View source for Engineering:Inductor

An '''inductor''', also called a '''coil''', '''choke''', or '''reactor''', is a [[Engineering:Passivity (engineering)|passive]] two-terminal [[Engineering:Electronic component|electrical component]] that stores energy in a [[Magnetic field|magnetic field]] when an [[Physics:Electric current|electric current]] flows through it.<ref>{{cite book|last1=Alexander|first1=Charles K.|last2=Sadiku|first2=Matthew N. O.|title=Fundamentals of Electric Circuits|publisher=McGraw-Hill|page=226|edition=5|year=2013|isbn=978-0-07-338057-5}}</ref>  An inductor typically consists of an insulated wire wound into a [[Engineering:Electromagnetic coil|coil]].

When the current flowing through the coil changes, the time-varying magnetic field induces an [[Physics:Electromotive force|electromotive force]] (''emf''), or [[Physics:Voltage|voltage]], in the conductor, described by [[Physics:Faraday's law of induction|Faraday's law of induction]]. According to [[Physics:Lenz's law|Lenz's law]], the induced voltage has a polarity (direction) which opposes the change in current that created it. As a result, inductors oppose any changes in current through them.

An inductor is characterized by its [[Physics:Inductance|inductance]], which is the ratio of the voltage to the rate of change of current. In the [[International System of Units]] (SI), the unit of inductance is the [[Henry (unit)|henry]] (H) named for 19th century American scientist [[Biography:Joseph Henry|Joseph Henry]]. In the measurement of magnetic circuits, it is equivalent to {{Sfrac|[[Weber (unit)|weber]]|[[Ampere|ampere]]}}. Inductors have values that typically range from 1{{nbsp}}μH (10<sup>−6</sup>{{nbsp}}H) to 20{{nbsp}}H.  Many inductors have a [[Physics:Magnetic core|magnetic core]] made of iron or [[Physics:Ferrite (magnet)|ferrite]] inside the coil, which serves to increase the magnetic field and thus the inductance. Along with [[Engineering:Capacitor|capacitor]]s and [[Engineering:Resistor|resistor]]s, inductors are one of the three passive [[Engineering:Linear circuit|linear]] circuit elements that make up electronic circuits. Inductors are widely used in [[Physics:Alternating current|alternating current]] (AC) electronic equipment, particularly in [[Social:Radio|radio]] equipment.  They are used to block AC while allowing DC to pass; inductors designed for this purpose are called [[Engineering:Choke (electronics)|choke]]s.  They are also used in [[Engineering:Electronic filter|electronic filter]]s to separate signals of different [[Physics:Frequency|frequencies]], and in combination with capacitors to make tuned circuits, used to tune radio and TV receivers.

The term inductor seems to come from Heinrich Daniel Ruhmkorff, who called the [[Engineering:Induction coil|induction coil]] he invented in 1851 an inductorium.<ref>{{cite book |last=Urbanitzky|first=Alfred Ritter von |title=Electricity in the Service of Man |year=1886 |publisher=Macmillan and Company |page=[https://archive.org/details/electricityinse01wormgoog/page/194/mode/2up 195]}}</ref>

==Description==
[[File:Electromagnetism.svg|thumb|An electric current {{math|I}} creates a magnetic field {{math|B}} around it]]

An electric current flowing through a [[Physics:Electrical conductor|conductor]] generates a magnetic field surrounding it. The [[Physics:Magnetic flux|magnetic flux linkage]] <math>\Phi_\mathbf{B}</math> generated by a given current <math>I</math> depends on the geometric shape of the circuit. Their ratio defines the inductance <math>L</math>.<ref name="Singh">{{cite book
  | last = Singh
  | first = Yaduvir 
  | title = Electro Magnetic Field Theory
  | publisher = Pearson India
  | date = 2011
  | pages = 65
  | url = https://books.google.com/books?id=0-PfbT49tJMC&q=inductance&pg=PA65
  | isbn = 978-8131760611}}</ref><ref name="Wadhwa">{{cite book
  | last = Wadhwa
  | first = C. L. 
  | title = Electrical Power Systems
  | publisher = New Age International
  | date = 2005
  | pages = 18
  | url = https://books.google.com/books?id=Su3-0UhVF28C&q=inductance&pg=PA18
  | isbn = 978-8122417227}}</ref><ref name="Pelcovits">{{cite book
  | last = Pelcovits
  | first = Robert A. 
  |author2=Josh Farkas
  | title = Barron's AP Physics C
  | publisher = Barron's Educational Series
  | date = 2007
  | pages = 646
  | url = https://books.google.com/books?id=yON684oSjbEC&q=inductance&pg=PA646
  | isbn = 978-0764137105}}</ref><ref name="Purcell">{{cite book
  | last = Purcell
  | first = Edward M.
  |author2=David J. Morin
  | title = Electricity and Magnetism
  | publisher = Cambridge Univ. Press
  | date = 2013
  | pages = 364
  | url = https://books.google.com/books?id=A2rS5vlSFq0C&pg=PA364
  | isbn = 978-1107014022
}}</ref> Thus
:<math>L := \frac{\Phi_\mathbf{B}}{I}</math>.

The inductance of a circuit depends on the geometry of the current path as well as the magnetic permeability of nearby materials. An inductor is a [[Engineering:Electronic component|component]] consisting of a wire or other conductor shaped to increase the magnetic flux through the circuit, usually in the shape of a coil or [[Helix|helix]], with two [[Engineering:Terminal (electronics)|terminals]]. Winding the wire into a [[Engineering:Electromagnetic coil|coil]] increases the number of times the [[Physics:Magnetic flux|magnetic flux]] lines link the circuit, increasing the field and thus the inductance. The more turns, the higher the inductance.  The inductance also depends on the shape of the coil, separation of the turns, and many other factors.  By adding a "magnetic core" made of a [[Physics:Ferromagnetic|ferromagnetic]] material like iron inside the coil, the magnetizing field from the coil will induce [[Physics:Magnetization|magnetization]] in the material, increasing the magnetic flux. The high permeability of a ferromagnetic core can increase the inductance of a coil by a factor of several thousand over what it would be without it.

===Constitutive equation===
Any change in the current through an inductor creates a changing flux, inducing a voltage across the inductor. By [[Physics:Faraday's law of induction|Faraday's law of induction]], the voltage <math>\mathcal{E}</math> induced by any change in magnetic flux through the circuit is given by<ref name="Purcell" /> 
:<math>\mathcal{E} = -\frac{d\Phi_\mathbf{B}}{dt}</math>.

Reformulating the definition of {{mvar|L}} above, we obtain<ref name="Purcell" />
:<math> \Phi_\mathbf{B} = LI</math>.
It follows that
:<math>\mathcal{E} = -\frac{d\Phi_\mathbf{B}}{dt} = -\frac{d}{dt}(LI)</math> 
{{Equation box 1 |indent =: |cellpadding = 0 |border = 2 |border colour = blue |background colour = transparent
|equation = <math>\; \mathcal{E} = -L\frac{dI}{dt} \;</math>
}}
if {{mvar|L}} is independent of time, current and magnetic flux linkage. Thus, inductance is also a measure of the amount of [[Physics:Electromotive force|electromotive force]] (voltage) generated for a given rate of change of current. This is usually taken to be the constitutive relation (defining equation) of the inductor.

[[File:Inductor-positive-voltage-with-input-current-function-of-time.svg|thumb|205x205px|Schematic using current's exit terminal as reference for voltage]]
Because the induced voltage is positive at the current's entrance terminal, the inductor's current–voltage relationship is often expressed without a negative sign by using the current's exit terminal as the reference point for the voltage <math>V(t)</math> at the current's entrance terminal (as labeled in the schematic). The current–voltage relationship is then:
{{Equation box 1 |indent =: |cellpadding = 0 |border = 2 |border colour = blue |background colour = transparent
|equation = <math>\; V(t) = L\frac{\mathrm{d}I(t)}{\mathrm{d}t} \;</math>
|title=Current–voltage relationship (derivative form)}}which can be rewritten as:{{Equation box 1 |indent =: |cellpadding = 0 |border = 2 |border colour = blue |background colour = transparent
|equation = <math>\; I(t) = I(t_0) + \frac{1}{L}\int_{t_0}^t V(\tau) \, \mathrm{d}\tau \;</math>
|title=Current–voltage relationship (integral form)}}As with any [[Antiderivative|antiderivative]], a [[Constant of integration|constant of integration]] is added to represent the initial current ''I''(''t''<sub>0</sub>). The [[Philosophy:Duality (electrical circuits)|dual]] of the inductor is the [[Engineering:Capacitor|capacitor]], which [[Engineering:Capacitor#Energy stored in a capacitor|stores energy in an electric field]] rather than a magnetic field. [[Engineering:Capacitor#Current–voltage relation|Its current–voltage relation]] replaces {{mvar|L}} with the capacitance {{mvar|C}} and has current and voltage swapped from these equations.

