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_\text{in} is the voltage across
C (from the op amp's
virtual ground negative terminal). • V_\text{out} is the voltage across
R (also from the op amp's virtual ground negative terminal). • I is the current flowing from the input through both
R and
C to the circuit's output. • No current flows into the ideal op amp's inputs because they have
very high input impedance. According to the
capacitor's current–voltage relation, this current I as it flows from the input through the capacitor to the virtual ground will be proportional to the derivative of the input voltage: :I = C \, \frac{dV_\text{in}}{dt} \, . This same current I is converted into a voltage when it travels from the virtual ground through the resistor to the output, according to
ohm's law: :0 - V_\text{out} = IR \, . Inserting the capacitor's equation for I provides the output voltage as a function of the input voltage: :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 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 = 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 , 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 \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 f_\text{0dB} {=} \tfrac{1}{2\pi RC} \, .
Advantages 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 . 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 :\frac{0 - V_o}{R} + \frac{0 - V_o}{\frac{1}{sC_1}} + \frac{0 - V_i}{R_1 + \frac{1}{sC}} = 0, :-V_o \left(\frac{1}{R} + sC_1\right) = \frac{V_i}{R_1 + \frac{1}{sC}}. Therefore, :\frac{V_o}{V_i} = \frac{-sRC}{(1 + sR_1C)(1 + sRC_1)}. Hence, there occurs one zero at s {=} 0 and one pole at s {=} \tfrac{\text{-}1}{R_1C} (corresponding to a corner frequency of f_1 {=} \tfrac{1}{2\pi R_1C}) and another pole at s {=} \tfrac{\text{-}1}{RC_1} (corresponding to a corner frequency of 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. A
straight-line approximation of its Bode plot when normalized with R_1 C {=} 10^{1} and R C_1 {=} 10^{\text{-}1} is: For the above plot: • Below \omega_1, the circuit attenuates, and well below \omega_1 acts like a differentiator. • Between \omega_1 and \omega_2, the circuit acts as a
voltage follower or buffer. • Above \omega_2, the circuit attenuates, and well above \omega_2 acts like an
integrator. Setting RC_1 {=} R_1C {=} RC will produce one zero at s {=} 0 and two poles at s {=} \tfrac{\text{-}1}{RC} (corresponding to one corner frequency of \omega_1 {=} \tfrac{1}{RC}), resulting in the following frequency response (normalized using RC {=} 1): For the above plot: • Well below \omega_1, the circuit acts like a differentiator; • Well above \omega_1, the circuit acts like 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. ==See also==