The Integrator Amplifier

In the previous tutorials we have seen circuits which show how an operational amplifier can be used as part of a positive or negative feedback amplifier or as an adder or subtractor type circuit using just resistors in both the input and the feedback loop. But what if we were to change the purely Resistive (Rf) feedback element of an inverting amplifier to that of a Frequency Dependant Impedance, (Z) type element, such as a Capacitor, C. We now have a resistor and capacitor combination forming an RC Network across the operational amplifier as shown below.

Integrator Amplifier Circuit

As its name implies, the Integrator Amplifier is an operational amplifier circuit that performs the mathematical operation of Integration, that is we can cause the output to respond to changes in the input voltage over time and the integrator amplifier produces a voltage output which is proportional to that of its input voltage with respect to time. In other words the magnitude of the output signal is determined by the length of time a voltage is present at its input as the current through the feedback loop charges or discharges the capacitor.

When a voltage, Vin is firstly applied to the input of an integrating amplifier, the uncharged capacitor C has very little resistance and acts a bit like a short circuit (voltage follower circuit) giving an overall gain of less than 1, thus resulting in zero output. As the feedback capacitor C begins to charge up, its reactance Xc decreases and the ratio of Zf/Rin increases producing an output voltage that continues to increase until the capacitor is fully charged. At this point the ratio of feedback capacitor to input resistor (Zf/Rin) is infinite resulting in infinite gain and the output of the amplifier goes into saturation as shown below. (Saturation is when the output voltage of the amplifier swings heavily to one voltage supply rail or the other with no control in between).

The rate at which the output voltage increases (the rate of change) is determined by the value of the resistor and the capacitor, "RC time constant". By changing this RC time constant value, either by changing the value of the Capacitor, C or the Resistor, R, the time in which it takes the output voltage to reach saturation can also be changed for example.

If we apply a constantly changing input signal such as a square wave to the input of an Integrator Amplifier then the capacitor will charge and discharge in response to changes in the input signal. This results in the output signal being that of a sawtooth waveform whose frequency is dependant upon the RC time constant of the resistor/capacitor combination. This type of circuit is also known as a Ramp Generator and the transfer function is given below.

Ramp Generator

Since the node voltage of the integrating op-amp at its inverting input terminal is zero, the current Iin flowing through the input resistor is given as:

The current flowing through the feedback capacitor C is given as:

Assuming that the input impedance of the op-amp is infinite (ideal op-amp), no current flows into the op-amp terminal. Therefore, the nodal equation at the inverting input terminal is given as:

From which we have an ideal voltage output for the Integrator Amplifier as:

This can also be re-written as:

Where jω = 2πƒ and the output voltage Vout is a constant 1/RC times the integral of the input voltage Vin with respect to time. The minus sign (-) indicates a 180⁰ phase shift because the input signal is connected directly to the inverting input terminal of the op-amp.

The AC or Continuous Integrator

If we changed the above square wave input signal to that of a sine wave of varying frequency the Integrator Amplifier begins to behave like an active "Low Pass Filter", passing low frequency signals while attenuating the high frequencies. However, at DC (0Hz) the capacitor acts like an open circuit blocking any feedback voltage resulting in zero negative feedback from the output back to the input of the amplifier. Then the amplifier effectively is connected as a normal open-loop amplifier with very high open-loop gain resulting in the output voltage saturating.

The addition of a large value resistor, R2 across the capacitor, C gives the circuit the characteristics of an inverting amplifier with finite closed-loop gain of Rf/Rin at very low frequencies while acting as an integrator at higher frequencies. This then forms the basis of a Active Low Pass Filter as seen before in the filters section tutorials.

The AC Integrator with DC Gain Control