===Lenz's law===
The polarity (direction) of the induced voltage is given by [[Physics:Lenz's law|Lenz's law]], which states that the induced voltage will be such as to oppose the change in current.<ref name="Shamos">{{Cite book|url=https://books.google.com/books?id=J0fCAgAAQBAJ&q=1834+Lenz%E2%80%99s+Law&pg=PT238|title=Great Experiments in Physics: Firsthand Accounts from Galileo to Einstein|last=Shamos|first=Morris H.|date=2012-10-16|publisher=Courier Corporation|isbn=9780486139623|language=en}}</ref> For example, if the current through an inductor is increasing, the induced potential difference will be positive at the current's entrance point and negative at the exit point, tending to oppose the additional current.<ref name="Schmitt">{{cite book
 | last1  = Schmitt
 | first1 = Ron 
 | title  = Electromagnetics Explained: A Handbook for Wireless/ RF, EMC, and High-Speed Electronics
 | publisher = Elsevier
 | date   = 2002
 | pages  = 75–77
 | url    = https://books.google.com/books?id=7gJ4RocvEskC&q=%22lenz%27s+law%22+energy&pg=PA75
 | isbn   = 978-0080505237
 }}</ref><ref name="Jaffe">{{cite book
 | last1  = Jaffe
 | first1 = Robert L.
 | last2  = Taylor
 | first2 = Washington
 | title  = The Physics of Energy
 | publisher = Cambridge Univ. Press
 | date   = 2018
 | pages  = 51
 | url    = https://books.google.com/books?id=RMhJDwAAQBAJ&q=%22lenz%27s+law%22+energy+stored+inductor+current&pg=PA51
 | isbn   = 978-1108547895
 }}</ref><ref name="Lerner">{{cite book
 | last1  = Lerner
 | first1 = Lawrence S. 
 | title  = Physics for Scientists and Engineers, Vol. 2
 | publisher = Jones and Bartlet Learning
 | date   = 1997
 | pages  = 856
 | url    = https://books.google.com/books?id=Nv5GAyAdijoC&q=inductor+energy+%22magnetic+field%22+current&pg=PA856
 | isbn   = 978-0763704605
 }}</ref> The energy from the external circuit necessary to overcome this potential "hill" is being stored in the magnetic field of the inductor. If the current is decreasing, the induced voltage will be negative at the current's entrance point and positive at the exit point, tending to maintain the current. In this case energy from the magnetic field is being returned to the circuit.

=== Energy stored in an inductor ===
One intuitive explanation as to why a potential difference is induced on a change of current in an inductor goes as follows:

When there is a change in current through an inductor there is a change in the strength of the magnetic field.  For example, if the current is increased, the magnetic field increases.  This, however, does not come without a price.  The magnetic field contains [[Physics:Potential energy|potential energy]], and increasing the field strength requires more energy to be stored in the field.  This energy comes from the electric current through the inductor.  The increase in the magnetic potential energy of the field is provided by a corresponding drop in the electric potential energy of the charges flowing through the windings.  This appears as a voltage drop across the windings as long as the current increases.  Once the current is no longer increased and is held constant, the energy in the magnetic field is constant and no additional energy must be supplied, so the voltage drop across the windings disappears.

Similarly, if the current through the inductor decreases, the magnetic field strength decreases, and the energy in the magnetic field decreases.  This energy is returned to the circuit in the form of an increase in the electrical potential energy of the moving charges, causing a voltage rise across the windings.

====Derivation====
The [[Physics:Work|work]] done per unit charge on the charges passing through the inductor is <math>-\mathcal{E}</math>. The negative sign indicates that the work is done ''against'' the emf, and is not done ''by'' the emf.  The current <math>I</math> is the charge per unit time passing through the inductor.  Therefore, the rate of work <math>W</math> done by the charges against the emf, that is the rate of change of energy of the current, is given by
:<math>\frac{dW}{dt} = -\mathcal{E}I </math>
From the constitutive equation for the inductor, <math>-\mathcal{E} = L\frac{dI}{dt}</math> so
:<math>\frac{dW}{dt}= L\frac{dI}{dt} \cdot I = LI \cdot \frac{dI}{dt}</math>

:<math>dW = L I \cdot dI</math>

In a ferromagnetic core inductor, when the magnetic field approaches the level at which the core saturates, the inductance will begin to change, it will be a function of the current <math>L(I)</math>. Neglecting losses, the [[Physics:Energy|energy]] <math>W</math> stored by an inductor with a current <math>I_0</math> passing through it is equal to the amount of work required to establish the current through the inductor:

:<math>W = \int_0^{I_0} L_d(I) \, I \, dI \, ,</math>

where <math>L_d(I)</math> is the so-called "differential inductance": <math>L_d = \frac{d\Phi_{\mathbf{B}}}{dI} \, .</math>

In an air core inductor or a ferromagnetic core inductor below saturation, the inductance is constant (and equal to the differential inductance), so the stored energy is
:<math>W = L\int_0^{I_0} I \, dI</math>
{{Equation box 1 |indent =: |cellpadding = 0 |border = 2 |border colour = blue |background colour = transparent
|equation = <math>\; W = \frac{1}{2}L {I_0}^2 \;</math>
}}  
For inductors with magnetic cores, the above equation is only valid for [[Engineering:Linear circuit|linear]] regions of the magnetic flux, at currents below the saturation level of the inductor, where the inductance is approximately constant. Where this is not the case, the integral form must be used with <math>L_d</math> variable.

=== Voltage step response ===
When a [[Step function|voltage step]] is applied to an inductor:
* In the short-time limit, since the current cannot change instantaneously, the initial current is zero. The equivalent circuit of an inductor immediately after the step is applied is an [[Physics:Electrical circuit|open circuit]].
* As time passes, the current increases at a constant rate with time until the inductor starts to saturate.
* In the long-time limit, the transient response of the inductor will die out, the magnetic flux through the inductor will become constant, so no voltage would be induced between the terminals of the inductor. Therefore, assuming the resistance of the windings is negligible, the equivalent circuit of an inductor a long time after the step is applied is a [[Physics:Short circuit|short circuit]].

===Ideal and real inductors===
<!-- NB. Section header used in redirects to this page -->
<!-- -->
The constitutive equation describes the behavior of an ''ideal inductor'' with inductance <math>L</math>, and without [[Physics:Electrical resistance|resistance]], [[Physics:Capacitance|capacitance]], or energy dissipation. In practice, inductors do not follow this theoretical model; ''real inductors'' have a measurable resistance due to the resistance of the wire and energy losses in the core, and [[Engineering:Parasitic element (electrical networks)|parasitic capacitance]] between turns of the wire.<ref name="Bowick">{{cite book
 | last1  = Bowick
 | first1 = Christopher 
 | title  = RF Circuit Design, 2nd Ed.
 | publisher = Newnes
 | date   = 2011
 | pages  = 7–8
 | url    = https://books.google.com/books?id=zpTnMsiUkmwC&q=inductor+%22parasitic+capacitance%22&pg=PA7
 | isbn   = 978-0080553429
 }}</ref><ref name="Kaiser">{{cite book
 | last1  = Kaiser
 | first1 = Kenneth L. 
 | title  = Electromagnetic Compatibility Handbook
 | publisher = CRC Press
 | date   = 2004
 | pages  = 6.4–6.5
 | url    = https://books.google.com/books?id=nZzOAsroBIEC&q=inductor+%22parasitic+capacitance%22
 | isbn   = 978-0849320873
 }}</ref>

A real inductor's capacitive reactance rises with frequency, and at a certain frequency, the inductor will behave as a resonant circuit. Above this self-resonant frequency, the capacitive reactance is the dominant part of the inductor's impedance. At higher frequencies, resistive losses in the windings increase due to the [[Physics:Skin effect|skin effect]] and [[Engineering:Proximity effect (electromagnetism)|proximity effect]].

