Inductor

An inductor is a passive electrical device employed in electrical circuits for its property of inductance. Passivity is a property of engineering systems most commonly used in electronic engineering and control systems An electrical network is an interconnection of Electrical elements such as Resistors Inductors Capacitors Transmission lines Voltage In Electrical circuits, any Electric current i produces a Magnetic field and hence generates a total Magnetic flux \Phi acting An inductor can take many forms.

Common inductors.

Symbol used to denote an inductor

1 Physics
2 Applications
3 Inductor construction
4 In electric circuits
- 4.1 Laplace circuit analysis (s-domain)
- 4.2 Inductor networks
5 Q factor
6 Formulae
7 See also
8 Synonyms
9 External links

Physics

Overview

Inductance (L, measured in henrys) is an effect which results from the magnetic field that forms around a current-carrying conductor. In Electrical circuits, any Electric current i produces a Magnetic field and hence generates a total Magnetic flux \Phi acting The henry (symbol H is the SI unit of Inductance. It is named after Joseph Henry (1797-1878 the American scientist who discovered electromagnetic In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Science and engineering, a conductor is a material which contains movable Electric charges. Electric current through the conductor creates a magnetic flux proportional to the current. Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. Magnetic flux, represented by the Greek letter Φ ( Phi) is a measure of quantity of Magnetism, taking into account the strength and the extent of a Magnetic A change in this current creates a change in magnetic flux that, in turn, generates an electromotive force (EMF) that acts to oppose this change in current. Electromotive force ( emf, \mathcal{E} is a term used to characterize electrical devices such as Voltaic cells thermoelectric devices electrical Inductance is a measure of the amount of EMF generated for a unit change in current. For example, an inductor with an inductance of 1 henry produces an EMF of 1 volt when the current through the inductor changes at the rate of 1 ampere per second. The number of loops, the size of each loop, and the material it is wrapped around all affect the inductance. For example, the magnetic flux linking these turns can be increased by coiling the conductor around a material with a high permeability. In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field.

Stored energy

The energy (measured in joules, in SI) stored by an inductor is equal to the amount of work required to establish the current through the inductor, and therefore the magnetic field. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity This is given by:

$E_\mathrm{stored} = {1 \over 2} L I^2$

where L is inductance and I is the current through the inductor.

Hydraulic model

Electric current can be modeled by the hydraulic analogy. The electronic&ndash hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Heaviside) is the most widely used analogy for "electron fluid" The inductor can be modeled by the flywheel effect of a turbine rotated by the flow. A flywheel is a mechanical device with significant Moment of inertia used as a storage device for Rotational energy. A turbine is a rotary Engine that extracts Energy from a Fluid flow As can be demonstrated intuitively and mathematically, this mimics the behavior of an electrical inductor; voltage is proportional to the derivative of current with respect to time. Thus a rapid change in current will cause a big voltage spike. Likewise, in cases of a sudden interruption of water flow the turbine will generate a high pressure across the blockage, etc. Magnetic interactions such as in transformers are not modeled hydraulically. A transformer is a device that transfers Electrical energy from one circuit to another through inductively coupled Electrical conductors

Applications

A choke with two 47mH windings, such as what might be found in a power supply. A choke is an Inductor designed to have a high reactance to a particular frequency when used in a signal-carrying circuit

Inductors are used extensively in analog circuits and signal processing. Analogue electronics (or analog in American English) are those electronic systems with a continuously Variable signal Inductors in conjunction with capacitors and other components form tuned circuits which can emphasize or filter out specific signal frequencies. A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors Electronic filters are Electronic circuits which perform Signal processing functions specifically intended to remove unwanted signal components and/or enhance wanted This can range from the use of large inductors as chokes in power supplies, which in conjunction with filter capacitors remove residual hum or other fluctuations from the direct current output, to such small inductances as generated by a ferrite bead or torus around a cable to prevent radio frequency interference from being transmitted down the wire. A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors Ferrites are a class of Chemical compounds with the formula AB2O4 where A and B represent various metal Cations usually including In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar Smaller inductor/capacitor combinations provide tuned circuits used in radio reception and broadcasting, for instance. An LC circuit is a variety of resonant circuit or tuned circuit and consists of an Inductor, represented by the letter L and a Capacitor, represented

Two (or more) inductors which have coupled magnetic flux form a transformer, which is a fundamental component of every electric utility power grid. A transformer is a device that transfers Electrical energy from one circuit to another through inductively coupled Electrical conductors A public utility (usually just utility) is an organization that maintains the Infrastructure for a public service (often also providing a service using The efficiency of a transformer decreases as the frequency increases but size can be decreased as well; for this reason, aircraft use 400 hertz alternating current rather than the usual 50 or 60 hertz, allowing a great saving in weight from the use of smaller transformers.

