Operational amplifier (op-amp) possesses an open-loop gain factor A_{0} and a close-circuit gain factor A.

The negative terminal of the op-amp is called “inverting” terminal and the positive terminal is called “non-inverting”. In addition to two input terminals, there are two power terminal V_{+vcc} and V_{-vcc}.

The output V_{out} is calculated as

**V _{out} = A * (V+in – V-in)**.

The ideal op-amp has the following characteristics:

Input impedance r_{in} approaches infinity.

Meanwhile, output r_{out} impedance approaches zero.

## Unity gain amplifier

These characteristics see applications in **voltage follower** (also known as **unity gain amplifier**), which produces an output voltage equal to input voltage but with much lower impedance. Low impedance circuits allow more current draw than high impedance ones.

This unity gain is possible because the gain factor used here is open loop

**V _{out} / Vin = A_{0 }/ (1+A_{0})**

as A_{0 }approaches infinity

**V _{out} / Vin = 1**

## Gain of closed-loop amplifiers

In other cases, the gain is determined by the values of the voltage divider overlaying the feedback loop. For example

In this case, the gain is determined using Kirchhoff’s law. Since Kirchhoff’s current law states that the same current must leave a node as enter it and since the impedance into the (−) pin is near infinity, we can assume practically all of the same current *i* flows through *R*_{f}, creating an output voltage

Finally, we have the following closed-loop gain (applicable only to the above circuit)

## High-frequency characteristics

The gains and formulas above are only applicable for low-frequency circuits. In high frequencies, the following will happen:

Above certain frequencies, voltage gain and current gain factors become diminished in inverse proportion to frequency increase.

The output signal becomes lagged behind the input signal (phase shifting) as the op-amp cannot react fast enough to the frequency changes.

Very high input amplitude (peak-to-peak voltage swing of the electrical signal) causes distortions and the higher the frequency, the lower this upper amplitude threshold is (distortion appears at lower amplitudes at high frequencies).

**The cut-off frequency** is the frequency at which the voltage gain is 1/sqrt(2) of the low-frequency gain, meaning, the power at the output is effectively halved in voltage follower setup. The cut-off frequency is unique to each op-amp component. It can be easier determined using an oscilloscope as **the frequency at which the phase shift is 180 ^{o}**.

Textbook definition:

The frequency where the voltage falls to 0.707 of its intended value is the **cutoff** or **-3 dB** frequency, *fc*. (Gain in decibels = 20∙log(0.707) = -3dB.)

As a result of these special high-frequency characteristics, op-amps can be unsuitable for high-frequency applications.

## Read more

https://en.wikipedia.org/wiki/Operational_amplifier

http://www.learningaboutelectronics.com/Articles/Voltage-follower

http://www.ecircuitcenter.com/Circuits/op_bandwidth1/op_bandwidth1.htm