Inductors with ferromagnetic cores experience additional energy losses due to [[Hysteresis|hysteresis]] and [[Physics:Eddy current|eddy current]]s in the core, which increase with frequency. At high currents, magnetic core inductors also show sudden departure from ideal behavior due to nonlinearity caused by magnetic saturation of the core.

Inductors radiate electromagnetic energy into surrounding space and may absorb electromagnetic emissions from other circuits, resulting in potential [[Engineering:Electromagnetic interference|electromagnetic interference]].

An early solid-state electrical switching and amplifying device called a [[Engineering:Saturable reactor|saturable reactor]] exploits saturation of the core as a means of stopping the inductive transfer of current via the core.

====''Q'' factor====
The winding resistance appears as a resistance in series with the inductor; it is referred to as DCR (DC resistance).  This resistance dissipates some of the reactive energy. The [[Physics: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 inductor.  High Q inductors are used with capacitors to make resonant circuits in radio transmitters and receivers.  The higher the Q is, the narrower the [[Bandwidth (signal processing)|bandwidth]] of the resonant circuit.

The Q factor of an inductor is defined as

:<math>Q = \frac{\omega L}{R}</math>

where <math>L</math> is the inductance, <math>R</math> is the DC resistance, and the product  <math>\omega L</math> is the inductive reactance

''Q'' increases linearly with frequency if ''L'' and ''R'' are constant. Although they are constant at low frequencies, the parameters vary with frequency. For example, skin effect, [[Engineering:Proximity effect (electromagnetism)|proximity effect]], and core losses increase ''R'' with frequency; winding capacitance and variations in [[Physics:Permeability (electromagnetism)|permeability]] with frequency affect ''L''.

At low frequencies and within limits, increasing the number of turns ''N'' improves ''Q'' because ''L'' varies as ''N''<sup>2</sup> while ''R'' varies linearly with ''N''. Similarly increasing the radius ''r'' of an inductor improves (or increases) ''Q'' because ''L'' varies with ''r''<sup>2</sup> while ''R'' varies linearly with ''r''.  So high ''Q'' air core inductors often have large diameters and many turns.  Both of those examples assume the diameter of the wire stays the same, so both examples use proportionally more wire. If the total mass of wire is held constant, then there would be no advantage to increasing the number of turns or the radius of the turns because the wire would have to be proportionally thinner.

Using a high permeability [[Physics:Ferromagnetic|ferromagnetic]] core can greatly increase the inductance for the same amount of copper, so the core can also increase the Q. Cores however also introduce losses that increase with frequency. The core material is chosen for best results for the frequency band.  High Q inductors must avoid saturation; one way is by using a (physically larger) air core inductor. At VHF or higher frequencies an air core is likely to be used. A well designed air core inductor may have a Q of several hundred.

==Applications==
[[File:InductorSignalFilter1.png|150px|thumb|Example of signal filtering. In this configuration, the inductor blocks AC current, while allowing DC current to pass.]]
[[File:InductorSignalFilter2.png|150px|thumb|Example of signal filtering. In this configuration, the inductor [[Decoupling (electronics)|decouples]] DC current, while allowing AC current to pass.]]
Inductors are used extensively in analog circuits and signal processing.  Applications range from the use of large inductors in power supplies, which in conjunction with filter [[Engineering:Capacitor|capacitor]]s remove [[Ripple (electrical)|ripple]] which is a multiple of the mains frequency (or the switching frequency for switched-mode power supplies) from the direct current output, to the small inductance of the [[Physics:Ferrite bead|ferrite bead]] or [[Torus|torus]] installed around a cable to prevent radio frequency interference from being transmitted down the wire.

Inductors are used as the energy storage device in many [[Engineering:Switched-mode power supply|switched-mode power supplies]] to produce DC current.  The inductor supplies energy to the circuit to keep current flowing during the "off" switching periods and enables topographies where the output voltage is higher than the input voltage.

A tuned circuit, consisting of an inductor connected to a [[Engineering:Capacitor|capacitor]], acts as a [[Electrical resonance|resonator]] for oscillating current.  Tuned circuits are widely used in [[Engineering:Radio frequency|radio frequency]] equipment such as radio transmitters and receivers, as narrow bandpass filters to select a single frequency from a composite signal, and in [[Engineering:Electronic oscillator|electronic oscillator]]s to generate sinusoidal signals.

Two (or more) inductors in proximity that have coupled magnetic flux (mutual inductance) form a [[Engineering:Transformer|transformer]], which is a fundamental component of every [[Finance:Electric utility|electric utility]] power grid. The efficiency of a transformer may decrease as the frequency increases due to eddy currents in the core material and skin effect on the windings. The size of the core can be decreased at higher frequencies. 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.<ref>{{cite web |title=What is an inductor? What is the definition of an inductor? Overview of inductor characteristics, parameters, and types. |url=https://www.gotrend.com.tw/ec99/rwd1606/news.asp?newsno=55 |access-date=2010-09-24 |publisher=GOTREND Technology |archive-date=2024-04-22 |archive-url=https://web.archive.org/web/20240422114936/https://www.gotrend.com.tw/ec99/rwd1606/news.asp?newsno=55 |url-status=dead }}</ref> Transformers enable switched-mode power supplies that [[Engineering:Galvanic isolation|galvanically isolate]] the output from the input.

Inductors are also employed in electrical transmission systems, where they are used to limit switching currents and fault currents. In this field, they are more commonly referred to as reactors.

Inductors have parasitic effects which cause them to depart from ideal behavior. They create and suffer from [[Engineering:Electromagnetic interference|electromagnetic interference]] (EMI). Their physical size prevents them from being integrated on semiconductor chips. So the use of inductors is declining in modern electronic devices, particularly compact portable devices.  Real inductors are increasingly being replaced by active circuits such as the [[Gyrator|gyrator]] which can synthesize inductance using capacitors.
{{clear}}

==Inductor construction==
{{multiple image
| align = right
| direction = horizontal
| total_width = 550
| header   =
| image1   = Common mode choke 2A with 20mH inductance.jpg
| caption1 = A ferrite core inductor with two 20&nbsp;mH windings.
| image2   = Ferrite bead no shell.jpg
| caption2 = A [[Physics:Ferrite bead|ferrite "bead"]] [[Engineering:Choke (electronics)|choke]], consisting of an encircling [[Physics:Ferrite (magnet)|ferrite]] cylinder, suppresses electronic noise in a computer power cord.
| image3   = Drosselspule im Umspannwerk Bisamberg.jpg
| caption3 = Large 50 [[Volt-ampere reactive|Mvar]] [[Physics:Three-phase electric power|three-phase]] iron-core loading inductor at a utility substation
| footer   =
}}

An inductor usually consists of a coil of conducting material, typically insulated copper wire, wrapped around a [[Physics:Magnetic core|core]] either of plastic (to create an air-core inductor) or of a [[Physics:Ferromagnetic|ferromagnetic]] (or [[Physics:Ferrimagnetism|ferrimagnetic]]) material; the latter is called an "iron core" inductor. The high [[Physics:Permeability (electromagnetism)|permeability]] of the ferromagnetic core increases the magnetic field and confines it closely to the inductor, thereby increasing the inductance. Low frequency inductors are constructed like transformers, with cores of [[Chemistry:Electrical steel|electrical steel]] laminated to prevent [[Physics:Eddy current|eddy current]]s. 'Soft' [[Physics:Ferrite (magnet)|ferrites]] are widely used for cores above [[Physics:Audio frequency|audio frequencies]], since they do not cause the large energy losses at high frequencies that ordinary iron alloys do. Inductors come in many shapes.  Some inductors have an adjustable core, which enables changing of the inductance. Inductors used to block very high frequencies are sometimes made by stringing a ferrite bead on a wire.

