Op-amp Differentiator

Differentiator

The circuit which produces the differentiation of the input voltage at its output is called differentiator. The differentiator circuit which does not use any active device is called passive differentiator. While the differentiator using an active device like op-amp is called an active differentiator. Let us discuss first the operation of ideal active op-amp differentiator circuit.

 

1. Ideal Active Op-amp Differentiator

The active differentiator circuit can be obtained by exchanging the positions of R and C in the basic active integrator circuit. The op-amp differentiator circuit is shown in the Fig. 2.30.1.

The active differentiator circuit can be obtained by exchanging the positions of R and C in the basic active integrator circuit. The op-amp differentiator circuit is shown in the Fig. 2.30.1.

The node B is grounded. The node A is also at the ground potential hence V_A = 0.

As input current of op-amp is zero, entire current I flows through the resistance Rf. 

From the input side we can write,

The equation shows that the output is C₁R_f times the differentiation of the input and product C₁R_f is called time constant of the differentiator. 

The negative sign indicates that there is a phase shift of 180° between input and output. The main advantage of such an active differentiator is the small time constant required for differentiation.

By Miller's theorem, the effective resistance between input node A and ground becomes R_f / 1 + A_v ≈ R_f / A_v where A_v is the gain of the op-amp which is very large. Hence effective Rf becomes very very small and hence the condition R_f C₁ << T gets satisfied at all the frequencies.

In practice a resistance R_comp = R_f  is connected to the non-inverting terminal to provide the bias compensation. This is shown in the Fig. 2.30.2.

 

2. Input and Output Waveforms

Let us study the output waveforms, for various input signals.

For simplicity of understanding, assume that the values of R_f and C₁ are selected to have time constant (R_fC₁) as unity. 

 i) Step input signal

Let the input waveform is of step type with a magnitude of A units. Mathematically it is expressed as,

V_in (t) = A for t ≥ 0 ... (2.30.5)

Now mathematically, the output of the differentiator must be,

Vo (t) = - dV_in / dt = - d(A) / dt = 0 …. (2.30.6)

This is because A is constant.

Actually the step input takes a finite time to rise from 0 to A volts.

Due to this finite time, the differentiator output is not zero but appears in the form of a spike at t = 0.

As the circuit acts as an inverting differentiator, the negative going spike or impulse appears at t = 0 and after that output remains zero.

Both input and output waveforms of the differentiator with a step input, are shown in the Fig. 2.30.3.

ii) Square wave input signal

Input and output for square wave input

The square wave is made of steps i.e. step of A volts from t = 0 to t = T/2, while a step of -A volts from t = T/2tot = T and so on.

Mathematically it can be expressed as,

V_in (t) = A   0 < t < T/2

= - A T/2 < t < T  … (2.30.7)

The differentiator behaves similar to its behaviour to step input.

For positive going impulse, the output shows negative going impulse and for negative going input, the output shows positive going impulse.

Hence the total output for the square wave input is in the form of train of impulses or spikes.

The input and output waveforms are shown in the Fig. 2.30.4

iii) Sine wave input

Let the input waveform be purely

sinusoidal with a frequency of ω rad/sec. Mathematically it can be expressed as,

V_in(t) = V_m sin ωt …. (2.30.8)

where V_m is the amplitude of the sine wave and T is the period of the waveform. Let us find out the expression for the output.

and so on.

Thus the output of the differentiator is a cosine waveform, for a sine wave input. The input and output waveform is shown in the Fig. 2.30.5.

 

3. Disadvantages of an Ideal Differentiator

The gain of the differentiator increases as frequency increases. Thus at some high frequency, the differentiator may become unstable and break into the oscillations. There is possibility that op-amp may go into the saturation.

Also the input impedance X_c1= (1 /2π f C₁) decreases as frequency increases. This makes the circuit very much sensitive to the noise. Thus when such noise gets amplified due to high gain at high frequency, noise may completely override the differentiated output.

