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.
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.
[1] This circuit operates by passing a current that charges or discharges the capacitor
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.
through the resistor producing a compensating current flow through the series capacitor to maintain the virtual ground.
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: For an ideal op-amp,
amps, so: Furthermore, the capacitor has a voltage-current relationship governed by the equation: Substituting the appropriate variables: For an ideal op-amp,
volts, so: Integrating both sides with respect to time: If the initial value of
Real op-amps have a finite open-loop gain, an input offset voltage
, both the output offset voltage and the input bias current
To negate the effect of the input bias current, it is necessary for the non-inverting terminal to include a resistor
Well-matched input bias currents then cause the same voltage drop of
Also, in a DC steady state, an ideal capacitor acts as an open circuit.
Any DC (or very low frequency) component may then cause the op amp output to drift into saturation.
[3] To prevent this, the DC gain can be limited to a finite value by inserting a large resistor
Note that some op amps have a large internal feedback resistor, and many real capacitors have leakage that is effectively a large feedback resistor.
[4] The addition of these resistors turns the output drift into a finite, preferably small, DC error voltage: Notes on offset: a variation of this circuit simply uses an adjustable voltage source instead of
[5] Offset correction is a bigger concern for older op amps, particularly BJT types.
Another variation circuit to avoid offset correction that works for AC signals only is to capacitively-couple the input with large input capacitor before
Additionally, because offset may drift over time and temperature, some op amps provide null offset pins, which can be connected to a potentiometer whose wiper connects to the negative supply to allow readjusting when conditions change.
corresponds to the time-domain integration property of the Laplace transform.
turns the circuit into an active low-pass filter with a pole at the -3 dB cutoff frequency: The frequency response has a relatively constant gain up to
While this circuit is no longer an integrator for low frequencies around and below
and the response approaches that of an ideal integrator as the frequency increases.
[3] Real op amps also have a limited gain-bandwidth product (GBWP), which adds an additional high frequency pole.