Small inductors can be etched directly onto a [[Engineering:Printed circuit board|printed circuit board]] by laying out the trace in a [[Spiral|spiral]] pattern. Some such planar inductors use a [[Physics:Magnetic core#Planar core|planar core]].  Small value inductors can also be built on [[Engineering:Integrated circuit|integrated circuit]]s using the same processes that are used to make [[Engineering:Interconnects (integrated circuits)|interconnect]]s. Aluminium interconnect is typically used, laid out in a spiral coil pattern. However, the small dimensions limit the inductance, and it is far more common to use a circuit called a ''[[Gyrator|gyrator]]'' that uses a [[Engineering:Capacitor|capacitor]] and active components to behave similarly to an inductor. Regardless of the design, because of the low inductances and low power dissipation on-die inductors allow, they are currently only commercially used for high frequency RF circuits.
{{clear}}

===Shielded inductors===
Inductors used in power regulation systems, lighting, and other systems that require low-noise operating conditions, are often partially or fully shielded.<ref name="Ott">{{cite book
 | last1  = Ott
 | first1 = Henry W.
 | title  = Electromagnetic Compatibility Engineering
 | publisher = John Wiley and Sons
 | date   = 2011
 | pages  = 203
 | url    = https://books.google.com/books?id=2-4WJKxzzigC&q=shielding+transformer+inductor&pg=PA203
 | isbn   = 978-1118210659
 }}</ref><ref name="Violette">{{cite book
 | last1  = Violette
 | first1 = Norman 
 | title  = Electromagnetic Compatibility Handbook
 | publisher = Springer
 | date   = 2013
 | pages  = 515–516
 | url    = https://books.google.com/books?id=K7_nCAAAQBAJ&q=shielding+transformer+inductor&pg=PA516
 | isbn   = 978-9401771443
 }}</ref> In [[Engineering:Telecommunication circuit|telecommunication circuit]]s employing induction coils and repeating transformers shielding of inductors in close proximity reduces circuit cross-talk.

==Types==
===Air-core inductor===
{{multiple image
| align = right
| direction = horizontal
| header   =
| image1   = Radio transmitter tank coil.png
| caption1 = High Q tank coil in tuned circuit of radio transmitter
| width1   = 220
| image2   = Antenna tuning coil - station WOR.jpg
| caption2 = An [[Physics:Antenna tuner|antenna tuning]] coil at an AM radio station. 
| width2   = 140
| footer   = These coils illustrate high power [[Physics:Q factor|high Q]] construction: single layer winding with turns spaced apart to reduce [[Engineering:Proximity effect (electromagnetism)|proximity effect]] losses, made of silver-plated wire or tubing to reduce [[Physics:Skin effect|skin effect]] losses, supported by narrow insulating strips to reduce dielectric losses
}}

The term ''air core coil'' describes an inductor that does not use a [[Physics:Magnetic core|magnetic core]] made of a ferromagnetic material. The term refers to coils wound on plastic, ceramic, or other nonmagnetic forms, as well as those that have only air inside the windings. Air core coils have lower inductance than ferromagnetic core coils, but are often used at high frequencies because they are free from energy losses called core losses that occur in ferromagnetic cores, which increase with frequency. A side effect that can occur in air core coils in which the winding is not rigidly supported on a form is 'microphony': mechanical vibration of the windings can cause variations in the inductance.

====Radio-frequency inductor====
[[Image:Hf spoler og transformatorer.jpg|thumb|upright=1.5|Collection of RF inductors, showing techniques to reduce losses. The three top left and the [[loop antenna|ferrite loopstick]] or rod antenna,<ref>{{cite web
  | title = An Unassuming Antenna – The Ferrite Loopstick
  | publisher = Radio Time Traveller
  | date = January 23, 2011
  | url = http://radio-timetraveller.blogspot.com/2011/01/unassuming-antenna-ferrite-loopstick.html
  | access-date = March 5, 2014 
}}</ref><ref name="Frost">{{cite web
  | last = Frost
  | first = Phil
  | title = What's an appropriate core material for a loopstick antenna?
  | work = Amateur Radio beta
  | publisher = Stack Exchange, Inc.
  | date = December 23, 2013
  | url = http://ham.stackexchange.com/questions/1156/whats-an-appropriate-core-material-for-a-loopstick-antenna
  | access-date = March 5, 2014
}}</ref><ref name="Poisel">{{cite book
  | last = Poisel
  | first = Richard 
  | title = Antenna Systems and Electronic Warfare Applications
  | publisher = Artech House
  | date = 2011
  | pages = 280
  | url = https://books.google.com/books?id=1YA1NZuo6u0C&q=%22ferrite+rod+loop+antenna&pg=PA280
  | isbn = 978-1608074846
}}</ref><ref name="Yadava">{{cite book
  | last = Yadava
  | first = R. L. 
  | title = Antenna and Wave Propagation
  | publisher = PHI Learning Pvt. Ltd
  | date = 2011
  | pages = 261
  | url = https://books.google.com/books?id=MMtjYYrE2r8C&q=%22ferrite+loop+antenna&pg=PA261
  | isbn = 978-8120342910
}}</ref> bottom, have basket windings.]]

At [[Physics:High frequency|high frequencies]], particularly [[Engineering:Radio frequency|radio frequencies]] (RF), inductors have higher resistance and other losses. In addition to causing power loss, in resonant circuits this can reduce the [[Physics:Q factor|Q factor]] of the circuit, broadening the [[Bandwidth (signal processing)|bandwidth]]. In RF inductors specialized construction techniques are used to minimize these losses. The losses are due to these effects:
*'''Skin effect''': The resistance of a wire to [[Physics:High frequency|high frequency]] current is higher than its resistance to [[Physics:Direct current|direct current]] because of [[Physics:Skin effect|skin effect]].<ref name="Zurek1">{{cite web
  | last = Zurek
  | first = Stan
  | title = Skin effect
  | website = Encyclopedia Magnetica website
  | publisher = 
  | date = 2023
  | url = http://www.e-magnetica.pl/doku.php/proximity_effect
  | format = 
  | doi = 
  | accessdate = 21 May 2024}}</ref><ref name="Kazimierczuk">{{cite book
  | last = Kazimierczuk
  | first = Marian K.
  | title = High-Frequency Magnetic Components
  | publisher = John Wiley and Sons
  | date = 2011
  | location = 
  | pages = 
  | language = 
  | url = https://books.google.com/books?id=t2TgU-uuNQ0C&pg=PA141
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 978-1-119-96491-9
  | mr = 
  | zbl = 
  | jfm =}}</ref>{{rp|p.141}}  Due to induced [[Physics:Eddy current|eddy current]]s, radio frequency alternating current does not penetrate far into the body of a conductor but travels along its surface.  For example, at 6&nbsp;MHz the skin depth of copper wire is about 0.001 inches (25&nbsp;μm); most of the current is within this depth of the surface.  Therefore, in a solid wire, the interior portion of the wire may carry little current, effectively increasing its resistance.
*'''Proximity effect''': Another similar effect that also increases the resistance of the wire at high frequencies is [[Engineering:Proximity effect (electromagnetism)|proximity effect]], which occurs in parallel wires that lie close to each other.<ref name="Zurek2">{{cite web
  | last = Zurek
  | first = Stan
  | title = Proximity effect
  | website = Encyclopedia Magnetica website
  | publisher = 
  | date = 2023
  | url = http://www.e-magnetica.pl/doku.php/proximity_effect
  | format = 
  | doi = 
  | accessdate = 21 May 2024}}</ref><ref name="Kazimierczuk" />{{rp|p.98}} The individual magnetic field of adjacent turns induces [[Physics:Eddy current|eddy current]]s in the wire of the coil, which causes the current density in the conductor to be displaced away from the adjacent surfaces.  Like skin effect, this reduces the effective cross-sectional area of the wire conducting current, increasing its resistance.
*'''Dielectric losses''': The high frequency electric field near the conductors in a [[Engineering:LC circuit|tank coil]] can cause the motion of polar molecules in nearby insulating materials, dissipating energy as heat.  For this reason, coils used for tuned circuits may be suspended in air, supported by narrow plastic or ceramic strips rather than being wound on coil forms.
*'''Parasitic capacitance''': The capacitance between individual wire turns of the coil, called [[Engineering:Parasitic capacitance|parasitic capacitance]], does not cause energy losses but can change the behavior of the coil. Each turn of the coil is at a slightly different potential, so the [[Physics:Electric field|electric field]] between neighboring turns stores charge on the wire, so the coil acts as if it has a capacitor in parallel with it. At a high enough frequency this capacitance can resonate with the inductance of the coil forming a tuned circuit, causing the coil to become self-resonant.

