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Inductor  

A selection of lowvalue inductors  
Type  Passive 
Working principle  Electromagnetic induction 
First production  Michael Faraday (1831) 
Electronic symbol  
This article needs additional citations for verification. (December 2008) 
Inductor  

A selection of lowvalue inductors  
Type  Passive 
Working principle  Electromagnetic induction 
First production  Michael Faraday (1831) 
Electronic symbol  
An inductor, also called a coil or reactor, is a passive twoterminal electrical component which resists changes in electric current passing through it. It consists of a conductor such as a wire, usually wound into a coil. When a current flows through it, energy is stored temporarily in a magnetic field in the coil. When the current flowing through an inductor changes, the timevarying magnetic field induces a voltage in the conductor, according to Faraday’s law of electromagnetic induction, which opposes the change in current that created it.
An inductor is characterized by its inductance, the ratio of the voltage to the rate of change of current, which has units of henries (H). Inductors have values that typically range from 1 µH (10^{−6}H) to 1 H. Many inductors have a magnetic core made of iron or ferrite inside the coil, which serves to increase the magnetic field and thus the inductance. Along with capacitors and resistors, inductors are one of the three passive linear circuit elements that make up electric circuits. Inductors are widely used in alternating current (AC) electronic equipment, particularly in radio equipment. They are used to block AC while allowing DC to pass; inductors designed for this purpose are called chokes. They are also used in electronic filters to separate signals of different frequencies, and in combination with capacitors to make tuned circuits, used to tune radio and TV receivers.
Inductance (L) results from the magnetic field around a currentcarrying conductor; the electric current through the conductor creates a magnetic flux. Mathematically speaking, inductance is determined by how much magnetic flux φ through the circuit is created by a given current i^{[1]}^{[2]}^{[3]}^{[4]}
For materials that have constant permeability with magnetic flux (which does not include ferrous materials) L is constant and (1) simplifies to
Any wire or other conductor will generate a magnetic field when current flows through it, so every conductor has some inductance. The inductance of a circuit depends on the geometry of the current path as well as the magnetic permeability of nearby materials. In inductors, the wire or other conductor is shaped to increase the magnetic field. Winding the wire into a coil increases the number of times the 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 winding the coil on a "magnetic core" made of a ferromagnetic material like iron, the magnetizing field from the coil will induce 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.
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 induced by any change in magnetic flux through the circuit is^{[4]}
From (1) above^{[4]}
So inductance is also a measure of the amount of electromotive force (voltage) generated for a given rate of change of 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. This is usually taken to be the constitutive relation (defining equation) of the inductor.
The dual of the inductor is the capacitor, which stores energy in an electric field rather than a magnetic field. Its currentvoltage relation is obtained by exchanging current and voltage in the inductor equations and replacing L with the capacitance C.
The polarity (direction) of the induced voltage is given by Lenz's law, which states that it will be such as to oppose the change in current. For example, if the current through an inductor is increasing, the induced voltage will be positive at the terminal through which the current enters and negative at the terminal through which it leaves. The energy from the external circuit necessary to overcome this potential "hill" is being stored in the magnetic field of the inductor; the inductor is said to be "charging" or "energizing". If the current is decreasing, the induced voltage will be negative at the terminal through which the current enters. Energy from the magnetic field is being returned to the circuit; the inductor is said to be "discharging".
In circuit theory, inductors are idealized as obeying the mathematical relation (2) above precisely. An "ideal inductor" has inductance, but no resistance or capacitance, and does not dissipate or radiate energy. However real inductors have side effects which cause their behavior to depart from this simple model. They have resistance (due to the resistance of the wire and energy losses in core material), and parasitic capacitance (due to the electric field between the turns of wire which are at slightly different potentials). At high frequencies the capacitance begins to affect the inductor's behavior; at some frequency, real inductors behave as resonant circuits, becoming selfresonant. Above the resonant frequency the capacitive reactance becomes the dominant part of the impedance. At higher frequencies, resistive losses in the windings increase due to skin effect and proximity effect.
Inductors with ferromagnetic cores have additional energy losses due to hysteresis and eddy currents in the core, which increase with frequency. At high currents, iron core inductors also show gradual departure from ideal behavior due to nonlinearity caused by magnetic saturation of the core. An inductor may radiate electromagnetic energy into surrounding space and circuits, and may absorb electromagnetic emissions from other circuits, causing electromagnetic interference (EMI). Realworld inductor applications may consider these parasitic parameters as important as the inductance.
