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.

differentiator.jpg

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.

differentiator_freq_domain.jpg

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

differentiator_example.jpg
© 2024 Microchip Technology, Inc.
Notice: ARM and Cortex are the registered trademarks of ARM Limited in the EU and other countries.
Information contained on this site regarding device applications and the like is provided only for your convenience and may be superseded by updates. It is your responsibility to ensure that your application meets with your specifications. MICROCHIP MAKES NO REPRESENTATIONS OR WARRANTIES OF ANY KIND WHETHER EXPRESS OR IMPLIED, WRITTEN OR ORAL, STATUTORY OR OTHERWISE, RELATED TO THE INFORMATION, INCLUDING BUT NOT LIMITED TO ITS CONDITION, QUALITY, PERFORMANCE, MERCHANTABILITY OR FITNESS FOR PURPOSE. Microchip disclaims all liability arising from this information and its use. Use of Microchip devices in life support and/or safety applications is entirely at the buyer's risk, and the buyer agrees to defend, indemnify and hold harmless Microchip from any and all damages, claims, suits, or expenses resulting from such use. No licenses are conveyed, implicitly or otherwise, under any Microchip intellectual property rights.