Differentiator Time Constant

The differentiator time constant is a crucial parameter in the analysis of differentiator circuits. It is defined as the time required for the output voltage of a differentiator circuit to reach 63.2% of its steady-state value in response to an input signal. The time constant is determined by the values of the resistor (R) and capacitor (C) used in the circuit.

Expression for Differentiator Time Constant

The differentiator circuit is a high-pass filter that produces an output voltage proportional to the rate of change of the input signal. The circuit consists of a capacitor in series with a resistor, as shown in the figure below.

![Differentiator Circuit](

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Using Kirchhoff's voltage law, we can write the equation for the voltage across the capacitor as:

Vc = -RC(dVin/dt)

where Vc is the output voltage, Vin is the input voltage, and dVin/dt is the rate of change of the input voltage. The negative sign indicates that the output voltage is 180 degrees out of phase with the input voltage.

Taking the Laplace transform of the above equation, we get:

Vc(s) = -RCsVin(s)

where Vc(s) and Vin(s) are the Laplace transforms of the output and input voltages, respectively, and s is the Laplace variable.

The transfer function of the differentiator circuit is given by:

Vc(s)/Vin(s) = -RCs

The magnitude of the transfer function is:

|Vc(s)/Vin(s)| = RCω

where ω is the frequency of the input signal.

The time constant of the differentiator circuit is defined as the reciprocal of the cutoff frequency, which is the frequency at which the magnitude of the transfer function is equal to 1/sqrt(2). Therefore, we can write:

τ = 1/ωc

where c is the cutoff frequency.

For the differentiator circuit, the cutoff frequency is equal to 1/RC. Therefore, the expression for the differentiator time constant is:

τ = RC

Hence, the correct answer is option 'A', i.e., CR.