Inductor 14896 225505485 2008-07-14T01:25:29Z Copeland.James.H 781638 /* Applications */ depress voltages from lightning strikes and to limit switching currents {{Unreferenced|date=January 2008}} An '''inductor''' is a [[Passive component|passive]] electrical device employed in [[Electrical network|electrical circuits]] for its property of [[inductance]]. An inductor can take many forms. [[Image:Electronic component inductors.jpg|thumb|right|250px|Common inductors.]] [[Image:Inductor.svg|thumb|right|150px|Symbol used to denote an inductor]] == Physics == === Overview === [[Inductance]] (''L'') (measured in [[Henry (unit)|henrys]]) is an effect which results from the [[magnetic field]] that forms around a current-carrying [[Electrical conductor|conductor]]. [[Electric current]] through the conductor creates a [[magnetic flux]] proportional to the current. 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. 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 (electromagnetism)|permeability]]. === Stored energy === The [[energy]] (measured in [[joule]]s, 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. This is given by: :<math> E_\mathrm{stored} = {1 \over 2} L I^2 </math> where ''L'' is inductance and ''I'' is the current through the inductor. === Hydraulic model === Electric current can be modeled by the [[hydraulic analogy]]. The inductor can be modeled by the [[flywheel]] effect of a [[turbine]] rotated by the 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 [[transformer#An analogy|transformers]] are not modeled hydraulically. == Applications == [[Image:Choke electronic component Epcos 2x47mH 600mA common mode.jpg|thumb|A [[Choke (electronics)|choke]] with two 47mH windings, such as what might be found in a power supply.]] Inductors are used extensively in [[analog circuit]]s and signal processing. Inductors in conjunction with [[capacitor]]s and other components form tuned circuits which can emphasize or [[electronic filter|filter]] out specific signal frequencies. This can range from the use of large inductors as chokes in power supplies, which in conjunction with filter [[capacitor]]s remove residual [[hum]] or other fluctuations from the direct current output, to such small inductances as generated by a [[Ferrite (magnet)|ferrite]] bead or [[torus]] around a cable to prevent [[radio frequency interference]] from being transmitted down the wire. Smaller inductor/capacitor combinations provide [[tuned circuit]]s used in radio reception and broadcasting, for instance. Two (or more) inductors which have coupled magnetic flux form a [[transformer]], which is a fundamental component of every electric [[Public utility|utility]] power grid. 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 supply|switched-mode power supplies]]. 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''<sub>L</sub> 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 depress voltages from lightning strikes and to limit switching currents and [[fault current]]. 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. == Inductor construction == [[Image:Coils.jpg|thumb|150px|Inductors. Major scale in centimetres.]] An inductor is usually constructed as a [[coil]] of [[Electrical conductor|conducting]] material, typically copper wire, wrapped around a [[magnetic core|core]] either of air or of [[ferromagnetic]] material. Core materials with a higher [[Permeability (electromagnetism)|permeability]] than air confine the magnetic field closely to the inductor, thereby increasing the inductance. Inductors come in many shapes. Most are constructed as enamel coated wire wrapped around a [[Ferrite (magnet)|ferrite]] [[bobbin]] with wire exposed on the outside, while some enclose the wire completely in ferrite and are called "shielded". 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. Some such planar inductors use a [[magnetic core#Planar core | planar core]]. Small value inductors can also be built on [[integrated circuit]]s using the same processes that are used to make [[transistor]]s. 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. == In electric circuits == While a [[capacitor]] opposes changes in voltage, an inductor opposes changes in current. An ideal inductor would offer no resistance to a constant [[direct current]]; however, only [[superconductor|superconducting]] inductors have truly zero [[electrical resistance]]. 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]]: :<math>v(t) = L \frac{di}{dt}</math> When there is a [[sinusoidal]] [[alternating current]] (AC) through an inductor, a sinusoidal voltage is induced. The amplitude of the voltage is proportional to the product of the amplitude (<math>I_P</math>) of the current and the frequency ('' f '') of the current. :<math>i(t) = I_P \sin(2 \pi f t)\,</math> :<math>\frac{di(t)}{dt} = 2 \pi f I_P \cos(2 \pi f t)</math> :<math>v(t) = 2 \pi f L I_P \cos(2 \pi f t)\,</math> In this situation, the [[Phase (waves)|phase]] of the current lags that of the voltage by 90 degrees. === 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: :<math>Z(s) = Ls\, </math> ::: 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: :<math> L I_0 \,</math> (''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: :<math> \frac{I_0}{s} </math> ::: where :::: ''L'' is the inductance, and :::: ''<math>I_0</math> is the initial current in the inductor. === Inductor networks === {{main|Series and parallel circuits}} Inductors in a [[Series and parallel circuits|parallel]] configuration each have the same potential difference (voltage). To find their total equivalent inductance (''L''<sub>eq</sub>): : [[Image:inductors in parallel.svg|A diagram of several inductors, side by side, both leads of each connected to the same wires]] :<math> \frac{1}{L_\mathrm{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + \cdots + \frac{1}{L_n}</math> The current through inductors in [[Series and parallel circuits|series]] stays the same, but the voltage across each inductor can be different. The sum of the potential differences (voltage) is equal to the total voltage. To find their total inductance: : [[Image:inductors in series.svg|A diagram of several inductors, connected end to end, with the same amount of current going through each]] :<math> L_\mathrm{eq} = L_1 + L_2 + \cdots + L_n \,\! </math> 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 [[Q factor|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. 