Hence the differentiator circuit suffers from the limitations on its stability and noise problems, at high frequencies. These problems can be corrected using some additional parameters in the basic differentiator circuit. Such a differentiator circuit is called

 

4. Practical Differentiator

The noise and stability at high frequency can be corrected, in the practical differentiator circuit using  the resistance R₁ in series with C1 and the capacitor C_f in parallel with resistance R_f.

The circuit is shown in the Fig. 2.30.6. The resistance R_comp is used for bias compensation.

 

5. The Analysis of the Practical Differentiator

As the input current of op-amp is zero, there is no current input at node B. Hence it is at the ground potential. From the concept of the virtual ground, node A is also at the ground potential and hence V_B = V_A = 0 V.

For the current I, we can write

I = V_in -V_A / Z₁ = V_in / Z₁          …. (2.30.10)

where Z₁ = R₁ in series with C₁

So in Laplace domain we can write,

The time constant R_fC₁ is much greater than R_fC₁ or R_fC_f  and hence the equation (2.30.20) reduces to,

Thus the output voltage is the RfC 1 times the differentiation of the input.

It may be noted that though R_fC₁ is much larger than R_fC_f  or R₁C₁ it is less than or equal to the time period T of the input, for the true differentiation.

R_fC₁ ≤  T   ….. (2.30.22)

 

6. Frequency Response of Practical Differentiator

To determine the frequency response, let us obtain the expression for the gain of the practical differentiator interms of the frequency.

From the equation (2.30.20) we can write,

Now as R_fC₁ is much larger than R₁C₁ we can write

f_a < f_b  ... (2.30.29)

Hence as frequency increases, the gain increases till f = f_b at a rate of +20 dB/decade. However after f = f_b the gain decreases at a rate of 20 dB/decade. This 40 dB/decade change at f = f_b occurs due to the combination of R₁C₁ and R_fC_f.

So for R_fC₁ << T, the true differentiation results.

The frequency response is shown in the Fig. 2.30.7. Refer Fig. 2.30.7.

Key Point It can be observed from the frequency response that the gain reduces as frequency increases greater than fb- Hence the problem of instability at high frequency gets eliminated.

Also the combination of RiCi and RfCf help to reduce effectively the impact of high frequency noise and offsets. 

It is important to remember that if fc is the Unity Gain Bandwidth (UGB) then the values of f_a and f_b must be selected in such a way that,

F_a < f_b < f_c  ….. (2.30.30)

where f_c is UGB of op-amp in the open loop configuration.

7. Steps to Design Practical Differentiator

By using following steps, a good practical differentiator can be designed :

i) Choose f_a as the highest frequency of the input signal.

ii) Choose C₁ to be less than 1 µF and calculate the value of R_f.

iii) Choose f_b as 10 times f_a which ensures that f_a < f_b

iv) Finally calculate the values of R₁ and C_f from the expression R₁C₁ = R_fC_f.

v) The R_comp can be selected as R₁ || R_f but practically it is almost equal to R₁.

 

8. Applications of Practical Differentiator

The practical differentiator circuits are most commonly used in :

i) In the wave shaping circuits to detect the high frequency components in the input signal.

ii) As a rate-of-change detector in the FM demodulators.

The differentiator circuit is avoided in the analog computers.

 

Example 2.30.1 Design a practical differentiator circuit that will differentiate an input signal with the f_max = 100 Hz.

April-10, Dec.-14, Marks 8 

Solution :   Refer to the steps for design.

But generally Rcomp is selected equal to R1. The designed differentiator circuit is shown in the Fig. 2.30.8.

The frequency response is as shown in the Fig. 2.30.7

Review Questions

1. What are the limitations of an ordinary differentiator? Draw the circuit of practical differentiator and explain.

May-08, Dec.-15, Marks 12

2. Explain the application of op-amp as a differentiator.

May-10,11,16, Dec.-11,12, Marks 4

3. Derive an expression for the output of a practical differentiator.

Dec.-12, Marks 6

4. Explain the application of Op-Amp as differentiator.

May-16, Marks 8