Editing Inductor (section)

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

An electric current flowing through a [[Electrical conductor|conductor]] generates a magnetic field surrounding it. The [[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 [[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]], with two [[terminal (electronics)|terminals]]. Winding the wire into a [[Electromagnetic coil|coil]] increases the number of times the [[magnetic flux]] [[Field Lines|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 [[ferromagnetic]] material like iron inside the coil, the magnetizing field from the coil will induce [[magnetization]] in the material, increasing the magnetic flux.  The high [[magnetic permeability|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 [[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 = black |background colour = transparent
|equation = <math>\quad\mathcal{E} = -L\frac{dI}{dt}\quad</math>
}}
if {{mvar|L}} is independent of time, current and magnetic flux linkage. Thus, inductance is also a measure of the amount of [[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:
:<math>V(t) = L\frac{\mathrm{d}I(t)}{\mathrm{d}t} \, .</math>

The [[duality (electrical circuits)|dual]] of the inductor is the [[capacitor]], which [[Capacitor#Energy stored in a capacitor|stores energy in an electric field]] rather than a magnetic field. [[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===
{{Main|Lenz's Law}}
The polarity (direction) of the induced voltage is given by [[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 [[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 [[Work (physics)|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 [[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.

This is given by:
<math>W = \int_0^{I_0} L_d(I) \, I \, dI</math>, where <math>L_d(I)</math> is the so-called "differential inductance" and is defined as: <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 = black |background colour = transparent
|equation = <math>\quad W = \frac{1}{2}L {I_0}^2\quad</math>
}}  
For inductors with magnetic cores, the above equation is only valid for [[linear circuit|linear]] regions of the magnetic flux, at currents below the [[magnetic saturation|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 [[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 [[short circuit]].

===Ideal and real inductors===
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The [[#Constitutive_equation|constitutive equation]] describes the behavior of an ''ideal inductor'' with inductance <math>L</math>, and without [[Electrical resistance|resistance]], [[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 [[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 [[skin effect]] and [[proximity effect (electromagnetism)|proximity effect]].

Inductors with ferromagnetic cores experience additional energy losses due to [[hysteresis]] and [[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 [[electromagnetic interference]].

An early solid-state electrical switching and amplifying device called a [[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 [[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, [[proximity effect (electromagnetism)|proximity effect]], and core losses increase ''R'' with frequency; winding capacitance and variations in [[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 [[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.