Integrator

Integrator

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An integrator is a device to perform the mathematical operation known as integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

, a fundamental operation in calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

.

The integration function is often part of engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, mechanical
Machine
A machine manages power to accomplish a task, examples include, a mechanical system, a computing system, an electronic system, and a molecular machine. In common usage, the meaning is that of a device having parts that perform or assist in performing any type of work...

, chemical and scientific
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

 calculations.

Integrator in signal processing


See also Integrator at op amp applications

Electronic
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

 devices have been constructed to perform integration (usually with respect to time) of signals. This operation is a form of first-order low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

, which can be performed in the continuous-time (analog) domain or approximated (simulated) in the discrete-time (digital) domain. An integrator will have a low pass filtering effect but when given an offset it will accumulate a value building it until it reaches a limit of the system or overflows.

A voltage integrator is an electronic device performing a time integration of an electric voltage, thus measuring a total electric flux.

A current integrator is an electronic device performing a time integration of an electric 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...

, thus measuring a total electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

. Charge amplifier
Charge amplifier
A charge amplifier is a current integrator driven by an electrical source with capacitive nature such as a piezoelectric sensor. Contrary to what its name may suggest, a charge amplifier does not amplify the electric charge present at its input...

 is an example of current integrator. A current integrator is also used to measure the electric charge on a Faraday cup
Faraday cup
A Faraday cup is a metal cup designed to catch charged particles in vacuum. The resulting current can be measured and used to determine the number of ions or electrons hitting the cup...

 in a residual gas analyzer
Residual gas analyzer
A residual gas analyzer is a small and usually rugged mass spectrometer, typically designed for process control and contamination monitoring in vacuum systems. Utilizing quadrupole technology, there exists two implementations, utilizing either an open ion source or a closed ion source...

 to measure partial pressures of gasses in a vacuum. Another application of current integration is in ion beam deposition
Ion beam deposition
Ion Beam Deposition is a process of applying materials to a target through the application of an ion beam.thumb|Ion beam deposition setup with mass separator...

, where the measured charge directly corresponds to the number of ions deposited on a substrate, assuming the charge state of the ion
Ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...

s is known. The two current-carrying electrical leads must to be connected to the ion source
Ion source
An ion source is an electro-magnetic device that is used to create charged particles. These are used primarily to form ions for mass spectrometers, optical emission spectrometers, particle accelerators, ion implanters and ion engines.- Electron ionization :...

 and the substrate, closing the electric circuit
Electrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...

 which in part is given by the ion beam.

Integrator in computer simulation


In some computational physics
Computational physics
Computational physics is the study and implementation of numerical algorithms to solve problems in physics for which a quantitative theory already exists...

 computer simulation
Computer simulation
A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...

s, such as numerical weather prediction
Numerical weather prediction
Numerical weather prediction uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic...

, molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...

, flight simulator
Flight simulator
A flight simulator is a device that artificially re-creates aircraft flight and various aspects of the flight environment. This includes the equations that govern how aircraft fly, how they react to applications of their controls and other aircraft systems, and how they react to the external...

s, reservoir simulation
Reservoir simulation
Reservoir simulation is an area of reservoir engineering in which computer models are used to predict the flow of fluids through porous media.-Uses:...

, noise barrier
Noise barrier
A noise barrier is an exterior structure designed to protect sensitive land uses from noise pollution...

 design, architectural acoustics
Architectural acoustics
Architectural acoustics is the science of noise control within buildings. The first application of architectural acoustics was in the design of opera houses and then concert halls. More widely, noise suppression is critical in the design of multi-unit dwellings and business premises that generate...

, and electronic circuit simulation
Electronic circuit simulation
Electronic circuit simulation uses mathematical models to replicate the behavior of an actual electronic device or circuit.Simulation software allows for modeling of circuit operation and is an invaluable analysis tool...

, an integrator is a numerical method for integrating trajectories from forces (and thereby accelerations) that are only calculated at discrete time steps.

There are a variety of explicit and implicit methods
Explicit and implicit methods
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes....

 used in computer simulations.
The most basic and least accurate kind of numerical integration is Euler integration
Euler integration
In mathematics and computational science, the Euler method, named after Leonhard Euler, is a first-order numerical procedure for solving ordinary differential equations with a given initial value...

.
Verlet integration
Verlet integration
Verlet integration is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics...

 improves the accuracy of the integration to within fourth-order Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 terms, and the Runge-Kutta method which is gaining popularity further improves this accuracy to within fifth-order Taylor series terms.

Mechanical integrators


Mechanical integrators were key elements in the mechanical differential analyser
Differential analyser
The differential analyser is a mechanical analogue computer designed to solve differential equations by integration, using wheel-and-disc mechanisms to perform the integration...

, used to solve practical physical problems. Mechanical integration mechanisms were also used in control systems such as regulating flows or temperature in industrial processes.

See also

  • Analog computer
    Analog computer
    An analog computer is a form of computer that uses the continuously-changeable aspects of physical phenomena such as electrical, mechanical, or hydraulic quantities to model the problem being solved...

  • Differentiator
    Differentiator
    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.-Theory:...

  • Digital Differential Analyzer
    Digital Differential Analyzer
    A digital differential analyzer , also sometimes called a digital integrating computer, is a digital implementation of a differential analyzer...

  • Fractional order integrator
    Fractional order integrator
    A fractional order integrator or just simply fractional integrator is an integrator device that calculates the fractional order integral or derivative of an input. Differentiation or integration is a real or complex parameter...

  • Lowpass filter
  • Operational amplifier
    Operational amplifier
    An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...

  • Integrating ADC
    Integrating ADC
    An integrating ADC is a type of analog-to-digital converter that converts an unknown input voltage into a digital representation through the use of an integrator. In its most basic implementation, the unknown input voltage is applied to the input of the integrator and allowed to ramp for a fixed...

  • Planimeter
    Planimeter
    A planimeter is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.-Construction:There are several kinds of planimeters, but all operate in a similar way. The precise way in which they are constructed varies, with the main types of mechanical planimeter being...

  • Signal processing
    Signal processing
    Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

  • Charge amplifier
    Charge amplifier
    A charge amplifier is a current integrator driven by an electrical source with capacitive nature such as a piezoelectric sensor. Contrary to what its name may suggest, a charge amplifier does not amplify the electric charge present at its input...


External links