An inductor is used as the energy storage device in some switched-mode power supplies. A switched-mode power supply, switching-mode power supply or SMPS, is an electronic Power supply unit (PSU that incorporates a switching regulator The inductor is energized for a specific fraction of the regulator's switching frequency, and de-energized for the remainder of the cycle. This energy transfer ratio determines the input-voltage to output-voltage ratio. This X_L is used in complement with an active semiconductor device to maintain very accurate voltage control.

Inductors are also employed in electrical transmission systems, where they are used to intentionally depress system voltages or limit fault current. A fault current is an abnormal current in an Electric circuit due to a fault (usually a Short circuit or abnormally low impedance path In this field, they are more commonly referred to as reactors.

As inductors tend to be larger and heavier than other components, their use has been reduced in modern equipment; solid state switching power supplies eliminate large transformers, for instance, and circuits are designed to use only small inductors, if any; larger values are simulated by use of gyrator circuits. The gyrator or positive impedance inverter is an Electric circuit which inverts an impedance.

Inductor construction

Inductors. Major scale in centimetres.

An inductor is usually constructed as a coil of conducting material, typically copper wire, wrapped around a core either of air or of ferromagnetic material. A coil is a series of loops A coiled coil is a structure where the coil itself is in turn also looping In Science and engineering, a conductor is a material which contains movable Electric charges. The magnetic core is a key component in electrical and electromechanical devices such as Electromagnets Transformers and Inductors A magnetic core is a Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it Core materials with a higher permeability than air confine the magnetic field closely to the inductor, thereby increasing the inductance. In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field. Inductors come in many shapes. Most are constructed as enamel coated wire wrapped around a ferrite bobbin with wire exposed on the outside, while some enclose the wire completely in ferrite and are called "shielded". Ferrites are a class of Chemical compounds with the formula AB2O4 where A and B represent various metal Cations usually including A bobbin is a spindle or cylinder with or without flanges on which Wire, Yarn, thread or film is wound Some inductors have an adjustable core, which enables changing of the inductance. Inductors used to block very high frequencies are sometimes made with a wire passing through a ferrite cylinder or bead.

Small inductors can be etched directly onto a printed circuit board by laying out the trace in a spiral pattern. A printed circuit board, or PCB, is used to mechanically support and electrically connect Electronic components using conductive pathways or traces In Mathematics, a spiral is a Curve which emanates from a central point getting progressively farther away as it revolves around the point Some such planar inductors use a planar core. The magnetic core is a key component in electrical and electromechanical devices such as Electromagnets Transformers and Inductors A magnetic core is a

Small value inductors can also be built on integrated circuits using the same processes that are used to make transistors. Microchipsjpg|right|thumb|200px|Microchips ( EPROM memory with a transparent window showing the integrated circuit inside In Electronics, a transistor is a Semiconductor device commonly used to amplify or switch electronic signals In these cases, aluminium interconnect is typically used as the conducting material. However, practical constraints make it far more common to use a circuit called a "gyrator" which uses a capacitor and active components to behave similarly to an inductor. The gyrator or positive impedance inverter is an Electric circuit which inverts an impedance. A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors

In electric circuits

While a capacitor opposes changes in voltage, an inductor opposes changes in current. A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors An ideal inductor would offer no resistance to a constant direct current; however, only superconducting inductors have truly zero electrical resistance. Direct current ( DC) is the unidirectional flow of Electric charge. Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Electrical resistance is a ratio of the degree to which an object opposes an Electric current through it measured in Ohms Its reciprocal quantity is

In general, the relationship between the time-varying voltage v(t) across an inductor with inductance L and the time-varying current i(t) passing through it is described by the differential equation:

$v(t) = L \frac{di}{dt}$

When there is a sinusoidal alternating current (AC) through an inductor, a sinusoidal voltage is induced. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the An alternating current ( AC) is an Electric current whose direction reverses cyclically as opposed to Direct current, whose direction remains constant The amplitude of the voltage is proportional to the product of the amplitude ( $I P$ ) of the current and the frequency ( f ) of the current.

