Editing Negative feedback (section)

===Operational amplifier circuits===
{{main|Operational amplifier applications}}

[[File:Feedback op-amp voltage amplifier.png|thumb|200px|A feedback voltage amplifier using an op amp with finite gain but infinite input impedances and zero output impedance.<ref name=Franco>See, for example, Figure 1.4, p. 7 ''Ideal op amp model'' in {{cite book |title=Design with operational amplifiers and analog integrated circuits |edition=3rd |author=Sergio Franco |url=https://books.google.com/books?id=em1BnAEACAAJ |publisher=McGraw-Hill |year=2002 |isbn=978-0078028168}} or {{cite book |title=Fundamentals of Circuits and Filters |editor=Wai-Kai Chen |author1=David G Nair |author2=Sergio B Franco |chapter=Figure 16.2: ''The four possible op-amp configurations'' |pages=16–2 |isbn=9781420058888 |year=2009 |publisher=CRC Press |edition=The Circuits and Filters Handbook, 3rd |chapter-url=https://books.google.com/books?id=_UVb4cxL0c0C&pg=SA16-PA2}}</ref>]]

The operational amplifier was originally developed as a building block for the construction of [[analog computers]], but is now used almost universally in all kinds of applications including [[audio signal|audio]] equipment and [[control systems]].

Operational amplifier circuits typically employ negative feedback to get a predictable transfer function. Since the open-loop gain of an [[Operational amplifier|op-amp]] is extremely large, a small differential input signal would drive the output of the amplifier to one rail or the other in the absence of negative feedback. A simple example of the use of feedback is the op-amp voltage amplifier shown in the figure.

The idealized model of an operational amplifier assumes that the gain is infinite, the input impedance is infinite, output resistance is zero, and input offset currents and voltages are zero. Such an ideal amplifier draws no current from the resistor divider.<ref name=Schitter>
{{cite book |title=The Design of High Performance Mechatronics |author1=G. Schitter |author2=A. Rankers |page=499 |chapter-url=https://books.google.com/books?id=3WvnAgAAQBAJ&pg=PA499 |chapter=§6.3.4 Linear amplifiers with operational amplifiers |isbn=9781614993681 |year=2014 |publisher=IOS Press}}
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Ignoring dynamics (transient effects and [[propagation delay]]), the infinite gain of the ideal op-amp means this feedback circuit drives the voltage difference between the two op-amp inputs to zero.<ref name=Schitter/> Consequently, the voltage gain of the circuit in the diagram, assuming an ideal op amp, is the reciprocal of feedback [[Voltage divider|voltage division]] ratio &beta;:

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

A real op-amp has a high but finite gain ''A'' at low frequencies, decreasing gradually at higher frequencies. In addition, it exhibits a finite input impedance and a non-zero output impedance. Although practical op-amps are not ideal, the model of an ideal op-amp often suffices to understand circuit operation at low enough frequencies. 
As discussed in the previous section, the feedback circuit stabilizes the closed-loop gain and desensitizes the output to fluctuations generated inside the amplifier itself.<ref name= Jung>
{{cite book |title=Op Amp Applications Handbook |author=Walter G Jung |chapter=Noise gain (NG) |pages=12 ''ff'' |isbn=9780750678445 |year=2005 |publisher=Newnes |chapter-url=https://books.google.com/books?id=dunqt1rt4sAC&q=%22Including+the+effects+of+finite+op+amp+gain,%22&pg=PA12}} 
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