In
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, 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
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
) of the input. 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. The differentiator circuit is essentially a
high-pass filter
A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
.
An active differentiator includes some form of amplifier, while a passive differentiator is made only of
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
s,
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s and
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s.
Passive differentiator
The simple four-terminal passive circuits depicted in figure, consisting of a
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
and a
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
, or alternatively a resistor and an
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
, behave as differentiators.
Indeed, according to
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
, the voltages at the two ends of the ''capacitive differentiator'' are related by a
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
that has a zero in the origin and a pole in −1/''RC'' and that is consequently a good approximation of an ideal differentiator at frequencies below the natural frequency of the pole:
:
Similarly, the transfer function of the ''inductive differentiator'' has a zero in the origin and a pole in −''R''/''L''.
Active differentiator
Ideal differentiator
![Op-Amp Differentiating Amplifier](https://upload.wikimedia.org/wikipedia/commons/5/5f/Op-Amp_Differentiating_Amplifier.svg)
A differentiator circuit (also known as a differentiating amplifier or inverting differentiator) consists of an
operational amplifier
An operational amplifier (often op amp or opamp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to c ...
in which a
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
''R'' provides
negative feedback
Negative feedback (or balancing feedback) occurs when some function (Mathematics), function of the output of a system, process, or mechanism is feedback, fed back in a manner that tends to reduce the fluctuations in the output, whether caused by ...
and a
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
is used at the input side. The circuit is based on the capacitor's
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
to
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
relationship
:
:
where ''I'' is the
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
through the capacitor, ''C'' is the
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
of the capacitor, and ''V'' is the
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
across the capacitor. The current flowing through the capacitor is then proportional to the derivative of the voltage across the capacitor. This current can then be connected to a resistor, which has the current to voltage relationship
:
where ''R'' is the
resistance of the resistor.
Note that the op-amp input has a very high input impedance (it also forms a
virtual ground due to the presence of negative feedback), so the entire input current has to flow through ''R''.
If ''V''
out is the voltage across the resistor and ''V''
in is the voltage across the capacitor, we can rearrange these two equations to obtain the following equation:
:
From the above equation following conclusions can be made:
* Output is proportional to the time derivative of the input. Hence, the op amp acts as a differentiator.
* Above equation is true for any frequency signal.
* The negative sign indicates that there is 180° phase shift in the output with respect to the input,
Thus, it can be shown that in an ideal situation the voltage across the resistor will be proportional to the derivative of the voltage across the capacitor with a
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
of ''RC''.
Operation
Input signals are applied to the capacitor ''C''. Capacitive
reactance is the important factor in the analysis of the operation of a differentiator. Capacitive reactance is ''X''
c = . Capacitive reactance is inversely proportional to the rate of change of input voltage applied to the capacitor. At low frequency, the reactance of a capacitor is high, and at high frequency reactance is low. Therefore, at low frequencies and for slow changes in input voltage, the gain, , is low, while at higher frequencies and for fast changes the gain is high, producing larger output voltages.
If a constant DC voltage is applied as input, then 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.
The active differentiator isolates the load of the succeeding stages, so it has the same response independent of the load.
Frequency response
The
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
of an ideal differentiator is
, and the
Bode plot of its magnitude is:
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 noise, that is, when amplified, noise dominates the input/message signal.
Practical differentiator
![Practical Differentiator Circuit Diagram](https://upload.wikimedia.org/wikipedia/commons/a/a2/Practical_Differentiator_Circuit_Diagram.png)
In order to overcome the limitations of the ideal differentiator, an additional small-value capacitor ''C''
1 is connected in parallel with the feedback resistor ''R'', which prevents the differentiator circuit from oscillating, and a resistor ''R''
1 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 = voltage at the non-inverting terminal = 0.
Applying nodal analysis, we get
:
:
Therefore,
:
Hence, there occurs one zero at
and two poles at
and
.
Frequency response
![Bode Plot of Practical Differentiator](https://upload.wikimedia.org/wikipedia/commons/8/80/Bode_Plot_of_Practical_Differentiator.png)
From the above plot, it can be seen that:
* when
, the circuit acts as a differentiator;
* when
, the circuit acts as a
voltage follower
A buffer amplifier (sometimes simply called a buffer) is one that provides electrical impedance transformation from one circuit to another, with the aim of preventing the signal source from being affected by whatever currents (or voltages, for a cu ...
or buffer;
* when
, the circuit acts as an
integrator
An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output.
Integration is an importan ...
.
If
(say), there occurs one zero at
and two poles at
.
For such a differentiator circuit, the frequency response would be
![Bode Plot of Practical Differentiator when RC1=R1C](https://upload.wikimedia.org/wikipedia/commons/9/9f/Bode_Plot_of_Practical_Differentiator_when_RC1%3DR1C.png)
From the above plot, we observe that:
* when
, the circuit acts as a differentiator;
* when
, the circuit acts as an
integrator
An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output.
Integration is an importan ...
.
Applications
The differentiator circuit is essentially a
high-pass filter
A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
. It can generate a
square wave
A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions b ...
from a
triangle wave
A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.
Like a square wave, the triangle wave contains only odd harmonics. However, the ...
input and produce alternating-direction voltage spikes when a square wave is applied. In ideal cases, a differentiator reverses the effects of an
integrator
An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output.
Integration is an importan ...
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 computer
An analog computer or analogue computer is a type of computer that uses the continuous variation aspect of physical phenomena such as electrical, mechanical, or hydraulic quantities (''analog signals'') to model the problem being solved. In ...
s and analogue
PID controller
A proportional–integral–derivative controller (PID controller or three-term controller) is a control loop mechanism employing feedback that is widely used in industrial control systems and a variety of other applications requiring continuou ...
s. They are also used in
frequency modulators as rate-of-change detectors.
A passive differentiator circuit is one of the basic
electronic circuit
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. It is a type of electrical ...
s, being widely used in circuit analysis based on the
equivalent circuit In electrical engineering and science, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that simplifies calculation, and more broadly ...
method.
See also
*
Integrator
An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output.
Integration is an importan ...
*
Inverting differentiator at op amp applications
References
{{reflist
Analog circuits