Op amp integrator

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 integrator 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 impedance of the circuit is almost zero because of the Miller effect. Hence all the stray capacitances (the cable capacitance, the amplifier input capacitance, etc.) are virtually grounded and they have no influence on the output signal.
Ideal circuit.
This circuit operates by passing a current that charges or discharges the capacitor formula_1 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 formula_2 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 ).
The circuit can be analyzed by applying Kirchhoff's current law at the inverting input:
For an ideal op-amp, formula_4 amps, so:
Furthermore, the capacitor has a voltage-current relationship governed by the equation:
Substituting the appropriate variables:
For an ideal op-amp, formula_8 volts, so:
Integrating both sides with respect to time:
If the initial value of formula_11 is assumed to be 0 volts, the output voltage will simply be proportional to the integral of the input voltage:
Practical circuit.
This practical integrator attempts to address a number of flaws of the ideal integrator circuit:
Real op-amps have a finite open-loop gain, an input offset voltage formula_13 and input bias currents formula_14, which may not be well-matched and may be distinguished as formula_15 going into the inverting input and formula_16 going into the non-inverting input. This can cause several issues for the ideal design; most importantly, if formula_17, both the output offset voltage and the input bias current formula_15 can cause current to pass through the capacitor, causing the output voltage to drift over time until the op-amp saturates. Similarly, if formula_19 were a signal centered about zero volts (i.e. without a DC component), no drift would be expected in an ideal circuit, but may occur in a real circuit.
To negate the effect of the input bias current, it is necessary for the non-inverting terminal to include a resistor formula_20 which simplifies to formula_21 provided that formula_21 is much smaller than the load resistance formula_23 and the feedback resistance formula_24. Well-matched input bias currents then cause the same voltage drop of formula_25 at both the inverting and non-inverting terminals, to effectively cancel out the effect of bias current at those inputs.
Also, in a DC steady state, the capacitor acts as an open circuit. The DC gain of the ideal circuit is therefore infinite (or in practice, the open-loop gain of a non-ideal op-amp). To counter this, a large resistor formula_26 is inserted in parallel with the feedback capacitor. This limits the DC gain of the circuit to a finite value.
The addition of these resistors turns the output drift into a finite, preferably small, DC error voltage:
Frequency response.
The frequency responses of the practical and ideal integrator are shown in the above figure. For both circuits, the crossover frequency formula_28, at which the gain is 0 dB, is given by:
The 3 dB cutoff frequency formula_30 of the practical circuit is given by:
The practical integrator circuit is equivalent to an active first-order low-pass filter. The gain is relatively constant up to the cutoff frequency and decreases by 20 dB per decade beyond it. The integration operation occurs for frequencies in the range formula_32, provided that formula_33 (i.e., formula_34). This condition can be achieved by appropriate choice of formula_35 and formula_36 time constants.