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 <math>\omega{}L</math> is Capacitive or Inductive reactance at resonance: :<math>Q = \frac{\omega{}L}{R}</math> By using a [[ferromagnetic]] core the inductance is increased for the same amount of copper, raising the Q. 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. Inductors wound around a ferromagnetic core may [[saturation (magnetic)|saturate]] at high currents, causing a dramatic decrease in inductance (and Q). 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 [[superconductor|superconducting]] [[alloy]] in [[liquid helium]] or [[liquid nitrogen]]. 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]]). == Formulae == The table below lists some common formulae for calculating the theoretical inductance of several inductor constructions. {| class="wikitable" ! Construction ! Formula ! Dimensions |- ! Cylindrical coil | <math>L=\frac{\mu_0\mu_rN^2A}{l}</math> | *''L'' = inductance in [[Henry (unit)|henries]] (H) *''μ<sub>0</sub>'' = [[permeability of free space]] = 4''<math>\pi</math>'' × 10<sup>-7</sup> H/m *''μ<sub>r</sub>'' = [[Permeability (electromagnetism)#Relative permeability|relative permeability]] of core material *''N'' = number of turns *''A'' = area of cross-section of the coil in [[square metre]]s (m<sup>2</sup>) *''l'' = length of coil in metres (m) |- ! rowspan="2"|Straight wire conductor | <math>L = l\left(\ln\frac{4l}{d}-1\right) \cdot 200 \times 10^{-9}</math> | *''L'' = inductance (H) *''l'' = length of conductor (m) *''d'' = diameter of conductor (m) |- | <math>L = 5.08 \cdot l\left(\ln\frac{4l}{d}-1\right)</math> | *''L'' = inductance (nH) *''l'' = length of conductor (in) *''d'' = diameter of conductor (in) |- ! Short air-core cylindrical coil | <math>L=\frac{r^2N^2}{9r+10l}</math> | *''L'' = inductance (µH) *''r'' = outer radius of coil (in) *''l'' = length of coil (in) *''N'' = number of turns |- ! Multilayer air-core coil | <math>L = \frac{0.8r^2N^2}{6r+9l+10d}</math> | *''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) |- ! rowspan="2"|Flat spiral air-core coil | <math>L=\frac{r^2N^2}{(2r+2.8d) \times 10^5}</math> | *''L'' = inductance (H) *''r'' = mean radius of coil (m) *''N'' = number of turns *''d'' = depth of coil (outer radius minus inner radius) (m) |- | <math>L=\frac{r^2N^2}{8r+11d}</math> | *''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) | <math>L=\mu_0\mu_r\frac{N^2r^2}{D}</math> | *''L'' = inductance (H) *''μ<sub>0</sub>'' = [[Permeability (electromagnetism)|permeability]] of [[vacuum|free space]] = 4''<math>\pi</math>'' × 10<sup>-7</sup> H/m *''μ<sub>r</sub>'' = relative permeability of core material *''N'' = number of turns *''r'' = radius of coil winding (m) *''D'' = overall diameter of toroid (m) |} == See also == <div style="-moz-column-count:3; column-count:3;"> * [[Electronic component]] * [[Capacitor]] * [[Resistor]] * [[Memristor]] * [[Electricity]] * [[Electronics]] * [[Gyrator]] * [[Inductance]] (including ''mutual inductance'') * [[Induction coil]] * [[induction cooker|Induction cooking]] * [[Induction loop]] * [[Magnetic core]] * [[Saturable reactor]] * [[Transformer]] * [[Reactance]] * [[RL circuit]] * [[RLC circuit]] </div> == Synonyms == <!-- Are some of these circular Wikilinks? Should they become hard-redirects back to here? --> * [[coil]] * [[Choke (electronics)]] * [[reactor]] == External links == ; General * [http://electronics.howstuffworks.com/inductor1.htm How stuff works] The initial concept, made very simple * [http://www.lightandmatter.com/html_books/4em/ch07/ch07.html Capacitance and Inductance] - A chapter from an online textbook * [http://www.mpdigest.com/issue/Articles/2005/aug2005/agilent/Default.asp Spiral inductor models]. Good article on inductor characteristics and modeling. * [http://www.66pacific.com/calculators/coil_calc.aspx Online coil inductance calculator]. Online calculator calculates the inductance of conventional and toroidal coils using formulas 3, 4, 5, and 6, above. * [http://www.phys.unsw.edu.au/~jw/AC.html AC circuits] * [http://www.mikroe.com/en/books/keu/03.htm]- Understanding coils and transforms * [http://library.thinkquest.org/10784/circuit_symbols.html] - Circuit Schematic Symbols [[Category:Electromagnetic components]] [[Category:Energy storage]] [[af:Induktor]] [[ar:مستحث]] [[bs:Zavojnica]] [[ca:Inductor]] [[cs:Cívka]] [[da:Elektrisk spole]] [[de:Spule (Elektrotechnik)]] [[et:Induktor]] [[es:Inductor]] [[eo:Induktilo]] [[fr:Bobine (électricité)]] [[ko:코일]] [[hr:Zavojnica]] [[id:Induktor]] [[is:Spanspóla]] [[it:Induttore]] [[he:סליל השראה]] [[lv:Induktivitātes spole]] [[hu:Tekercs (elektronika)]] [[ms:Peraruh]] [[mn:Индукцийн ороомог]] [[nl:Spoel]] [[ja:コイル]] [[no:Spole (induktans)]] [[nn:Spole]] [[pl:Cewka]] [[pt:Indutor]] [[ro:Bobină]] [[ru:Катушка индуктивности]] [[scn:Ruccheddu]] [[simple:Inductor]] [[sq:Induktori]] [[sk:Cievka (elektrická súčiastka)]] [[sl:Dušilka]] [[fi:Kela (komponentti)]] [[sv:Spole]] [[ta:மின்தூண்டி]] [[vi:Cuộn cảm]] [[tr:Bobin]] [[uk:Котушка індуктивності]] [[zh:电感元件]]