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 capacitors remove residual hums known as the mains hum or other fluctuations from the direct current output, to the small inductance of the ferrite bead or 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 switchedmode power supplies to produce DC current. The inductor supplies energy to the circuit to keep current flowing during the "off" switching periods.
An inductor connected to a capacitor forms a tuned circuit, which acts as a resonator for oscillating current. Tuned circuits are widely used in radio frequency equipment such as radio transmitters and receivers, as narrow bandpass filters to select a single frequency from a composite signal, and in electronic oscillators to generate sinusoidal signals.
Two (or more) inductors in proximity that have coupled magnetic flux (mutual inductance) form a transformer, which is a fundamental component of every 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.^{[5]}
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.
Because inductors have complicated side effects (detailed below) which cause them to depart from ideal behavior, because they can radiate electromagnetic interference (EMI), and most of all because of their bulk which prevents them from being integrated on semiconductor chips, 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 which can synthesize inductance using capacitors.
An inductor usually consists of a coil of conducting material, typically insulated copper wire, wrapped around a core either of plastic or of a ferromagnetic (or ferrimagnetic) material; the latter is called an "iron core" inductor. The high 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 electrical steel laminated to prevent eddy currents. 'Soft' ferrites are widely used for cores above audio frequencies, since they do not cause the large energy losses at high frequencies that ordinary iron alloys do. Inductors come in many shapes. Most are constructed as enamel coated wire (magnet wire) wrapped around a ferrite bobbin with wire exposed on the outside, while some enclose the wire completely in ferrite and are referred to as "shielded". 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 printed circuit board by laying out the trace in a spiral pattern. Some such planar inductors use a planar core.
Small value inductors can also be built on integrated circuits using the same processes that are used to make transistors. 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" that uses a capacitor and active components to behave similarly to an inductor.
The term air core coil describes an inductor that does not use a 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.
At high frequencies, particularly radio frequencies (RF), inductors have higher resistance and other losses. In addition to causing power loss, in resonant circuits this can reduce the Q factor of the circuit, broadening the bandwidth. In RF inductors, which are mostly air core types, specialized construction techniques are used to minimize these losses. The losses are due to these effects:
To reduce parasitic capacitance and proximity effect, 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 highpower 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 silverplated.
Ferromagneticcore or ironcore inductors use a magnetic core made of a ferromagnetic or ferrimagnetic material such as iron or 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:
Lowfrequency inductors are often made with laminated cores to prevent eddy currents, using construction similar to transformers. The core is made of stacks of thin steel sheets or laminations 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 lowcoercivity silicon steel, to reduce hysteresis losses.
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_{2}O_{4} where xx represents various metals. For inductor cores soft ferrites are used, which have low coercivity and thus low hysteresis losses. Another similar material is powdered iron cemented with a binder.
In an inductor wound on a straight rodshaped core, the magnetic field lines emerging from one end of the core must pass through the air to reenter 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. A higher magnetic field and inductance can be achieved by forming the core in a closed magnetic circuit. The magnetic field lines form closed loops within the core without leaving the core material. The shape often used is a toroidal or doughnutshaped 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.^{[13]}
A choke is designed specifically for blocking higherfrequency alternating current (AC) in an electrical circuit, while allowing lower frequency or DC current to pass. It usually consists of a coil of insulated wire often wound on a magnetic core, although some consist of a donutshaped "bead" of ferrite material strung on a wire. Like other inductors, chokes resist changes to the current passing through them, and so alternating currents of higher frequency, which reverse direction rapidly, are resisted more than currents of lower frequency; the choke's impedance increases with frequency. Its low electrical resistance allows both AC and DC to pass with little power loss, but it can limit the amount of AC passing through it due to its reactance.