{{multiple image
| align = right
| direction = horizontal
| header   =
| total_width = 300
| image1   = Spider coil.jpg
| image2   = Kreuzwickelspule.png
| footer   = ''(left)'' Spiderweb coil ''(right)'' Adjustable ferrite slug-tuned RF coil with basketweave winding and litz wire
}}
To reduce parasitic capacitance and proximity effect, [[Physics:Q factor|high Q]] RF coils are constructed to avoid having many turns lying close together, parallel to one another. The windings of RF coils are often limited to a single layer, and the turns are spaced apart. To reduce resistance due to skin effect, in high-power inductors such as those used in transmitters the windings are sometimes made of a metal strip or tubing which has a larger surface area, and the surface is silver-plated.
; Basket-weave coils: To reduce proximity effect and parasitic capacitance, multilayer RF coils are wound in patterns in which successive turns are not parallel but crisscrossed at an angle; these are often called ''honeycomb'' or ''[[Engineering:Basket winding|basket-weave]]'' coils.  These are occasionally wound on a vertical insulating supports with dowels or slots, with the wire weaving in and out through the slots.
; Spiderweb coils: Another construction technique with similar advantages is flat spiral coils. These are often wound on a flat insulating support with radial spokes or slots, with the wire weaving in and out through the slots; these are called ''spiderweb'' coils. The form has an odd number of slots, so successive turns of the spiral lie on opposite sides of the form, increasing separation. 
; Litz wire: To reduce skin effect losses, some coils are wound with a special type of radio frequency wire called litz wire. Instead of a single solid conductor, litz wire consists of a number of smaller wire strands that carry the current. Unlike ordinary stranded wire, the strands are insulated from each other, to prevent skin effect from forcing the current to the surface, and are twisted or braided together. The twist pattern ensures that each wire strand spends the same amount of its length on the outside of the wire bundle, so skin effect distributes the current equally between the strands, resulting in a larger cross-sectional conduction area than an equivalent single wire.

; Axial Inductor
Small inductors for low current and low power are made in molded cases resembling resistors.  These may be either plain (phenolic) core or ferrite core.  An ohmmeter readily distinguishes them from similar-sized resistors by showing the low resistance of the inductor.

===Ferromagnetic-core inductor===
{{See also|Physics:Magnetic core}}
[[Image:Aplikimi i feriteve.png|thumb|A variety of types of ferrite core inductors and transformers]]

Ferromagnetic-core or iron-core inductors use a magnetic core made of a [[Physics:Ferromagnetic|ferromagnetic]] or ferrimagnetic material such as iron or [[Physics:Ferrite (magnet)|ferrite]] to increase the inductance. A magnetic core can increase the inductance of a coil by a factor of several thousand, by increasing the magnetic field due to its higher magnetic permeability. However the magnetic properties of the core material cause several side effects which alter the behavior of the inductor and require special construction:
  {{glossary}}
  {{term|Core losses}}{{defn|A time-varying current in a ferromagnetic inductor, which causes a time-varying magnetic field in its core, causes energy losses in the core material that are dissipated as heat, due to two processes:{{glossary}}
  {{term|[[Physics:Eddy current|Eddy current]]s}}{{defn|From [[Physics:Faraday's law of induction|Faraday's law of induction]], the changing magnetic field can induce circulating loops of electric current in the conductive metal core. The energy in these currents is dissipated as heat in the [[Physics:Electrical resistance|resistance]] of the core material. The amount of energy lost increases with the area inside the loop of current.}}
{{term|Hysteresis}}{{defn|Changing or reversing the magnetic field in the core also causes losses due to the motion of the tiny [[Physics:Magnetic domain|magnetic domain]]s it is composed of. The energy loss is proportional to the area of the hysteresis loop in the BH graph of the core material. Materials with low [[Physics:Coercivity|coercivity]] have narrow hysteresis loops and so low hysteresis losses.}}
  {{glossary end}}
Core loss is non-linear with respect to both frequency of magnetic fluctuation and magnetic flux density. Frequency of magnetic fluctuation is the frequency of AC current in the electric circuit; magnetic flux density corresponds to current in the electric circuit. Magnetic fluctuation gives rise to hysteresis, and magnetic flux density causes eddy currents in the core. These nonlinearities are distinguished from the threshold nonlinearity of saturation. Core loss can be approximately modeled with [[Physics:Steinmetz's equation|Steinmetz's equation]]. At low frequencies and over limited frequency spans (maybe a factor of 10), core loss may be treated as a linear function of frequency with minimal error. However, even in the audio range, nonlinear effects of magnetic core inductors are noticeable and of concern. }}
{{term|Saturation}}{{defn|If the current through a magnetic core coil is high enough that the core [[Physics:Saturation (magnetic)|saturates]], the inductance will fall and current will rise dramatically. This is a nonlinear threshold phenomenon and results in distortion of the signal. For example, [[Engineering:Audio signal|audio signal]]s can suffer intermodulation distortion in saturated inductors. To prevent this, in [[Engineering:Linear circuit|linear circuit]]s the current through iron core inductors must be limited below the saturation level. Some laminated cores have a narrow air gap in them for this purpose, and powdered iron cores have a distributed air gap. This allows higher levels of magnetic flux and thus higher currents through the inductor before it saturates.<ref>{{cite web |url=http://www.newark.com/pdfs/techarticles/vishay/Inductors101.pdf |title=Inductors 101 |publisher=vishay |access-date=2010-09-24}}</ref>}}
{{term|Curie point demagnetization}}{{defn|If the temperature of a ferromagnetic or ferrimagnetic core rises to a specified level, the magnetic domains dissociate, and the material becomes paramagnetic, no longer able to support magnetic flux. The inductance falls and current rises dramatically, similarly to what happens during saturation. The effect is reversible: When the temperature falls below the Curie point, magnetic flux resulting from current in the electric circuit will realign the magnetic domains of the core and its magnetic flux will be restored. The Curie point of ferromagnetic materials (iron alloys) is quite high; iron is highest at 770{{nbsp}}°C. However, for some ferrimagnetic materials (ceramic iron compounds – [[Physics:Ferrite (magnet)|ferrite]]s) the Curie point can be close to ambient temperatures (below 100{{nbsp}}°C). }}
{{glossary end}}

====Laminated-core inductor====
[[Image:Vorschaltdrossel Kvg.jpg|thumb|upright=0.8|Laminated iron core [[ballast (electrical)|ballast]] inductor for a metal halide lamp ]]
Low-frequency inductors are often made with laminated cores to prevent eddy currents, using construction similar to [[Engineering:Transformer|transformer]]s. The core is made of stacks of thin steel sheets or [[Lamination|lamination]]s oriented parallel to the field, with an insulating coating on the surface. The insulation prevents eddy currents between the sheets, so any remaining currents must be within the cross sectional area of the individual laminations, reducing the area of the loop and thus reducing the energy losses greatly. The laminations are made of low-conductivity silicon steel to further reduce eddy current losses.

====Ferrite-core inductor====
{{main|Engineering:Ferrite core}}
For higher frequencies, inductors are made with cores of ferrite. Ferrite is a ceramic ferrimagnetic material that is nonconductive, so eddy currents cannot flow within it. The formulation of ferrite is xxFe<sub>2</sub>O<sub>4</sub> where xx represents various metals. For inductor cores soft ferrites are used, which have low coercivity and thus low hysteresis losses.

