Differentiators

Interchanging the location of the capacitor and resistor of the integrator circuit results in the differentiator circuit, which performs the mathematical function of differentiation.

Transfer Function
The transfer function of the differentiator circuit is as follows:

(1)
\begin{align} V_{out} = -RC\frac{\mathrm{d} V_{in}}{\mathrm{d} t} \end{align}

Frequency Domain
The frequency response of the differentiator can be thought of as that of a low pass filter with a cut-off frequency of zero.

Example
Design a differentiator that has a time constant of 10 ms and an input capacitance of 0.01 μf. What is the value of R? What is the gain magnitude and phase of this circuit at 10 Hz, and at 100 Hz? In order to limit the high frequency gain of the differentiator circuit to 100, a resistor is added in series with the capacitor. Find the required resistor value.

Given the differentiator transfer function, magnitude |Vout/Vin | = 2 π f R C and Phase φ = -90°

RC = 10 ms, C = 0.01 μf, R = 1 MΩ
At f =100 Hz, |Vout/Vin| = 2 π f R C = 6.28, φ = -90°
At f =1,000 Hz, |Vout/Vin| = 2 π f R C = 62.8, φ = -90°
At high frequency, |Vout/Vin| = R/R1 = 100, R1 = 10 kΩ