Differentiator using Op Amp:

A Differentiator using Op Amp circuit that performs the mathematical operation of differ­entiation. It produces an output voltage proportional to slope of the input volt­age. A common application of a differentiator is the detection of the leading and trailing edges of rectangular pulse.

Figure 14.11 shows a basic CR differentiator. If a rectangular pulse is applied, the output of the circuit is positive and negative spikes. The positive spike occurs at the same instant as the leading edge of the input and the negative spike at the same instant as the trailing edge.

When the input voltage changes from 0 to V, the capacitor begins to charge exponentially, and the output voltage suddenly jumps from 0 to V (after 5 time constants, the capacitor voltage is within 1% of its final voltage V) and then decays exponentially.

Over the trailing edge of a pulse, the input voltage steps negatively and we get a negative spike. For a CR Differentiator to produce narrow spikes the time constant must be 10 times smaller than period T.

In Fig. 14.12 (b), when the input voltage varies, the capacitor charges or discharges. Because of the virtual ground, the capacitor current passes through the feedback resistor, producing a voltage. This voltage is proportional to the slope of the input voltage.

A rectangular input, shown in Fig. 14.12 (a), is used to obtain positive and negative spikes. The leading edge of the pulse is approximately a positive ramp, so that the output is a negative spike of very short duration. Similarly, the trailing edge of the input pulse is approximately negative ramp, so that the out­put is a very narrow positive spike. Similarly, if a triangular waveform is given as input, a square waveform is obtained at the output.

A basic differentiator using Op Amp is constructed from a basic inverting amplifier by replac­ing the input resistor R₁ by a capacitor C₁ as shown in Fig. 14.13. Using Kirchoff’s current equation at node V₂ is I_c = I_b + I_f. Since I_b ≡ 0,

But

and

As

But V₁ = V₂ = 0, because A is very large therefore

From the above equation it can be seen that a differentiator performs the opposite function of integration. If a cosine wave is applied as the input, a sine wave will be produced at the output, and a triangular wave as the input will produce a square wave at the output.

The Differentiator using Op Amp of Fig. 14.13 has a tendency to oscillate which is undesirable.

The gain R_F/X_C1 of the circuit increases at a rate of 20 db/decade with in­creases in frequency, which makes the circuit unstable. As the frequency in­creases, the input impedance (X_C1) decreases, making the circuit more suscep­tible to high frequency noise. This noise is superimposed after amplification, on the differentiated output signal. This high frequency noise and stability can be corrected by the use of R₁ and C_F, as shown in Fig. 14.14, which is a practical differentiator.

A typical value for R₁ is between 0.01R_F and 0.1R_F. With this the closed loop voltage gain is reduced (to between -10 and -100).

The differentiating circuit is most commonly used in wave shaping circuits to detect high frequency components in an input signal and as a rate of change detector in FM modulators. It is also used in triggering the time base generator in an oscilloscope.

To design a practical differentiator using Op Amp the following parameters are required.

(a) f_a is the highest frequency range to be differentiated, and is given by f_a = 1/(2πR_FC₁).

(b) f_b is the gain limiting frequency at which the gain begins to decrease at a rate of 20 db/decade. This frequency is given by f_b = 1/2πR₁C₁, where R₁C₁ = R_FC_F and f_b = 20 f_a.