====Powdered-iron-core inductor <span class="anchor" id="powdered_iron_anchor"></span>====
{{see also|Chemistry:Carbonyl iron}}

====Toroidal-core inductor====
{{main|Engineering:Toroidal inductors and transformers}}
[[File:3Com OfficeConnect ADSL Wireless 11g Firewall Router 2012-10-28-0869.jpg|thumb|Toroidal inductor in the power supply of a wireless router]]

In an inductor wound on a straight rod-shaped core, the magnetic field lines emerging from one end of the core must pass through the air to re-enter the core at the other end. This reduces the field, because much of the magnetic field path is in air rather than the higher permeability core material and is a source of [[Engineering:Electromagnetic interference|electromagnetic interference]]. A higher magnetic field and inductance can be achieved by forming the core in a closed [[Physics:Magnetic circuit|magnetic circuit]]. The magnetic field lines form closed loops within the core without leaving the core material. The shape often used is a [[Toroid|toroid]]al or doughnut-shaped ferrite core. Because of their symmetry, toroidal cores allow a minimum of the magnetic flux to escape outside the core (called ''leakage flux''), so they radiate less electromagnetic interference than other shapes. Toroidal core coils are manufactured of various materials, primarily ferrite, powdered iron and laminated cores.<ref>{{cite web|url=http://www.vishay.com/docs/34053/definit.pdf |title=Inductor and Magnetic Product Terminology |publisher=Vishay Dale |access-date=2012-09-24}}</ref>

===Variable inductor===
{{multiple image
| align = right
| direction = horizontal
| total_width = 230
| image1   = Ferrite slug tuned inductor with pot core.JPG
| image2   = Variometer.jpg
| footer   = ''(left)'' Inductor with a threaded ferrite slug ''(visible at top)'' that can be turned to move it into or out of the coil, 4.2&nbsp;cm high. ''(right)'' A variometer used in radio receivers in the 1920s
}}
[[Image:Rollspule.jpg|thumb|A "roller coil", an adjustable air-core RF inductor used in the tuned circuits of radio transmitters. One of the contacts to the coil is made by the small grooved wheel, which rides on the wire. Turning the shaft rotates the coil, moving the contact wheel up or down the coil, allowing more or fewer turns of the coil into the circuit, to change the inductance.]]

Probably the most common type of variable inductor today is one with a moveable ferrite magnetic core, which can be slid or screwed in or out of the coil. Moving the core farther into the coil increases the [[Physics:Permeability (electromagnetism)|permeability]], increasing the magnetic field and the inductance. Many inductors used in radio applications (usually less than 100&nbsp;MHz) use adjustable cores in order to tune such inductors to their desired value, since manufacturing processes have certain tolerances (inaccuracy). Sometimes such cores for frequencies above 100&nbsp;MHz are made from highly conductive non-magnetic material such as aluminium.<ref>{{cite web |url=https://www.coilcraft.com/pdfs/uni5.pdf |series=Coilcraft catalog |title=page with aluminum cores |access-date=10 July 2015 }} </ref> They decrease the inductance because the magnetic field must bypass them.

Air core inductors can use sliding contacts or multiple taps to increase or decrease the number of turns included in the circuit, to change the inductance. A type much used in the past but mostly obsolete today has a spring contact that can slide along the bare surface of the windings. The disadvantage of this type is that the contact usually [[Physics:Short circuit|short-circuits]] one or more turns. These turns act like a single-turn short-circuited transformer secondary winding; the large currents induced in them cause power losses.

A type of continuously variable air core inductor is the ''variometer''. This consists of two coils with the same number of turns connected in series, one inside the other. The inner coil is mounted on a shaft so its axis can be turned with respect to the outer coil. When the two coils' axes are collinear, with the magnetic fields pointing in the same direction, the fields add and the inductance is maximum. When the inner coil is turned so its axis is at an angle with the outer, the mutual inductance between them is smaller so the total inductance is less. When the inner coil is turned 180° so the coils are collinear with their magnetic fields opposing, the two fields cancel each other and the inductance is very small. This type has the advantage that it is continuously variable over a wide range. It is used in [[Physics:Antenna tuner|antenna tuner]]s and matching circuits to match low frequency transmitters to their antennas.

Another method to control the inductance without any moving parts requires an additional DC current bias winding which controls the permeability of an easily saturable core material. ''See'' [[Engineering:Magnetic amplifier|Magnetic amplifier]].

===Choke===
[[File:Two inductors (437342545).jpg|thumbnail|An MF or HF radio choke for tenths of an ampere, and a ferrite bead VHF choke for several amperes.]]
A [[Engineering:Choke (electronics)|choke]] is an inductor designed specifically for blocking high-frequency alternating current (AC) in an electrical circuit, while allowing DC or low-frequency signals to pass. Because the inductor restricts or "chokes" the changes in current, this type of inductor is called a choke. It usually consists of a coil of insulated wire wound on a magnetic core, although some consist of a donut-shaped "bead" of ferrite material strung on a wire. Like other inductors, chokes resist changes in current passing through them increasingly with frequency. The difference between chokes and other inductors is that chokes do not require the high [[Physics:Q factor|Q factor]] construction techniques that are used to reduce the resistance in inductors used in tuned circuits.

==Circuit analysis==
The effect of an inductor in a circuit is to oppose changes in current through it by developing a voltage across it proportional to the rate of change of the current. An ideal inductor would offer no resistance to a constant [[Physics:Direct current|direct current]]; however, only [[Physics:Superconductor|superconducting]] inductors have truly zero [[Physics:Electrical resistance|electrical resistance]].

The relationship between the time-varying voltage {{math|''v''(''t'')}} across an inductor with inductance {{mvar|L}} and the time-varying current {{math|''i''(''t'')}} passing through it is described by the [[Differential equation|differential equation]]:

:<math> v(t) = L\ \frac{~ \mathrm{d}\;\! i(t) ~}{ \mathrm{d}\;\! t } </math>
 
When there is a sinusoidal [[Physics:Alternating current|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 angular frequency (<math>\omega</math>) of the current.

:<math>\begin{align}
              i(t) ~&=~ I_\mathsf P\ \sin( \omega\ t ) \\
  \frac{~ \mathrm{d}\;\! i(t) ~}{ \mathrm{d}\;\! t } ~&=~ I_\mathsf P\ \omega\ \cos( \omega\ t ) \\
              v(t) ~&=~ L\ I_\mathsf P\ \omega\ \cos( \omega\ t ) ~.
\end{align}</math>

In this situation, the [[Physics:Phase (waves)|phase]] of the current lags that of the voltage by {{math| {{sfrac|π|2}} (90°).}} For sinusoids, as the voltage across the inductor goes to its maximum value, the current goes to zero, and as the voltage across the inductor goes to zero, the current through it goes to its maximum value.

If an inductor is connected to a direct current source with value {{mvar|I}} via a resistance {{mvar|R}} (at least the DCR of the inductor), and then the current source is short-circuited, the differential relationship above shows that the current through the inductor will discharge with an [[Exponential decay|exponential decay]]:

:<math> i(t) = I\ e^{ -\tfrac{R}{L}\ t } </math>

===Reactance===
The ratio of the peak voltage to the peak current in an inductor energised from an AC source is called the [[Physics:Electrical reactance|reactance]] and is denoted {{mvar|X}}<sub>L</sub>.

:<math>X_\mathsf L = \frac{\ V_\mathsf P }{\ I_\mathsf P } = \frac{\ \omega\ L\ I_\mathsf P }{\ I_\mathsf P } </math>

Thus,

:<math> X_\mathsf L = \omega\ L </math>

where {{mvar|ω}} is the [[Angular frequency|angular frequency]].

Reactance is measured in ohms but referred to as ''impedance'' rather than resistance; energy is stored in the magnetic field as current rises and discharged as current falls.  Inductive reactance is proportional to frequency.  At low frequency the reactance falls; at DC, the inductor behaves as a short circuit.  As frequency increases the reactance increases and at a sufficiently high frequency the reactance approaches that of an open circuit.

===Corner frequency===
In filtering applications, with respect to a particular load impedance, an inductor has a corner frequency defined as:

:<math>f_\mathsf{ 3\ dB } = \frac{ R }{ 2\pi\ L }</math>

===Laplace circuit analysis ({{mvar|s}}-domain)===
When using the [[Laplace transform]] in circuit analysis, the impedance of an ideal inductor with no initial current is represented in the {{mvar|s}} domain by:
:<math> Z(s) = L\ s\ </math>

where
: <math> L </math> is the inductance, and
: <math> s </math> is the complex frequency.

