Differentiator

In electronics, a differentiator is a circuit that is designed such that the output of the circuit is approximately directly proportional to the rate of change (the time derivative) of the input. An active differentiator includes some form of amplifier. A passive differentiator circuit is made of only resistors and capacitors.

A true differentiator cannot be physically realized, because it has infinite gain at infinite frequency. A similar effect can be achieved, however, by limiting the gain above some frequency. Therefore, a passive differentiator circuit can be made using a simple first-order high-pass filter, with the cut-off frequency set to be far above the highest frequency in the signal. This is a four-terminal network consisting of two passive elements as shown in Figures 1 and 2.

The analysis here is for the capacitive circuit in Figure 1. The inductive case in Figure 2 can be handled in a similar way.

The transfer function shows the dependence of the network gain on the signal frequency for sinusoidal signals.

According to Ohm's law,

where ${\displaystyle X}$ and ${\displaystyle Y}$ are input and output signals' amplitudes respectively, and ${\displaystyle Z_{R}}$ and ${\displaystyle Z_{C}}$ are the resistor's and capacitor's impedances. Therefore, the complex transfer function is

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