Differentiator
Encyclopedia
A Differentiator is a circuit that is designed such that the output of the circuit is proportional to the time derivative
of the input. There are two types of differentiator circuits, active and passive
.
, resistor
s are used at feedback side and capacitor
s are used at the input side. The circuit is based on the capacitors current
to voltage
relationship:
where I is the current
through the capacitor, C is the capacitance
of the capacitor, and V is the voltage
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 opamp input has a very high input impedance (it also forms a virtual ground
) so the entire input current has to flow through R.
If Vout is the voltage across the resistor and Vin 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:
Here output is nothing but the time derivative of the input. Such differentiator is called as a perfect differentiator.
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
of RC.
of capacitor is high and at high frequency reactance is low. Therefore, at low frequency gain(Rf/Xc) is low while at higher frequency gain is high with high output voltage.But at high frequencies differentiator circuit becomes unstable and start to oscillate.This high frequency gain of circuit is reduced by adding small value capacitor across feedback resistor Rf.
If there is constant DC voltage applied as input then output voltage is 0 .If input voltage is changing from 0 to negative going voltage output voltage is positive DC.If input voltage applied is changing from 0 to positive going voltage then output is negative DC.
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of the input. There are two types of differentiator circuits, active and passive
Passive differentiator circuit
A passive differentiator circuit is a simple four-terminal network consisting of two passive elements as shown in Figures 1 and 2. It is a simple first-order high-pass filter.- Transfer function :...
.
Theory
A differentiator circuit consists of an operational amplifierOperational amplifier
An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...
, resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s are used at feedback side and capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s are used at the input side. The circuit is based on the capacitors current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
to voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
relationship:
where I is the current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
through the capacitor, C is the capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
of the capacitor, and V is the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
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
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
of the resistor.
Note that the opamp input has a very high input impedance (it also forms a virtual ground
Virtual ground
Virtual ground is a node of the circuit that is maintained at a steady reference potential, without being connected directly to the reference potential...
) so the entire input current has to flow through R.
If Vout is the voltage across the resistor and Vin 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 opamp acts as a differentiator.
- The minus sign indicates the phase difference of 180 degrees between the output and the input.
- Above equation is true for any frequency signal.
- If we select R and C in such a manner that R×C=1 then the above equation becomes
Here output is nothing but the time derivative of the input. Such differentiator is called as a perfect differentiator.
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
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...
of RC.
Operation
Input signal applied to the capacitor C. Capacitive reactance is the important in analysis of operation of differentiator.Capacitive reactance is Xc=1/2πfc.Capacitive reactance is directly proportional to the rate of change of input voltage applied to the capacitor. At low frequency reactanceReactance
Reactance is the opposition of a circuit element to a change of electric current or voltage, due to that element's capacitance or inductance. A built-up electric field resists the change of voltage on the element, while a magnetic field resists the change of current...
of capacitor is high and at high frequency reactance is low. Therefore, at low frequency gain(Rf/Xc) is low while at higher frequency gain is high with high output voltage.But at high frequencies differentiator circuit becomes unstable and start to oscillate.This high frequency gain of circuit is reduced by adding small value capacitor across feedback resistor Rf.
If there is constant DC voltage applied as input then output voltage is 0 .If input voltage is changing from 0 to negative going voltage output voltage is positive DC.If input voltage applied is changing from 0 to positive going voltage then output is negative DC.
See also
- IntegratorIntegratorAn integrator is a device to perform the mathematical operation known as integration, a fundamental operation in calculus.The integration function is often part of engineering, physics, mechanical, chemical and scientific calculations....
- Differentiator at op amp applications
- Passive differentiator circuitPassive differentiator circuitA passive differentiator circuit is a simple four-terminal network consisting of two passive elements as shown in Figures 1 and 2. It is a simple first-order high-pass filter.- Transfer function :...