**Introduction**

An op-amp differentiator or a differentiating amplifier is a circuit configuration which produces output voltage amplitude that is proportional to the rate of change of the applied input voltage.

A differentiator with only RC network is called a passive differentiator, whereas a differentiator with active circuit components like transistors and operational amplifiers is called an active differentiator. Active differentiators have higher output voltage and much lower output resistance than simple RC differentiators.

An op-amp differentiator is an inverting amplifier, which uses a capacitor in series with the input voltage. Differentiating circuits are usually designed to respond for triangular and rectangular input waveforms. For a sine wave input, the output of a differentiator is also a sine wave, which is out of phase by 180^{o} with respect to the input (cosine wave).

Differentiators have frequency limitations while operating on sine wave inputs; the circuit attenuates all low frequency signal components and allows only high frequency components at the output. In other words, the circuit behaves like a high-pass filter.

**Ideal Op-Amp Differentiator Circuit**

An op-amp differentiating amplifier uses a capacitor in series with the input voltage source, as shown in the figure below.

For DC input, the input capacitor C_{1} remains uncharged and behaves like an open-circuit. The non-inverting input terminal of the op-amp is connected to ground through a resistor R_{comp}, which provides input bias compensation, and the inverting input terminal is connected to the output through the feedback resistor R_{f}. Thus, the circuit behaves like a voltage follower.

When the input is a positive-going voltage, a current *I* flows into the capacitor C_{1}, as shown in the figure. Since the current flowing into the op-amp internal circuit is zero, effectively all of the current *I* flows through the resistor R_{f}. The output voltage is,

**V _{out} = – (I x R_{f})**

Here this output voltage is directly proportional to the rate of change of the input voltage.

From the figure, node X is virtually grounded and node Y is also at ground potential.

i.e. V_{x} = V_{y} = 0

From the input side, the current I can be given as,

I = C_{1} [d(V_{in}-V_{x})/dt] = C_{1} [d(V_{in})/dt]

From the output side, the current I is given as,

I = -{(V_{out}-V_{x})/R_{f}} = -{V_{out}/R_{f}}

Equating the above two equations of current we get

C_{1}{d(V_{in})/dt} = -V_{out}/R_{f}

**V _{out} = -C_{1}.R_{f }{d(V_{in})/dt}**

Above equation indicates that the output is C_{1}.R_{f} times the differentiation of the input voltage. The product C_{1}.R_{f} is called as the RC time constant of the differentiator circuit. The negative sign indicates the output is out of phase by 180^{o} with respect to the input.

The main advantage of such an active differentiating amplifier circuit is the small time constant required for differentiation.

**Input and Output Waveforms**

When a step input with amplitude V_{m} is applied to an op-amp differentiator, the output can be mathematically expressed as,

**V _{out} = – C_{1}.R_{f} {d(V_{m})/dt}**

For simplicity, assume the product (C_{1}.R_{f}) is unity.

Therefore, V_{out }= 0 because the amplitude V is constant.

But practically, the output is not zero since the input step wave takes a finite amount of time to rise from 0 volts to V_{m} volts. Hence the output appears like a spike at time t = 0, as shown in the figure below.

If the input to the differentiator is changed to a square wave, the output will be a waveform consisting of positive and negative spikes, corresponding to the charging and discharging of the capacitor, as shown in the figure below.

For sine wave input, which is mathematically represented as V (t) = V_{m} sin ωt, where V_{m} is the amplitude of the input signal and t is the period, the output of the differentiator is given as,

**V _{out} = -C_{1}.R_{f} {d(V_{m} sin ωt)/dt}**

For simplicity, let us assume the product (C_{1}.R_{f}) is unity.

**V _{out }= – V_{m}. ω. cos ωt**

Thus the output of a differentiator for a sine wave input is a cosine wave and the input-output waveforms are shown in the figure below.

**Frequency Response of Ideal Differentiator**

The gain of an op-amp differentiator is directly dependent on the frequency of the input signal. Hence, for DC inputs where f = 0, the output is also zero. As the frequency of the input signal increases, the output also increases. The frequency response of an ideal differentiator is as shown in the figure below.

The frequency f_{1} is the frequency for which the gain of the differentiator becomes unity. It can be seen from the figure that for frequency less than f_{1}, the gain is less than unity. For f_{1}, the gain becomes the unity (0 dB) and beyond f_{1}, the gain increases at 20dB per decade.

**Practical Op-amp Differentiator Circuit**

For an ideal differentiator, the gain increases as frequency increases. Thus, at some higher frequencies, the differentiator may become unstable and cause oscillations which results in noise.

These problems can be avoided or corrected in a practical differentiator circuit which uses a resistor R1 in series with the input capacitor and a capacitor Cf in parallel with the feedback resistor, as shown in the figure below.

The output voltage of the practical op-amp differentiating amplifier circuit is given as,

**V _{out} = -R_{f}.C_{1} {d(V_{in})/dt}**

i.e. the output voltage is R_{f}.C_{1} times the differentiation of the input voltage.

The addition of resistor R_{1} and capacitor C_{f} stabilizes the circuit at higher frequencies, and also reduces the effect of noise on the circuit.

**Frequency Response of Practical Differentiator**

The gain of the practical differentiator increases with increasing frequency and at a particular frequency, f_{1}, the gain becomes the unity (0 dB). The gain continues to increase at a rate of 20dB per decade till the input frequency reaches a frequency, f_{2}.

Beyond this frequency of the input signal, the gain of the differentiator starts to decrease at a rate of 20dB per decade. This effect is due to the addition of the resistor R_{1} and capacitor C_{f}. The frequency response curve of a practical differentiator is as shown in the figure below.

**Applications of Op-amp Differentiator**

- Differentiating amplifiers are most commonly designed to operate on triangular and rectangular signals.
- Differentiators also find application as wave shaping circuits, to detect high frequency components in the input signal.

**Op-amp Differentiator Summary**

An op-amp differentiating amplifier is an inverting amplifier circuit configuration, which uses reactive components (usually a capacitor than inductor). The differentiator performs mathematical differentiation operation on the input signal with respect to time, i.e. the output voltage is proportional to the rate of change of the input signal.

Differentiating circuits are commonly used to operate on triangular and rectangular signals. While operating on sine wave inputs, differentiating circuits have frequency limitations.

dinesh devireddy says

good explanation

Raj Kumar Singh says

What is the angle of gain (Vo/Vin)???