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 permeability, increasing the magnetic field and the inductance. Many inductors used in radio applications (usually less than 100 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 MHz are made from highly conductive nonmagnetic material such as aluminum.^{[citation needed]} 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 shortcircuits one or more turns. These turns act like a singleturn shortcircuited 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 antenna tuners 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 Magnetic amplifier.
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 direct current; however, only superconducting inductors have truly zero electrical resistance.
The relationship between the timevarying voltage v(t) across an inductor with inductance L and the timevarying current i(t) passing through it is described by the differential equation:
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 (I_{P}) of the current and the frequency (f) of the current.
In this situation, the phase of the current lags that of the voltage by 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 I via a resistance R, and then the current source is shortcircuited, the differential relationship above shows that the current through the inductor will discharge with an exponential decay:
The ratio of the peak voltage to the peak current in an inductor energised from a sinusoidal source is called the reactance and is denoted X_{L}. The suffix is to distinguish inductive reactance from capacitive reactance due to capacitance.
Thus,
Reactance is measured in the same units as resistance (ohms) but is not actually a resistance. A resistance will dissipate energy as heat when a current passes. This does not happen with an inductor; rather, energy is stored in the magnetic field as the current builds and later returned to the circuit as the current falls. Inductive reactance is strongly frequency dependent. At low frequency the reactance falls, and for a steady current (zero frequency) the inductor behaves as a shortcircuit. At increasing frequency, on the other hand, the reactance increases and at a sufficiently high frequency the inductor approaches an open circuit.
When using the Laplace transform in circuit analysis, the impedance of an ideal inductor with no initial current is represented in the s domain by:
where
If the inductor does have initial current, it can be represented by:
where
(Note that the source should have a polarity that is aligned with the initial current)
where
Inductors in a parallel configuration each have the same potential difference (voltage). To find their total equivalent inductance (L_{eq}):
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:
These simple relationships hold true only when there is no mutual coupling of magnetic fields between individual inductors.
Neglecting losses, the energy (measured in joules, in SI) stored by an inductor is equal to the amount of work required to establish the current through the inductor, and therefore the magnetic field. This is given by:
where L is inductance and I is the current through the inductor.
This relationship is only valid for linear (nonsaturated) regions of the magnetic flux linkage and current relationship. In general if one decides to find the energy stored in a LTI inductor that has initial current in a specific time between and can use this:
An ideal inductor would have no resistance or energy losses. However, real 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 electric current through the coils into heat, thus causing a loss of inductive quality. The quality factor (or Q) of an inductor is the ratio of its inductive reactance to its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the inductor, the closer it approaches the behavior of an ideal, lossless, 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 of the resonant circuit.
The Q factor of an inductor can be found through the following formula, where L is the inductance, R is the inductor's effective series resistance, ω is the radian operating frequency, and the product ωL is the inductive reactance:
Notice that 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, and core losses increase R with frequency; winding capacitance and variations in permeability with frequency affect L.
Qualitatively, at low frequencies and within limits, increasing the number of turns N improves Q because L varies as N^{2} while R varies linearly with N. Similarly, increasing the radius r of an inductor improves Q because L varies as r^{2} 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 (copper). 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. At VHF or higher frequencies an air core is likely to be used.
Inductors wound around a ferromagnetic core may 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.
The table below lists some common simplified formulas for calculating the approximate inductance of several inductor constructions.
Construction  Formula  Notes 

Cylindrical aircore coil^{[14]} 
 The exact calculation of K is complex. K is approximately unity 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).^{[15]} 
Straight wire conductor^{[16]} 
 Exact if ω = 0 or ω = ∞ 
(when d² f ≫ 1 mm² MHz) (when d² f ≪ 1 mm² MHz)

 
Short aircore cylindrical coil^{[20]} 
 
Multilayer aircore coil^{[citation needed]} 
 
Flat spiral aircore coil^{[21]}^{[citation needed]} 
 
 accurate to within 5 percent for d > 0.2 r.^{[22]}  
Toroidal core (circular crosssection)^{[23]} 
 
 approximation when d < 0.1 D  
Toroidal core (rectangular crosssection)^{[22]} 

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