The operational amplifier integrator is an electronic integration circuit. Based on the operational amplifier (op-amp), it performs the mathematical operation of integration with respect to time; that is, its output voltage is proportional to the input voltage integrated over time.
Applications The typical circuit, and its common name, is a form of the Miller integrator, but based on an op-amp. The circuit is mostly used in analog computers, analog-to-digital converters and wave-shaping circuits. A common wave-shaping use is as a charge amplifier and they are usually constructed using an operational amplifier though they can use high gain discrete transistor configurations.
Design The input current is offset by a negative feedback current flowing in the capacitor, which is generated by an increase in output voltage of the amplifier. The output voltage is therefore dependent on the value of input current it has to offset and the inverse of the value of the feedback capacitor. The greater the capacitor value, the less output voltage has to be generated to produce a particular feedback current flow. The input capacitance of the circuit is almost zero because of the Miller effect. This ensures that the stray capacitances (the cable capacitance, the amplifier input capacitance, etc.) are virtually grounded and have no influence on the output signal.
Ideal circuit This circuit operates by passing a current that charges or discharges the capacitor C F {\displaystyle C_{\text{F}}} during the time under consideration, which strives to retain the virtual ground condition at the input by off-setting the effect of the input current:
Referring to the above diagram, if the op-amp is assumed to be ideal, then the voltage at the inverting (-) input is held equal to the voltage at the non-inverting (+) input as a virtual ground. The input voltage passes a current V in / R 1 {\displaystyle V_{\text{in}}/{R_{1}}} through the resistor producing a compensating current flow through the series capacitor to maintain the virtual ground. This 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, so a linear integration that works across all frequencies is achieved (unlike RC circuit § Integrator). The circuit can be analyzed by applying Kirchhoff's current law at the inverting input:
i 1 = I B + i F {\displaystyle i_{\text{1}}=I_{\text{B}}+i_{\text{F}}}
For an ideal op-amp, I B = 0 {\displaystyle I_{\text{B}}=0} amps, so:
i 1 = i F {\displaystyle i_{\text{1}}=i_{\text{F}}}
Furthermore, the capacitor has a voltage-current relationship governed by the equation:
i F = C F d ( V 2 − V o ) d t {\displaystyle i_{\text{F}}=C_{\text{F}}{\frac {d(V_{\text{2}}-V_{\text{o}})}{dt}}}
Substituting the appropriate variables:
V in − V 2 R 1 = C F d ( V 2 − V o ) d t {\displaystyle {\frac {V_{\text{in}}-V_{\text{2}}}{R_{\text{1}}}}=C_{\text{F}}{\frac {d(V_{\text{2}}-V_{\text{o}})}{dt}}}
For an ideal op-amp, V 2 = 0 {\displaystyle V_{2}=0} volts, so:
V in R 1 = − C F d V o d t {\displaystyle {\frac {V_{\text{in}}}{R_{\text{1}}}}=-C_{\text{F}}{\frac {dV_{\text{o}}}{dt}}}
Integrating both sides with respect to time:
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