Integrators in detail

As we have said, the choice of the feedback element used with an operational amplifier has much to do with the behavior of the circuit. Therefore, we should explore what happens if we use different types of feedback components. In this example, we'll replace the feedback resistor with a capacitor and note the results.

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In the circuit shown to the right, we have replaced the feedback resistor with a capacitor. Therefore, any feedback current must be based on a change in output voltage. As feedback current flows, the capacitor will gain an electric charge, which will change according to the cumulative effects of the output signal.

If the input voltage is zero, no input current will flow. Therefore no feedback current can flow and the output voltage will remain constant. If the input voltage is non-zero, the basic equation for the output voltage becomes V_out = -V_in/RC + K, where R is the input resistance in ohms, C is the feedback capacitance in farads, and K is a fixed constant representing the accumulated voltage from the past.

If the input voltage is constantly changing, the output voltage at any instant will be the integral of all past input voltage values. For example, a bipolar sine wave input will actually produce another sine wave as its output, at a phase angle of 90° from the input sine wave. Technically, the output will be an inverted cosine wave.

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A couple of factors are of interest with these circuits:

1. If the input is a constant positive dc voltage, the output will be a negative linear ramp. There is no exponential factor in an op amp integrator. The equation for the ramp will be V_out = -V_int/RC, where t is time in seconds.
2. In an analog computer, an "initial condition" can be applied as a starting voltage on the capacitor, at the beginning of the integration process.
3. The integrator has an automatic and natural tendency to damp out any high-frequency noise that may appear in the input signal.
4. It is essential to avoid any long-term dc offset in the input voltage; if such an offset is present, it will cause the output voltage to gradually shift toward one extreme or the other, and stay there. In an analog computer, such an offset problem is avoided by limiting the time during which the integration process is allowed to continue. At the end of that time, the circuit is reset back to its initial conditions before being allowed to repeat the operation.