en>Headbomb: Various citation cleanup and WP:AWB general fixes, added orphan tag using AWB

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Various citation cleanup and WP:AWB general fixes, added orphan tag using AWB

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{{merge to|Operational amplifier applications|Talk=Talk:Op amp integrator#Merge_proposal|date=December 2012}}

The '''operational amplifier integrator''' is an electronic integration circuit. Based around the [[operational amplifier]] (op-amp), it performs the mathematical operation of [[integration (mathematics)|integration]] with respect to time; that is, its output [[voltage]] is proportional to the input voltage integrated over time.

==Ideal circuit==
[[File:Integrator circuit.png|Integrator circuit]]

Intuitively, the circuit operates by passing a [[electric current|current]] that charges or discharges the [[capacitor]] over time. If the op-amp is assumed [[Operational amplifier#Ideal op-amps|ideal]], nodes ''v1'' and ''v2'' are held equal, and so ''v2'' is a [[virtual ground]]. The input voltage passes a current <math>\frac{v_{in}}{R_1}</math> through the [[resistor]] and series capacitor, which charges or discharges the capacitor over time. Because the resistor and capacitor are connected to a virtual ground, the input current does not vary with capacitor charge and a [[linearity|linear]] integration operation is achieved.

The circuit can be analyzed by applying [[Kirchhoff's current law]] at the node ''v2'', keeping ideal op-amp behaviour in mind.

:<math>i_{\text{1}} = I_{\text{B}} + i_{\text{F}}</math>

<math>I_{\text{B}} = 0</math> in an ideal op-amp, so:

:<math>i_{\text{1}} = i_{\text{F}}</math>

Furthermore, the capacitor has a voltage-current relationship governed by the equation:

:<math>I_{\text{C}} = C \frac{dV_{\text{c}}}{dt}</math>

Substituting the appropriate variables:

:<math>\frac{v_{\text{in}} - v_{\text{2}}}{R_{\text{1}}} = C_{\text{F}}\frac{d(v_{\text{2}} - v_{\text{o}})}{dt}</math>

<math>v_2 = v_1 = 0</math> in an ideal op-amp, resulting in:

:<math>\frac{v_{\text{in}}}{R_{\text{1}}} = -C_{\text{F}}\frac{dv_{\text{o}}}{dt}</math>

Integrating both sides with respect to time:

:<math> \int_0^t\frac{v_{\text{in}}}{R_{\text{1}}} \ dt\ = - \int_0^t C_{\text{F}} \frac{dv_{\text{o}}}{dt} \, dt</math>

If the initial value of ''vo'' is assumed to be 0 V, this results in a DC error of:<ref name="microchip-opa-dc" />

:<math>v_{\text{o}} = -\frac{1}{R_{\text{1}}C_{\text{F}}}\int_0^t v_{\text{in}}\, dt</math>

==Practical circuit==
[[File:Practical integrator.png|Practical integrator]]

The ideal integrator seen above is not a practical circuit design. Non-ideal op-amps have a finite [[open-loop gain]], an [[input offset voltage]] and [[input bias current]]s (<math>I_B</math> in the ideal circuit figure, above). This can cause several issues for the ideal design; most importantly, if <math>v_{\text{in}} = 0</math>, both the output offset voltage and the input bias current <math>I_B</math> can cause current to pass through the capacitor, causing the output voltage to drift over time until the op-amp saturates. Similarly, if <math>v_{\text{in}}</math> were a signal centered about zero volts (i.e. without a [[direct current|DC]] component), no drift would be expected in an ideal circuit, but may occur in a real circuit.

In DC [[steady state]], the capacitor acts as an open circuit. The DC gain of the ideal circuit is therefore infinite (or the open-loop gain of a non-ideal op-amp). A large resistor <math>R_F</math> can be inserted in parallel with the [[Feedback#Electronic engineering|feedback]] capacitor, as shown in the figure above. This limits the DC gain of the circuit to a finite value, and hence changes the output drift into a finite, preferably small, DC error:

:<math>V_\text{E} = \left( \frac{R_\text{f}}{R_1} + 1 \right) \left( V_{IOS} + I_{BI} \left( R_\text{f} \parallel R_1 \right) \right)</math>

where <math>V_{IOS}</math> is the input offset voltage and <math>I_{BI}</math> is the input bias current on the inverting terminal. <math>R_f \parallel R_1</math> indicates two resistance values in parallel.

To negate the effect of the input bias current, set <math>R_{\text{on}}=R_1 || R_f || R_L</math>. The error voltage then becomes:

:<math>V_\text{E} = \left( \frac{R_\text{f}}{R_1} + 1 \right) V_{IOS}</math>

The input bias current causes the same voltage drops at both the positive and negative terminals.

==Frequency response==
[[File:Frequency response of ideal and practical integrator.png|Frequency response of ideal and practical integrator.]]

The frequency responses of the practical and ideal integrator are shown in the above figure. For both circuits, the crossover frequency <math>f_\text{b}</math>, at which the gain is 0 dB, is given by:

:<math>f_{\text{b}}=\frac{1}{{2\pi}{R_{\text{1}}}{ C_{\text{F}}}}</math>

The 3 dB [[cutoff frequency]] <math>f_\text{a}</math> of the practical circuit is given by:

:<math>f_{\text{a}}=\frac{1}{{2\pi}{R_{\text{F}}}{ C_{\text{F}}}}</math>

The practical integrator circuit is equivalent to an active first-order [[low-pass filter]]. The gain is relatively constant up to the cutoff frequency decreases by 20 dB per decade beyond it. The integration operation occurs for frequencies in the range <math>\left[ f_\text{a}, f_\text{b} \right]</math>, provided that <math>f_\text{a} < f_\text{b}</math>. This condition can be achieved by appropriate choice of <math>R_\text{F}C_\text{F}</math> and <math>R_1 C_\text{F}</math> time constants.

==Applications==
The integrator is mostly used in [[analog computer]]s, [[analog-to-digital converter]]s and wave-shaping circuits.

==References==
*[http://www.mycircuits9.com/2012/12/op-amp-as-integrator-tutorial-with-design-derivation.html OP AMP Integrator Complete Tutorial]

{{Reflist|refs=
<ref name="microchip-opa-dc">{{cite web
| title = AN1177 Op Amp Precision Design: DC Errors
| publisher = Microchip
| date = 2 January 2008
| url = http://ww1.microchip.com/downloads/en/AppNotes/01177a.pdf
| archiveurl = http://www.webcitation.org/6Db011SaW
| archivedate = 2013-01-11
| deadurl = no
| format = PDF
| accessdate = 26 December 2012}}</ref>
}}

[[Category:Analog circuits]]

Actuarial polynomials - Revision history

en>Headbomb: Various citation cleanup and WP:AWB general fixes, added orphan tag using AWB