In this chapter we have examined the structure and use of integrator and differentiator circuits. Integrators produce a summing action whereas differentiators find the slope of the input. Both types are based on the general parallel-parallel inverting voltage feedback model. In order to achieve integration and differentiation, a capacitor is used in the feedback network in place of the standard resistor. Because the capacitor current is proportional to the rate of change of the capacitor's voltage, a differential or integral response is possible. For the integrator, the capacitor is placed in the normal \(R_f\) position, and for the differentiator, the capacitor is placed in the normal \(R_i\) position. As a result, the integrator exhibits a −6 dB per octave slope through the useful integration range. The differentiator exhibits the mirror image, or +6 dB per octave slope, throughout its useful range.
In both circuits, practical limitations require the use of additional components. In the case of the integrator, small DC offsets at the input can force the output into saturation. To avoid this, a resistor is placed in parallel with the integration capacitor in order to limit the low frequency gain. This has the unfortunate side effect of limiting the useful integration range to higher frequencies. In the case of the differentiator, noise, stability, and input impedance limits can pose problems. In order to minimize noise and aid in stability, a small capacitor may be placed in parallel with \(R_f\). This reduces the high frequency gain. In order to place a lower limit on the input impedance, a resistor may be placed in series with the differentiation capacitor. The addition of either component will limit the upper range of differentiation.
We examined two general techniques for determining the output signal. The first form is referred to as the time-continuous method and although it may be used with virtually any waveform, our use was with simple sine waves only. In the case of the integrator, it corresponds to the indefinite integral - a time-domain Equation for the output is the result. The second form is the time-discrete method and is useful for waves that may be easily broken into segments. Each segment is analyzed, and the results joined graphically. This has the advantage of an immediate graphical result, whereas the shape of the time-continuous result may not be immediately apparent. The time-discrete method is also useful for producing output tables and graphs with a digital computer. In the case of the integrator, the time-discrete method corresponds to the definite integral.
Integrators and differentiators may be used in combination with summers and amplifiers to form analog computers. Analog computers may be used to model a variety of physical systems in real time. Unlike their digital counterparts, programming an analog computer only requires the proper interconnection of the various building blocks and appropriate settings for the required physical constants.
Finally, it is noted that integrators and differentiators may be used as wave shaping circuits. Integrators may be used to turn square waves into triangle waves. Differentiators can be used to turn triangle waves into square waves. Indeed, as differentiators tend to “seek” rapid changes in the input signal, they can be quite useful as edge detectors.