A.P. Malvino, Electronic Principles (2nd Ed. 1979. ISBN 0-07-039867-4) p. 476.