$i(t) = I_P \sin(2 \pi f t)\,$

$\frac{di(t)}{dt} = 2 \pi f I_P \cos(2 \pi f t)$

$v(t) = 2 \pi f L I_P \cos(2 \pi f t)\,$

In this situation, the phase of the current lags that of the voltage by 90 degrees. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0

Laplace circuit analysis (s-domain)

When using the Laplace transform in circuit analysis, the transfer impedance of an ideal inductor with no initial current is represented in the s domain by:

$Z(s) = Ls\,$

where

L is the inductance, and

s is the complex frequency

If the inductor does have initial current, it can be represented by:

adding a voltage source in series with the inductor, having the value:

$L I_0 \,$

(Note that the source should have a polarity that opposes the initial current)

or by adding a current source in parallel with the inductor, having the value:

$\frac{I_0}{s}$

where

L is the inductance, and

$I 0$ is the initial current in the inductor. In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic

Inductor networks

Main article: Series and parallel circuits

Inductors in a parallel configuration each have the same potential difference (voltage). If two or more circuit components are connected end to end like a daisy chain it is said they are connected in series. If two or more circuit components are connected end to end like a daisy chain it is said they are connected in series. To find their total equivalent inductance (L_eq):

$\frac{1}{L_\mathrm{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + \cdots + \frac{1}{L_n}$

The current through inductors in series stays the same, but the voltage across each inductor can be different. If two or more circuit components are connected end to end like a daisy chain it is said they are connected in series. The sum of the potential differences (voltage) is equal to the total voltage. To find their total inductance:

$L_\mathrm{eq} = L_1 + L_2 + \cdots + L_n \,\!$

These simple relationships hold true only when there is no mutual coupling of magnetic fields between individual inductors.

Q factor

An ideal inductor will be lossless irrespective of the amount of current through the winding. However, typically inductors have winding resistance from the metal wire forming the coils. Since the winding resistance appears as a resistance in series with the inductor, it is often called the series resistance. The inductor's series resistance converts electrical current through the coils into heat, thus causing a loss of inductive quality. The quality factor (or Q) of an inductor is the ratio of its inductive reactance to its resistance at a given frequency, and is a measure of its efficiency. For other uses of the terms Q and Q factor see Q value. In Physics and Engineering the quality The higher the Q factor of the inductor, the closer it approaches the behavior of an ideal, lossless, inductor.

The Q factor of an inductor can be found through the following formula, where R is its internal electrical resistance and $ω L$ is Capacitive or Inductive reactance at resonance:

$Q = \frac{\omega{}L}{R}$

By using a ferromagnetic core the inductance is increased for the same amount of copper, raising the Q. Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it Cores however also introduce losses that increase with frequency. A grade of core material is chosen for best results for the frequency band. At VHF or higher frequencies an air core is likely to be used. Very high frequency (VHF is the Radio frequency range from 30 MHz to 300 MHz. Inductors wound around a ferromagnetic core may saturate at high currents, causing a dramatic decrease in inductance (and Q). Seen in some Magnetic materials saturation is the state reached when an increase in applied external Magnetizing field H cannot increase the This phenomenon can be avoided by using a (physically larger) air core inductor. A well designed air core inductor may have a Q of several hundred.

An almost ideal inductor (Q approaching infinity) can be created by immersing a coil made from a superconducting alloy in liquid helium or liquid nitrogen. Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance An alloy is a Solid solution or Homogeneous mixture of two or more elements, at least one of which is a Metal, which itself has Helium exists in Liquid form only at very low Temperatures The Boiling point and critical point depend on the Isotope Liquid nitrogen (liquid density at the Triple point is 0707 g/mL is the liquid produced industrially in large quantities by Fractional distillation of This supercools the wire, causing its winding resistance to disappear. Because a superconducting inductor is virtually lossless, it can store a large amount of electrical energy within the surrounding magnetic field (see superconducting magnetic energy storage). Superconducting Magnetic Energy Storage (SMES systems store energy in the Magnetic field created by the flow of Direct current in a superconducting coil

Formulae

The table below lists some common formulae for calculating the theoretical inductance of several inductor constructions.