If the inductor does have initial current, it can be represented by: {{bulleted list
| adding a voltage source in series with the inductor, having the value:
: <math> L\ I_0 </math>

where
: <math> L </math> is the inductance, and
: <math> I_0 </math> is the initial current in the inductor.

(The source should have a polarity that is aligned with the initial current.)

| or by adding a current source in parallel with the inductor, having the value:
:<math> \frac{\ I_0 }{ s } </math>

where
: <math> I_0 </math> is the initial current in the inductor.
: <math> s </math> is the complex frequency.
}}

===Inductor networks===
{{main|Engineering:Series and parallel circuits}}
Inductors in a parallel configuration each have the same potential difference (voltage). To find their total equivalent inductance ({{mvar|L}}<sub>eqv</sub>):

: A diagram of several inductors, side by side, both leads of each connected to the same wires

:<math> L_\mathsf{eqv} ~=~ \left( \sum_{i=1}^n \frac{ 1 }{\ L_i }\right)^{-1} ~=~ \left( \frac{ 1 }{\ L_1 } + \frac{ 1 }{\ L_2 } + \dots + \frac{ 1 }{\ L_n } \right)^{-1} ~.</math>
 
The current through inductors in 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|400px|A diagram of several inductors, connected end to end, with the same amount of current going through each]]

:<math> L_\mathsf{eqv} ~=~ \sum_{i=1}^n L_i ~\equiv~ 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.

====Mutual inductance====
Mutual inductance occurs when the magnetic field of an inductor induces a magnetic field in an adjacent inductor.  Mutual induction is the basis of transformer construction.
:<math> M = \sqrt{\ L_1\ L_2 ~} </math>
where {{mvar|M}} as shown is the maximum possible mutual inductance between 2&nbsp;inductors and {{mvar|L}}<sub>1</sub> and {{mvar|L}}<sub>2</sub> are the two inductors.
Hovever, in general 
:<math> M \leq \sqrt{\ L_1\ L_2 ~} </math>
as only a fraction of self flux is linked with the other. 
This fraction is called "coefficient of flux linkage" (<math>\ K\ </math>) or "coupling coefficient" (with <math>\ 0 \le K \le 1\ </math>).
:<math> M = K\ \sqrt{\ L_1\ L_2 ~} </math>

==Inductance formulas==
The table below lists some common simplified formulas for calculating the approximate inductance of several inductor constructions.

{| class="wikitable"
 ! Construction
 ! Formula
 ! Notes
 |-
 ! Cylindrical air-core coil<ref name=Nagaoka>
{{cite journal
  | date=1909-05-06
  | title=The inductance coefficients of solenoids
  | journal=Journal of the College of Science, Imperial University, Tokyo, Japan
  | volume=27   | page=18
  | url=http://www.g3ynh.info/zdocs/refs/Nagaoka1909.pdf
  | url-status=dead
  | access-date=2011-11-10
  | archive-url=https://web.archive.org/web/20191217051434/http://www.g3ynh.info/zdocs/refs/Nagaoka1909.pdf
  | archive-date=2019-12-17
}}
</ref>
 | <math> L = \mu_0\ K\ N^2\ \frac{ A }{\ \ell\ } </math>
* {{math|''L''}} = inductance in [[Henry (unit)|henries]] (H)
* {{math|''μ''<sub>0</sub>}} = permeability of free space = {{nobr| 4 <math>\pi</math> × {{10^|−7}} H/m }}
* {{math|''K''}} = Nagaoka coefficient<ref name=Nagaoka/><ref group=lower-alpha>Nagaoka's coefficient ({{mvar|K}}) is approximately 1 for a coil which is much longer than its diameter and is tightly wound using small gauge wire (so that it approximates a current sheet).</ref>
* {{math|''N''}} = number of turns
* {{math|''A''}} = area of cross-section of the coil in square metres (m<sup>2</sup>)
* {{math|''ℓ''}} = length of coil in metres (m)
 |<math>K\approx 1</math> Calculation of Nagaoka's coefficient ({{math|''K''}} is complicated; normally it must be looked up in a table.<ref>{{cite book |first=Kenneth L. |last=Kaiser |year=2004 |title=Electromagnetic Compatibility Handbook |page=30.64 |place=Boca Raton, FL |publisher=CRC Press |ISBN=0849320879 }}</ref>
 |-
 ! rowspan="2"| Straight wire conductor<ref>{{cite journal |first=Edward B. |last=Rosa |year=1908 |title=The Self and Mutual Inductances of Linear Conductors |journal=Bulletin of the Bureau of Standards |volume=4 |issue=2 |pages=301–344 |doi=10.6028/bulletin.088 |doi-access=free |url=http://www.g3ynh.info/zdocs/refs/NBS/Rosa1908.pdf |url-status=dead |access-date=2013-07-09 |archive-url=https://web.archive.org/web/20150422002647/http://www.g3ynh.info/zdocs/refs/NBS/Rosa1908.pdf |archive-date=2015-04-22 }}</ref>
 | <math> L = \frac{\mu_0}{2\pi}\ \ell \left( A \; - \; B \right) + C\ ,</math>

where:
:<math>\begin{align}
  A &= \ln\left(\ \frac{\ \ell\ }{ r } + \sqrt{    \left( \frac{ \ell }{ r } \right)^2 + 1\ }\ \right) \\
  B &= \frac{ 1 }{\ \frac{ r }{ \ell } + \sqrt{1 + \left( \frac{ r }{ \ell } \right)^2\ }\ } \\
  C &= \mathcal{I_m} \left(\ \frac{ n\ \rho\ J_0(\ n\ r\ ) }{\ 2 \pi\ \omega\ \mu\ r\ J_1(\ n\ r\ )\ }\ \right)
\end{align}</math>
* {{math|''L''}} = inductance
* {{math|''ℓ''}} = cylinder length
* {{math|''r''}} = cylinder radius
* {{math|''μ''<sub>0</sub>}} = permeability of free space = {{nobr| 4 <math>\pi</math> × {{10^|−7}} H/m }}
* {{math|''μ''}} = conductor permeability
* {{math|''ρ''}} = resistivity
* {{math|''ω''}} = angular frequency
* <math> n = \sqrt{ -i\frac{ \omega\mu }{ \rho }\ } = (-1 + i)\sqrt{ \frac{\omega\mu}{2\rho}\ } </math>
* <math>\ J_0, J_1\ </math> are Bessel functions.
* <math>\tfrac{\mu_0}{2\pi}</math> = 0.2&nbsp;μH/m, exactly.
 | The term {{math|''C''}} gives the ''internal'' inductance of the wire with skin-effect correction (the imaginary part of the internal impedance of the wire). If {{math| ''ω'' {{=}} 0 }} ([[Physics:Direct current|DC]]) then <math>\ C = \tfrac{\mu}{8\pi}\ ,</math> and as ω approaches {{math| ∞ , ''C''}} approaches 0.<ref>{{cite book |first1=Hugh Hildreth |last1=Skilling |year=1951 |title=Electric Transmission Lines: Distributed constants, theory, and application |pages=153–159  |publisher=Mcgraw-Hill}}</ref>

The term {{mvar|B}} subtracts rather than adds.

 |-
 | <math>L = \frac{\mu_0}{2\pi}\ \ell \left[\ln\left(\frac{4\ell}{d}\right) - 1\right] \quad</math> (when {{nowrap|''d''² ''f'' ≫ 1 mm² MHz}})

<math>L = \frac{\mu_0}{2\pi}\ \ell \left[\ln\left(\frac{4\ell}{d}\right) - \frac{3}{4}\right] \quad</math> (when {{nowrap|''d''² ''f'' ≪ 1 mm² MHz}})
* ''L'' = inductance (nH)<ref>{{Harvnb|Rosa|1908|loc=equation (11a)}}, subst. radius ''ρ'' = d/2 and cgs units</ref><ref name="Terman straight">{{Harvnb|Terman|1943|pp=48–49}}, convert to natural logarithms and inches to mm.</ref>
* ''ℓ'' = length of conductor (mm)
* ''d'' = diameter of conductor (mm)
* ''f'' = frequency
* <math>\tfrac{\mu_0}{2\pi}</math> = 0.2&nbsp;μH/m, exactly.
 | Requires ''ℓ''&nbsp;>&nbsp;100&nbsp;''d''<ref name="Terman adjust">{{Harvtxt|Terman|1943|p=48}} states for ''ℓ''&nbsp;<&nbsp;100 ''d'', include ''d''/2''ℓ'' within the parentheses.</ref>

