From a practical standpoint, the principal difficulty with the differcntiator is that it effectively amplifies an input in direct proportion to its frequency amiiereforc i””Il:ases the level of high-frequency noise in the output. Unlike the integrator, which “smooths” a signal by reducing the amplitude of high-frequency components, the differentiator intensifies the contamination of a signal by high-frequency noise.

For this reason, differentiators are rarely used in applications requiring high precision, such as analog computers. In a practical diffcrcntiator, the amplification of signals in direct proportion to their frequencies cannot continue indefinitely as frequency increases, because the amplifier has a finite bandwidth. As we have already discussed in Chapter 13, there is some frequency at which the output amplitude must begin to fall off. Nevertheless, it is often desirable to design a practical differentiator so that it will have a break frequency even lower than that determined by the upper cutoff frequency of the amplifier, that is, to roll off its gain characteristic at some relatively low frequency. This action is accomplished in a practical differentiator by connecting a resistor in

series with the input capacitor, as shown in Figure 14-27. We can understand how this modification achieves the stated goal by considering the net impedance of the U1-C combination at low and high frequencies:

At very small values of w, Zu, is dominated by the capacitive reactance component, so the combination is essentially the same as C alone, and differentiator action occurs. At very high values of w, l/wC is negligible, so Zjll is essentially the resistance Rio and the circuit behaves like an ordinary inverting amplifier (with gain RrI R,)le brca), :”C41’jCTi~y f,. Ut, .ind which differentiation no longer 0 curs.

Where Ih is the highest differentiation frequency. Figure 14-28 shows Bode plots for the gain of the ideal and practical differentiators In the low-frequency region where differentiation occurs, note that the gain nses with frequency at the rate of 20 dB/decade. The plot shows that the gain levels off beyond the break frequency In all • i. !i1 falls off at -20 dB/decade beyond the amplifier’s upper cutoff frequency. Recall that the closed-loop bandwidth, or upper cutoff frequency of the amplifier, is given by where f3 in this case is R,,(R1 + RI).

In some applications, where very wide bandwidth operational amplifiers are used; it may be necessary or desirable to roll the frequency response off even faster than that shown for the practical differentiator in Figure 14-28. This can be accomplished by connecting a capacitor Cf ill parallel with the feedback resistor RI. This modification will cause the response to roll off at -20 dB/decade beginning at the break frequency

Example 14-8 shows that increasing the frequency by a factor of 10 causes a decrease in output amplitude by a factor of 10. This familiar relationship implies that a Bode plot for the gain of an idc.il integrator will have slope -20 dB/decade, or -6 dB/octave. Gain magnitude is the ratio of the peak value of the output to the peak value of the input:

This equation clearly shows that gain is inversely proportional to frequency. A Bode plot for the case RI C = 0.001 is shown in Figure 14-21. Because the integrator’s output amplitude decreases with frequency. it is a kind of low-pass filter. It is sometimes called a smoothing circuit. because the amplitudes of high-frequency components in a complex waveform are reduced, thus smoothing the jagged appearance of the waveform. This feature is useful for reducing highfrequency noise in a signal. Integrators arc also used in 1I1l1l1o~computers to obtain real-rime solutions to differential (calculus) equations.

**Practical Integrators**

Although high-quality, precision integrators are constructed as shown in Figure 14.19 for use in low-frequency applications such as analog computers, these applications require high-quality amplifiers with extremely small offset voltages or chopper stabilization. As mentioned earlier, any input offset is integrated as if it were a dc signal input and will eventuaIly cause the amplifier to saturate. To eliminate this problem in practical integrators using general-purpose amplifiers, a resistor is connected in parallel with the feedback capacitor, as shown in Figure 14-22(a). Since the capacitor is an open circuit as far as de is concerned, the integrator responds to dc inputs just as if it were an inverting amplifier. In other words, the de closedloop gain of the integrator is -R,IRI• At high frequencies, the impedance of the capacitor is much smaller than R” so the parallel combination of C and R, is essentially the same as C alone, and signals are integrated as usual. While the feedback resistor in Figure 14-22(a) prevents integration of de inputs, it also degrades the integration of low-frequency signals.

At frequencies where the capacitive reactance of C is comparable in value to R” the net feedback impedance is not predominantly capacitive and true integration does not OCClIr.As a rule of thumb; we can say that satisfactory integration will occur at frequencies much greater than the frequency at which Xc = RI. That is, for integrator action we want defines a break frequency in the’ Bode plot of the practical integrator. As shown in Figure 14-22(b), at frequencies well above It, the gain falls off at the rate of – 20 dB/decade, like that of an ideal integrator, and at frequencies below fet the gain approaches its de value of R” RI•

**Design a practical integrator that**

1. integrates signals with frequencies down to 100 Hz; and

2. produces a peak output of 0.1 V when theinput is a 10-V-peak sine wave having frequency 10 kHz.

Find the de component in the output when there is a +50-mV dc input.

**Solution**. In order to integrate frequencies down to 100 Hz, we require fr « 100 Hz. Let us choose [.. one decade below 100 Hz: [.. = 10 Hz. Then, from equation 14-48a,

In closing our discussion of integrators, we should note that it is possible to scale and integrate several input signals simultaneously, using an arrangement similar to the linear combination circuit studied earlier. Figure 14-24 shows a practical, threeinput integrator that performs the following operation at frequencies above Ie:

1. Design a practical diffcrentiator that will differentiate signals with frequencies up to 200 Hz. The gain at 10 Hz should be 0.1. \

2. If the operational amplifier used in the design has a unity-gain frequency of 1 MHz, what is the upper cutoff frequency of the differentiator?

**Solution**

1. We must select R. and C to produce a break frequency fb that is well above fh = 200 Hz. Let us choose fb = 10 fi, = 2 kHz. Letting C = 0.1 JLF, we have, from equation 14

Use SPICE to verify the design of the differentiator in Example 14-10. Assume the amplifier is ideal.

**Solution**. The SPICE circuit and the input data file. The inverting operational amplifier is modeled like the one in Example 13-14, except that we assume zero output resistance and a (nearly) ideal voltage gain having the very large value 1 x 109•

Figure 14-31(b) shows the frequency response computed by SPICE over therange from 1 Hz to 100 kHz. Since the input signal has default amplitude 1 V, the plot represents values of voltage gain. Notice that the gain at 10 Hz is 0.0999 0.1, as required. The theoretical gain of the ideal differentiator at 200 Hz is cuRIC = (217″x 2(0)(15.9 kU)(O.l “,F) = 1.998

The plot shows that the gain at a frequency close to 200 Hz (199.5 Hz) is 1.983, so we’ conclude that satisfactory differentiation occurs up to 200 Hz. The plot shows that the high-frequency gain (at 100 kHz) is 19.97. At very high frequencies, the reactance of the O.l-“,F capacitor is negligible. Su the gain, for all practical purposes, equals the ratio of Rf to R1: 15.9 kU1796 U = 19.97 Since the break frequency f” in the design was selected to be 2 kHz, the gain at 2 kHz should be approximately (0.707)(19.97) = 14.11.The plot shows that the gain at a frequency