Differentiator

In electronics, a differentiator is a circuit that outputs a signal approximately proportional to the rate of change (i.e. the derivative with respect to time) of its input signal. Because the derivative of a sinusoid is another sinusoid whose amplitude is multiplied by its frequency, a true differentiator that works across all frequencies can't be realized (as its gain would have to increase indefinitely as frequency increase). Real circuits such as a 1st-order high-pass filter are able to approximate differentiation at lower frequencies by limiting the gain above its cutoff frequency. An active differentiator includes an amplifier, while a passive differentiator is made only of resistors, capacitors and inductors.

Passive differentiator

The four-terminal 1st-order passive high-pass filter circuits depicted in figure, consisting of a resistor and a capacitor, or alternatively a resistor and an inductor, approximate differentiation at frequencies well-below each filter's cutoff frequency.

According to Ohm's law, the voltages at the two ends of the capacitive differentiator are related by the following transfer function (which has a zero in the origin and a pole at

s

=

-

1

R

C

{\displaystyle s{=}{\tfrac {{\text{-}}1}{RC}}}

):

Y

=

Z

R

Z

R

+

Z

C

X

=

R

R

+

1

s

C

X

=

s

R

C

1

+

s

R

C

X

⟹

Y

≈

s

R

C

X

for

|

s

|

≪

1

R

C

{\displaystyle Y={\frac {Z_{R}}{Z_{R}+Z_{C}}}X={\frac {R}{R+{\frac {1}{sC}}}}X={\frac {sRC}{1+sRC}}X\implies Y\approx sRCX\quad {\text{for}}\ |s|\ll {\frac {1}{RC}}}

which is a good approximation of an ideal differentiator at frequencies well below the filter's cutoff frequency of

1

2

π

R

C

{\displaystyle {\tfrac {1}{2\pi RC}}}

in hertz or

1

R

C

{\displaystyle {\tfrac {1}{RC}}}

in radians.

Similarly, the transfer function of the inductive differentiator has a zero in the origin and a pole in

s

=

-

R

L

{\displaystyle s{=}{\tfrac {{\text{-}}R}{L}}}

, corresponding to a cutoff frequency of

R

2

π

L

{\displaystyle {\tfrac {R}{2\pi L}}}

in hertz or

R

L

{\displaystyle {\tfrac {R}{L}}}

in radians.

Active differentiator

Ideal differentiator

A differentiator circuit (also known as a differentiating amplifier or inverting differentiator) consists of an ideal operational amplifier with a resistor R providing negative feedback and a capacitor C at the input, such that:

V

in

{\displaystyle V_{\text{in}}}

is the voltage across C (from the op amp's virtual ground negative terminal).

V

out

{\displaystyle V_{\text{out}}}

is the voltage across R (also from the op amp's virtual ground negative terminal).