For relative permeability ''μ''<sub>r</sub>&nbsp;=&nbsp;1 (e.g., [[Chemistry:Copper|Cu]] or [[Chemistry:Aluminium|Al]]).
 |-
 ! Small loop or very short coil<ref>Burger, O. & Dvorský, M. (2015). ''Magnetic Loop Antenna''. Ostrava, Czech Republic: EDUCA TV o.p.s.</ref>
 | <math>L \approx \frac{\mu_0}{2\pi} N^2 \pi D \left[ \ln\left( \frac{D}{d} \right) + \left(\ln 8 - 2\right) \right]
 + \sqrt{\frac{\mu_0}{2\pi}}\; \frac{N D}{d} \sqrt{\frac{\mu_\text{r}}{2 f \sigma}}
</math>
* ''L'' = inductance in the same units as ''μ''<sub>0</sub>.
* ''D'' = Diameter of the coil (conductor center-to-center)
* ''d'' = diameter of the conductor
* ''N'' = number of turns
* ''f'' = operating frequency (regular ''f'', not ''&omega;'')
* ''&sigma;'' = specific conductivity of the coil conductor
* ''μ''<sub>r</sub> = relative permeability of the conductor 
* Total conductor length <math>\ell_\text{c} \approx N \pi D</math> should be roughly {{frac|1|10}}&nbsp;wavelength or smaller.<ref>Values of <math>\pi D</math> up to {{frac|1|3}}&nbsp;wavelength are feasible antennas, but for windings that long, this formula will be inaccurate.</ref>
* Proximity effects are not included: edge-to-edge gap between turns should be 2×''d'' or larger.
* <math>\tfrac{\mu_0}{2\pi}</math> = 0.2&nbsp;μH/m, exactly.
 | Conductor ''μ''<sub>r</sub> should be as close to 1 as possible &ndash; [[Chemistry:Copper|copper]] or [[Chemistry:Aluminium|aluminium]] rather than a magnetic or paramagnetic metal.
 |-
 ! Medium or long air-core [[Solenoid|cylindrical coil]]<ref>ARRL Handbook, 66th Ed. American Radio Relay League (1989).</ref><ref>{{Cite web|date=2014-07-09|title=Helical coil calculator|url=https://kaizerpowerelectronics.dk/calculators/helical-coil-calculator/|access-date=2020-12-29|website=Kaizer Power Electronics|language=en-US}}</ref>
 | <math>L = \frac{r^2 N^2}{23 r + 25 \ell}</math>
* ''L'' = inductance (μH)
* ''r'' = outer radius of coil (cm)
* ''ℓ'' = length of coil (cm)
* ''N'' = number of turns
 | Requires cylinder length ''ℓ''&nbsp;>&nbsp;0.4&nbsp;''r'': Length must be at least {{frac|1|5}} of the diameter. Not applicable to single-loop antennas or very short, stubby coils.
 |-
 ! Multilayer air-core coil<ref>{{cite journal|last1=Wheeler|first1=H.A.|title=Simple Inductance Formulas for Radio Coils|journal=Proceedings of the Institute of Radio Engineers|date=October 1928|volume=16|issue=10|page=1398|doi=10.1109/JRPROC.1928.221309|s2cid=51638679}}</ref>
 | <math>L = \frac{r^2 N^2}{19 r + 29 \ell + 32 d}</math>
* ''L'' = inductance (μH)
* ''r'' = mean radius of coil (cm)
* ''ℓ'' = physical length of coil winding (cm)
* ''N'' = number of turns
* ''d'' = depth of coil (outer radius minus inner radius) (cm)
 |
 |-
 ! rowspan="2"|Flat spiral air-core coil<ref>For the second formula, {{Harvtxt|Terman|1943|p=58}} which cites to {{Harvnb|Wheeler|1928}}.</ref><ref>{{Cite web|url=http://quantum-technologies.iap.uni-bonn.de/de/diplom-theses.html?task=download&file=302&token=fff191dcc4193aae12b9b5b0e9e199c9|title=A Magnetic Elevator for Neutral Atoms into a 2D State-dependent Optical Lattice Experiment|website=Uni-Bonn|access-date=2017-08-15|archive-date=2018-05-23|archive-url=https://web.archive.org/web/20180523063438/http://quantum-technologies.iap.uni-bonn.de/de/diplom-theses.html?task=download&file=302&token=fff191dcc4193aae12b9b5b0e9e199c9|url-status=dead}}</ref><ref>{{Cite web|date=2014-07-10|title=Spiral coil calculator|url=https://kaizerpowerelectronics.dk/calculators/spiral-coil-calculator/|access-date=2020-12-29|website=Kaizer Power Electronics|language=en-US}}</ref>
 | <math>L = \frac{r^2 N^2}{20 r + 28 d}</math>
* ''L'' = inductance (μH)
* ''r'' = mean radius of coil (cm)
* ''N'' = number of turns
* ''d'' = depth of coil (outer radius minus inner radius) (cm)
 |
 |-
 | <math>L = \frac{r^2 N^2}{8 r + 11 d}</math>
* ''L'' = inductance (μH)
* ''r'' = mean radius of coil (inches)
* ''N'' = number of turns
* ''d'' = depth of coil (outer radius minus inner radius) (inches)
 | Accurate to within 5&nbsp;percent for ''d''&nbsp;&gt;&nbsp;0.2&nbsp;''r''.<ref name="Terman1943">{{Harvnb|Terman|1943|p=58}}</ref>
 |-
 ! rowspan="2"|Toroidal air-core (circular cross-section)<ref>{{Harvnb|Terman|1943|p=57}}</ref>
 | <math>L = 2\pi N^2 \left(D - \sqrt{D^2 - d^2}\right)</math>
* ''L'' = inductance (nH)
* ''d'' = diameter of coil winding (cm)
* ''N'' = number of turns
* ''D'' = 2 * radius of revolution (cm)
 |
 |-
 | <math>L \approx \pi {d^2 N^2 \over D}</math>
* ''L'' = inductance (nH)
* ''d'' = diameter of coil winding (cm)
* ''N'' = number of turns
* ''D'' = 2 * radius of revolution (cm)
 | Approximation when ''d''&nbsp;&lt;&nbsp;0.1&nbsp;''D''
 |-
 ! Toroidal air-core (rectangular cross-section)<ref name="Terman1943" />
 | <math>L = 2 N^2 h \ln\left({\frac{d_2}{d_1}}\right)</math>
* ''L'' = inductance (nH)
* ''d<sub>1</sub>'' = inside diameter of toroid (cm)
* ''d<sub>2</sub>'' = outside diameter of toroid (cm)
* ''N'' = number of turns
* ''h'' = height of toroid (cm)
|
|}

==See also==
* Bellini–Tosi direction finder (radio goniometer)
* [[Physics:Hanna curve|Hanna curve]]
* [[Engineering:Induction coil|Induction coil]]
* [[Engineering:Induction cooking|Induction cooking]]
* [[Engineering:Induction loop|Induction loop]]
* [[Engineering:LC circuit|LC circuit]]
*[[Engineering:RLC circuit|RLC circuit]]
* [[Engineering:Saturable reactor|Saturable reactor]] – a type of adjustable inductor
* [[Engineering:Accumulator (energy)|Accumulator (energy)]]

==Notes==
{{notelist}}

==References==
{{reflist|25em}}

;Source
* {{Cite book |last=Terman |first=Frederick |title=Radio Engineers' Handbook |publisher=McGraw-Hill |year=1943}}

==External links==
{{wikibooks
 |1= Electronics
 |2= Inductors
 |3= Inductors
}}

{{Electronic components}}
{{Digital electronics}}
{{Machines}}

[[Category:Electromagnetic components]]
[[Category:Energy storage]]

{{Sourceattribution|Inductor}}