Construction	Formula	Dimensions
Cylindrical coil	$L=\frac{\mu_0\mu_rN^2A}{l}$	L = inductance in henries (H) μ₀ = permeability of free space = 4 $π$ × 10^-7 H/m μ_r = relative permeability of core material N = number of turns A = area of cross-section of the coil in square metres (m²) l = length of coil in metres (m)
Straight wire conductor	$L = l\left(\ln\frac{4l}{d}-1\right) \cdot 200 \times 10^{-9}$	L = inductance (H) l = length of conductor (m) d = diameter of conductor (m)
Straight wire conductor	$L = 5.08 \cdot l\left(\ln\frac{4l}{d}-1\right)$	L = inductance (nH) l = length of conductor (in) d = diameter of conductor (in)
Short air-core cylindrical coil	$L=\frac{r^2N^2}{9r+10l}$	L = inductance (µH) r = outer radius of coil (in) l = length of coil (in) N = number of turns
Multilayer air-core coil	$L = \frac{0.8r^2N^2}{6r+9l+10d}$	L = inductance (µH) r = mean radius of coil (in) l = physical length of coil winding (in) N = number of turns d = depth of coil (outer radius minus inner radius) (in)
Flat spiral air-core coil	$L=\frac{r^2N^2}{(2r+2.8d) \times 10^5}$	L = inductance (H) r = mean radius of coil (m) N = number of turns d = depth of coil (outer radius minus inner radius) (m)
Flat spiral air-core coil	$L=\frac{r^2N^2}{8r+11d}$	L = inductance (µH) r = mean radius of coil (in) N = number of turns d = depth of coil (outer radius minus inner radius) (in)
Toroidal core (circular cross-section)	$L=\mu_0\mu_r\frac{N^2r^2}{D}$	L = inductance (H) μ₀ = permeability of free space = 4 $π$ × 10^-7 H/m μ_r = relative permeability of core material N = number of turns r = radius of coil winding (m) D = overall diameter of toroid (m)

Synonyms

External links

General

How stuff works The initial concept, made very simple
Capacitance and Inductance - A chapter from an online textbook
Spiral inductor models. The henry (symbol H is the SI unit of Inductance. It is named after Joseph Henry (1797-1878 the American scientist who discovered electromagnetic The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field. M^2 redirects here For other uses see M². CM2 redirects here In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field. This vacuum means "absence of matter" or "an empty area or space" for the cleaning appliance see Vacuum cleaner. An electronic component is a basic electronic element usually packaged in a discrete form with two or more connecting leads or metallic pads A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors |- align = "center"| |width = "25"| | |- align = "center"| || Potentiometer |- align = "center"| | | |- align = "center"| Resistor| | Memristors /memˈrɪstɚ/ ("memory resistors" are a class of passive two-terminal Circuit elements that maintain a functional relationship between Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical The gyrator or positive impedance inverter is an Electric circuit which inverts an impedance. In Electrical circuits, any Electric current i produces a Magnetic field and hence generates a total Magnetic flux \Phi acting An induction coil or "spark coil" ( archaically known as a Ruhmkorff coil) is a type of Disruptive discharge Coil. An induction cooker uses Induction heating for Cooking. A conducting pot is placed above an Induction coil for the heating process to take place Induction loop is a term historically used to describe an Electromagnetic Communication system, which exploits a phenomenon that was (according to anecdote first The magnetic core is a key component in electrical and electromechanical devices such as Electromagnets Transformers and Inductors A magnetic core is a A saturable reactor in Electrical engineering is a special form of Inductor where the Magnetic core can be deliberately saturated by means of A transformer is a device that transfers Electrical energy from one circuit to another through inductively coupled Electrical conductors A resistor-inductor circuit (RL circuit, or RL filter or RL network, is one of the simplest analogue Infinite impulse response An RLC circuit (also known as a Resonant circuit tuned circuit or LCR circuit is an Electrical circuit consisting of a Resistor (R an A coil is a series of loops A coiled coil is a structure where the coil itself is in turn also looping A choke is an Inductor designed to have a high reactance to a particular frequency when used in a signal-carrying circuit Good article on inductor characteristics and modeling.
Online coil inductance calculator. Online calculator calculates the inductance of conventional and toroidal coils using formulas 3, 4, 5, and 6, above.
AC circuits
[1]- Understanding coils and transforms
[2] - Circuit Schematic Symbols

Dictionary

-noun

(physics) a passive device that introduces inductance into an electrical circuit
(medicine) an evocator or an organizer

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