Editing Operational amplifier (section)

===Negative-feedback applications===

==== Non-inverting amplifier ====
[[Image:Op-Amp Non-Inverting Amplifier.svg|frame|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
:<math>V_\text{out} = A_\text{OL} (V_+ - V_-).</math>

However, in this circuit ''V''<sub>−</sub> is a function of ''V''<sub>out</sub> because of the negative feedback through the ''R''<sub>1</sub> ''R''<sub>2</sub> network. ''R''<sub>1</sub> and ''R''<sub>2</sub> form a [[voltage divider]], and as ''V''<sub>−</sub> is a high-impedance input, it does not load it appreciably. Consequently
:<math>V_- = \beta V_\text{out},</math>

where
:<math>\beta = \frac{R_1}{R_1 + R_2}.</math>

Substituting this into the gain equation, we obtain
:<math>V_\text{out} = A_\text{OL} (V_\text{in} - \beta V_\text{out}).</math>

Solving for <math>V_\text{out}</math>:
:<math>V_\text{out} = V_\text{in} \left( \frac{1}{\beta + \frac{1}{A_\text{OL}}} \right).</math>

If <math>A_\text{OL}</math> is very large, this simplifies to
:<math>
  V_\text{out} \approx \frac{V_\text{in}}{\beta}
                     = \frac{V_\text{in}}{\frac{R_1}{R_1 + R_2}}
                     = V_\text{in} \left(1 + \frac{R_2}{R_1}\right).
</math>

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.<ref>An input bias current of 1&nbsp;μA through a DC source resistance of 10&nbsp;kΩ produces a 10&nbsp;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.</ref> The ideal value for the feedback resistors (to give minimal 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.<ref>{{cite web |author=Analog Devices |title=Op Amp Input Bias Current |date=2009 |id=Tutorial MT-038 |publisher=Analog Devices |url=http://www.analog.com/static/imported-files/tutorials/MT-038.pdf |access-date=2014-05-15 |archive-date=2015-02-13 |archive-url=https://web.archive.org/web/20150213055046/http://www.analog.com/static/imported-files/tutorials/MT-038.pdf |url-status=dead }}</ref>

==== Inverting amplifier ====
[[Image:Op-Amp Inverting Amplifier.svg|frame|right|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:

:<math>V_\text{out} = A_\text{OL} (V_+ - V_-).</math>

This time, ''V''<sub>−</sub> is a function of both ''V''<sub>out</sub> and ''V''<sub>in</sub> due to the voltage divider formed by ''R''<sub>f</sub> and ''R''<sub>in</sub>. Again, the op-amp input does not apply an appreciable load, so

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

Substituting this into the gain equation and solving for <math>V_\text{out}</math>:

:<math>V_\text{out} = - V_\text{in} \frac{A_\text{OL} R_\text{f}}{R_\text{f} + R_\text{in} + A_\text{OL} R_\text{in}}.</math>

If <math>A_\text{OL}</math> is very large, this simplifies to

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

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 [[Capacitive coupling|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 [[complex zero|zero]] and a low-frequency [[complex pole|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.{{cn|date=January 2025}}