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.
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, are called differentiators because they 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
Similarly, the transfer function of the inductive differentiator has a zero in the origin and a pole in
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: According to the capacitor's current–voltage relation, this current
as it flows from the input through the capacitor to the virtual ground will be proportional to the derivative of the input voltage: This same current
is converted into a voltage when it travels from the virtual ground through the resistor to the output, according to ohm's law: Inserting the capacitor's equation for
provides the output voltage as a function of the input voltage: Consequently, The op amp's low-impedance output isolates the load of the succeeding stages, so this circuit has the same response independent of its load.
If a square-wave input is applied to a differentiator, then a spike waveform is obtained at the output.
Treating the capacitor as an impedance with capacitive reactance of Xc = 1/2πfC allows analyzing the differentiator as a high pass filter.
Since the feedback configuration provides a gain of Rf/Xc, that means the gain is low at low frequencies (or for slow changing input), and higher at higher frequencies (or for fast changing input).
The transfer function of an ideal differentiator is
, resulting in the Bode plot of its magnitude having a positive +20 dB per decade slope over all frequencies and having unity gain at
A small time constant is sufficient to cause differentiation of the input signal.
At high frequencies: In order to overcome the limitations of the ideal differentiator, an additional small-value capacitor C1 is connected in parallel with the feedback resistor R, which prevents the differentiator circuit from oscillating, and a resistor R1 is connected in series with the capacitor C, which limits the increase in gain to a ratio of R/R1.
Since negative feedback is present through the resistor R, we can apply the virtual ground concept, that is, the voltage at the inverting terminal is the same 0 volts at the non-inverting terminal.
Applying nodal analysis, we get Therefore, Hence, there occurs one zero at
This practical differentiator's frequency response is a band-pass filter with a +20 dB per decade slope over frequency band for differentiation.
A straight-line approximation of its Bode plot when normalized with
), resulting in the following frequency response (normalized using
For the above plot: The differentiator circuit is essentially a high-pass filter.
It can generate a square wave from a triangle wave input and produce alternating-direction voltage spikes when a square wave is applied.
In ideal cases, a differentiator reverses the effects of an integrator on a waveform, and conversely.
Hence, they are most commonly used in wave-shaping circuits to detect high-frequency components in an input signal.
Differentiators are an important part of electronic analogue computers and analogue PID controllers.
They are also used in frequency modulators as rate-of-change detectors.
