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Differentiator

## Summary

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, are called differentiators because they approximate differentiation at frequencies well-below each filter's cutoff frequency.

Capacitive differentiator

Inductive differentiator

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 ${\displaystyle s{=}{\tfrac {{\text{-}}1}{RC}}}$ ):

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

which is a good approximation of an ideal differentiator at frequencies well below the filter's cutoff frequency of ${\displaystyle {\tfrac {1}{2\pi RC}}}$  in hertz or ${\displaystyle {\tfrac {1}{RC}}}$  in radians.

Similarly, the transfer function of the inductive differentiator has a zero in the origin and a pole in ${\displaystyle s{=}{\tfrac {{\text{-}}R}{L}}}$ , corresponding to a cutoff frequency of ${\displaystyle {\tfrac {R}{2\pi L}}}$  in hertz or ${\displaystyle {\tfrac {R}{L}}}$  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:

• ${\displaystyle V_{\text{in}}}$  is the voltage across C (from the op amp's virtual ground negative terminal).
• ${\displaystyle V_{\text{out}}}$  is the voltage across R (also from the op amp's virtual ground negative terminal).
• ${\displaystyle I}$  is the current flowing from the input through both R and C to the circuit's output.

According to the capacitor's current–voltage relation, this current ${\displaystyle I}$  as it flows from the input through the capacitor to the virtual ground will be proportional to the derivative of the input voltage:

${\displaystyle I=C\,{\frac {dV_{\text{in}}}{dt}}\,.}$

This same current ${\displaystyle I}$  is converted into a voltage when it travels from the virtual ground through the resistor to the output, according to ohm's law:

${\displaystyle 0-V_{\text{out}}=IR\,.}$

Inserting the capacitor's equation for ${\displaystyle I}$  provides the output voltage as a function of the input voltage:

${\displaystyle V_{\text{out}}=-RC{\frac {dV_{\text{in}}}{dt}}.}$

Consequently,

• The output voltage is proportional to the time derivative of the input voltage with a gain of ${\displaystyle RC.}$  Hence, the circuit acts as a differentiator and amplifier.
• The negative sign indicates the output has a 180° phase shift (inversion) with respect to the input.
• The equation is true for any frequency signal, assuming an ideal op amp (though a real op-amp has limited bandwidth).

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 constant DC voltage is applied as input, the output voltage is zero. If the input voltage changes from zero to negative, the output voltage is positive. If the applied input voltage changes from zero to positive, the output voltage is negative. If a square-wave input is applied to a differentiator, then a spike waveform is obtained at the output.

#### Operation as high pass filter

Treating the capacitor as an impedance with capacitive reactance of Xc = 1/2πfC allows analyzing the differentiator as a high pass filter. The inverse-proportionality to frequency means that at low frequency, the reactance of a capacitor is high, and at high frequency reactance is low. 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).

#### Frequency response

The transfer function of an ideal differentiator is ${\displaystyle {\tfrac {V_{\text{out}}}{V_{\text{in}}}}={\text{-}}sRC}$ , resulting in the Bode plot of its magnitude having a positive +20 dB per decade slope over all frequencies and having unity gain at ${\displaystyle f_{\text{0dB}}{=}{\tfrac {1}{2\pi RC}}\,.}$

A small time constant is sufficient to cause differentiation of the input signal.

#### Limitations

At high frequencies:

• this simple differentiator circuit becomes unstable and starts to oscillate;
• the circuit becomes sensitive to high frequency noise that, when amplified, dominates the input signal.
• the limited gain–bandwidth product of real op amps will put an upper frequency limit for differentiation

### Practical differentiator

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

${\displaystyle {\frac {0-V_{o}}{R}}+{\frac {0-V_{o}}{\frac {1}{sC_{1}}}}+{\frac {0-V_{i}}{R_{1}+{\frac {1}{sC}}}}=0,}$
${\displaystyle -V_{o}\left({\frac {1}{R}}+sC_{1}\right)={\frac {V_{i}}{R_{1}+{\frac {1}{sC}}}}.}$

Therefore,

${\displaystyle {\frac {V_{o}}{V_{i}}}={\frac {-sRC}{(1+sR_{1}C)(1+sRC_{1})}}.}$

Hence, there occurs one zero at ${\displaystyle s{=}0}$  and one pole at ${\displaystyle s{=}{\tfrac {{\text{-}}1}{R_{1}C}}}$  (corresponding to a corner frequency of ${\displaystyle f_{1}{=}{\tfrac {1}{2\pi R_{1}C}}}$ ) and another pole at ${\displaystyle s{=}{\tfrac {{\text{-}}1}{RC_{1}}}}$  (corresponding to a corner frequency of ${\displaystyle f_{2}{=}{\tfrac {1}{2\pi RC_{1}}}}$ ).

#### Frequency response

This practical differentiator's frequency response is a band-pass filter with a +20 dB per decade slope over frequency band for differentiation. Its Bode plot when normalized with ${\displaystyle R_{1}C{=}10^{1}}$  and ${\displaystyle RC_{1}{=}10^{{\text{-}}1}}$  is:

From the above plot, it can be seen that:

• Below ${\displaystyle \omega _{1}}$ , the circuit attenuates and acts as a differentiator.
• Between ${\displaystyle \omega _{1}}$  and ${\displaystyle \omega _{2}}$ , the circuit acts as a voltage follower or buffer.
• Above ${\displaystyle \omega _{2}}$ , the circuit attenuates and acts as an integrator.

Setting ${\displaystyle RC_{1}{=}R_{1}C{=}RC}$  will produce one zero at ${\displaystyle s{=}0}$  and two poles at ${\displaystyle s{=}{\tfrac {{\text{-}}1}{RC}}}$  (corresponding to one corner frequency of ${\displaystyle \omega _{1}{=}{\tfrac {1}{RC}}}$ ), resulting in the following frequency response (normalized using ${\displaystyle RC{=}1}$ ):

From the above plot, we observe that:

• Below ${\displaystyle \omega _{1}}$ , the circuit acts as a differentiator;
• Above ${\displaystyle \omega _{1}}$ , the circuit acts as an integrator.

### Applications

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.

A passive differentiator circuit is one of the basic electronic circuits, being widely used in circuit analysis based on the equivalent circuit method.