Operational amplifier

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Because Kirchhoff's current law states that the same current must leave a node as enter it, and because the impedance into the (−) pin is near infinity per assumption 2, we can assume practically all of the same current flows through , creating an output voltage

<math display="block">V_\text{out} = V_\text{in} + iR_\text{f} = V_\text{in} + \left(\frac{V_\text{in{R_\text{g R_\text{f}\right) = V_\text{in} + \frac{V_\text{in}R_\text{f {R_\text{g = V_\text{in} \left(1 + \frac{R_\text{f{R_\text{g\right).</math>

By combining terms, we determine the closed-loop gain :

Characteristics

Ideal op amps

thumb|250px|right|An equivalent circuit of an operational amplifier that models some resistive non-ideal parameters.

An ideal op amp is usually considered to have the following characteristics:

Arbitrarily high open-loop gain
Infinite input impedance , and thus zero input current
Zero input offset voltage
Unbounded output voltage range
Unrestricted bandwidth with zero phase shift and infinite slew rate
Zero output impedance , and thus ability to source or sink unbounded output current
Zero noise
No effect of common-mode voltages, as described by common-mode rejection ratio (CMRR)
No effect of supply variations on the output, i.e., perfect rejection of power supply variation.

These ideals can be summarized by the two :

In a negative feedback configuration the output does whatever is necessary to make the voltage difference between the inputs zero.
The inputs draw zero current.

The first rule only applies in the usual case where the op amp is used in a negative feedback design, where there is a signal path of some sort feeding back from the output to the inverting input. These rules are commonly used as a good first approximation for analyzing or designing op-amp circuits. flows into the inputs. When high resistances or sources with high output impedances are used in the circuit, these small currents can produce significant voltage drops. If the input currents are matched, the impedance looking of inputs are matched, then those voltages at each input will be equal. Because the operational amplifier operates on the between its inputs, these matched voltages will have no effect. It is more common for the input currents to be slightly mismatched. The difference is called input offset current, and even with matched resistances a small offset voltage (distinct from the below) can be produced. This offset voltage can create offsets or drift in the operational amplifier.

Non-linear imperfections

thumb|The input (yellow) and output (green) of a saturated op amp in an inverting amplifier

thumb|Simplified opamp internals. The first amplification stage multiplies the differential input voltage (<sub>in</sub>) times a [[transconductance (g<sub>m</sub>) to produce a current (). The next stage converts that current into a voltage () and provides frequency compensation by integrating that current through a miller capacitance (). The maximum current that can be drawn from that first stage will limit the slew rate in this integration stage to /. A final stage (not shown) buffers to provide high output current (both sinking and sourcing) for the output voltage.]]

thumb|Slew limiting may distort large or fast signals. A 250 kHz input sine (magenta) is buffered by an opamp with a 720 mV/μs slew limit. With a small input sine, the output (yellow) has almost no distortion. But as the input's amplitude increases, the output can't transition fast enough to reproduce the larger sine's steeper slope and looks more like a triangle wave.

Power considerations

Classification

Op amps may be classified by their construction:

discrete, built from individual transistors or tubes/valves,
hybrid, consisting of discrete and integrated components,
full integrated circuits — most common, having displaced the former two due to low cost.

IC op amps may be classified in many ways, including:

Device grade, including acceptable operating temperature ranges and other environmental or quality factors. For example: LM101, LM201, and LM301 refer to the military, industrial, and commercial versions of the same component. Military and industrial-grade components offer better performance in harsh conditions than their commercial counterparts but are sold at higher prices.
Classification by package type may also affect environmental hardiness, as well as manufacturing options; DIP, and other through-hole packages are tending to be replaced by surface-mount devices.
Classification by internal compensation: op amps may suffer from high frequency instability in some negative feedback circuits unless a small compensation capacitor modifies the phase and frequency responses. Op amps with a built-in capacitor are termed compensated, and allow circuits above some specified closed-loop gain to be stable with no external capacitor. In particular, op amps that are stable even with a closed loop gain of 1 are called unity gain compensated.
Single, dual and quad versions of many commercial op-amp IC are available, meaning 1, 2 or 4 operational amplifiers are included in the same package.
Rail-to-rail input (and/or output) op amps can work with input (and/or output) signals very close to the power supply rails.

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 John R. Ragazzini of Columbia University. In this same paper, a footnote mentions an op-amp design by a student that would turn out to be quite significant. This op amp, designed by Loebe Julie, has two major innovations. Its input stage uses a long-tailed triode pair with loads matched to reduce drift in the output and, far more importantly, it is the first op-amp design to have two inputs (one inverting, the other non-inverting). The differential input makes 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. 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 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 the 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 were not widely practiced until the 1960s when op-amp ICs became available.

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. 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. 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.

thumb|GAP/R 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 are effectively small circuit boards with packages such as edge connectors. They usually have hand-selected resistors in order to improve things such as voltage offset and drift. The P45 (1961) has a gain of 94 dB and runs on ±15 V rails. It was intended to deal with signals in the range of .

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. 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.

thumb|GAP/R 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. 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 (also designed by Bob Widlar) was released.

1968: Release of the μA741. The popularity of monolithic op amps was further improved with 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 manufacturing processes allowed them to include a 30 pF compensation capacitor inside the chip instead of requiring external compensation. This simple difference has made the 741 a canonical op amp, and a range of 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.

1970: First high-speed, low-input current FET design.

In the 1970s, high-speed, low-input current designs started to be made by using JFETs.

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.

Recent trends.

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.