# Ideal Differentiator by using Op-amp

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.

The active differentiator circuit can be obtained by exchanging the positions of R and C in the basic integrator circuit.

$\displaystyle I_1 = C_1 \frac {d(V_{in} - V_A)}{dt} = C_1 \frac {dV_{in}}{dt}$

$\displaystyle I = \frac {V_A - V_0}{R_f} = -\frac {V_0}{R_f}$

$\displaystyle V_0 = -\ C_1 R_f \frac {dV_{in}}{dt}$

• Here negative sign shows 180-degree phase difference between input and output voltage.
• Advantages of ideal Differentiator :
1. The main advantage of such ideal differentiator is the small time constant required for differentiation.

By Miller’s theorem, the effective resistance between input node A and ground becomes $\displaystyle \frac {R_f}{1 + A_v}$, where $\displaystyle A_v$ is the gain of Op-amp which is very large.

Hence effective resistance becomes very-very small and hence the condition $\displaystyle R_f C_1 << T$ gets satisfied for all the frequencies.

• Disadvantages of Ideal Differentiator :
1. The gain of the differentiator increases as frequency increases. Thus at some high frequency, the differentiator may become unstable and break into oscillations. There is a possibility that Op-amp may go into saturation.
2. Also, the input impedance decreases as frequency increases. This makes circuit very much sensitive to the noise.

Hence the differentiator circuit suffers from the limitations on its stability and noise problems at high frequencies.

• Examples of Differentiator input and output waveforms :

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