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Inductor

 


A selection of low-value inductors

 

Type
Passive

 

Working principle
Electromagnetic
induction

 

First production
Michael Faraday(1831)

 

Electronic
symbol

 

File:Inductor.svg

 

This box: view • talk

 

An inductor is a passive electrical
component that can store energy in a magnetic field created by
the electric current
passing through it. An inductor's ability to store magnetic energy is measured
by its inductance, in
units of henries.
Typically an inductor is a conducting wire shaped as a coil, the loops help
create a strong magnetic field inside the coil due to Faraday's law
of induction. Inductors are one of the basic electronic components used in
electronics where current and voltage change with time, due to the ability of
inductors to delay and reshape alternating currents.

 

An "ideal inductor" has inductance, but no resistance or capacitance, and does not
dissipate energy. A real inductor is equivalent to a combination of inductance,
some resistance due to the resistivity of the wire, and some capacitance. At
some frequency, usually much higher than the working frequency, a real inductor
behaves as a resonant
circuit (due to its self capacitance).
In addition to dissipating energy in the resistance of the wire, magnetic core
inductors may dissipate energy in the core due to hysteresis, and at high
currents may show other departures from ideal behavior due to nonlinearity.

 

 

[edit]
Physics

 

 

[edit]
Overview

 

Inductance (L)
(measured in henries) is an effect
resulting from the magnetic field that forms
around a current-carrying conductor that
tends to resist changes in the current. Electric current
through the conductor creates a magnetic flux proportional
to the current. A change in this current creates a change in magnetic flux that,
in turn, by Faraday's
law generates an electromotive force
(EMF) that acts to oppose this change in current. Inductance is a measure of the
amount of EMF generated for a unit change in current. For example, an inductor
with an inductance of 1 henry produces an EMF of 1 volt when the current through
the inductor changes at the rate of 1 ampere per second. The number of loops,
the size of each loop, and the material it is wrapped around all affect the
inductance. For example, the magnetic flux linking these turns can be increased
by coiling the conductor around a material with a high permeability
such as iron. This can increase the inductance by 2000 times, although less so
at high frequencies.

 

 

[edit]
Hydraulic model

 

Electric current can be modeled by the hydraulic analogy. An
inductor can be modeled by the flywheel effect of a heavy turbine rotated by the flow.
When water first starts to flow (current), the stationary turbine will cause an
obstruction in the flow and high pressure (voltage) opposing the flow until it
gets turning. Once it is turning, if there is a sudden interruption of water
flow the turbine will continue to turn by inertia, generating a high pressure to
keep the flow moving. Magnetic interactions such as in transformers are
not modeled hydraulically.

 

 

[edit]
Applications

 

 

 

 

An inductor with two 47mH windings, as may be found in a power supply.

 

Inductors are used extensively in analog circuits and
signal processing. Inductors in conjunction with capacitors and other
components form tuned circuits which can emphasize or filter out specific
signal frequencies. Applications range from the use of large inductors in power
supplies, which in conjunction with filter capacitors remove residual 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. Smaller
inductor/capacitor combinations provide tuned circuits used in
radio reception and broadcasting, for instance.

 

Two (or more) inductors which have coupled magnetic flux form a transformer, which is a
fundamental component of every electric 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. Size of the core
can be decreased at higher frequencies and, 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[1].

 

An inductor is used as the energy storage device in some switched-mode
power supplies. The inductor is energized for a specific fraction of the
regulator's switching frequency, and de-energized for the remainder of the
cycle. This energy transfer ratio determines the input-voltage to output-voltage
ratio. This XL is used in complement with an active
semiconductor device to maintain very accurate voltage control.

 

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

 

Larger value inductors may be simulated by use of gyrator circuits.

 

 

[edit]
Kind of coils

 

 

[edit]
Ferrite honeycomb coil:

 

The honeycomb coils is wounded in a crisscross manner to reduce distributed
capacitance. It is used in the circuits tuners radio in the ranges of medium and
long waves, thanks to the shape of the winding are achieved inductive high
values in low volume.

 

 

[edit]
Toroidal core coil:

 

A simple coil wound on a cylindrical form creates an external magnetic field
with a north and south pole. A toroidal coil can be created from a cylindrical
coil by bending it into a doughnut shape thereby merging the north and south
poles. In a toroidal coil, the magnetic flux is largely kept internal to the
coil. This results in less magnetic radiation from coil, and less sensitivity to
external fields.

 

 

[edit]
Inductor construction

 

 

 

 

Inductors. Major scale in centimetres.

 

An inductor is usually constructed as a coil of conducting
material, typically copper wire, wrapped around a core either of air or of
ferromagnetic material.
Core materials with a higher permeability
than air increase the magnetic field and confine 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 don't cause the large energy losses at high frequencies that ordinary
iron alloys do. This is because of their narrow hysteresis curves, and their
high resistivity prevents
eddy currents. Inductors
come in many shapes. Most are constructed as enamel coated wire wrapped around a
ferrite bobbin with wire exposed on the
outside, while some enclose the wire completely in ferrite and are called
"shielded". 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 cylinder or 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" which uses a capacitor and active
components to behave similarly to an inductor.

 

 

[edit]
In electric circuits

 

An inductor opposes changes in current. An ideal inductor would offer no
resistance to a constant direct current; however,
only superconducting
inductors have truly zero electrical
resistance.

 

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

 

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

 

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
(IP) of the current and the frequency ( f ) of
the current.

 

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

 

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

 

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

 

In this situation, the phase of the current
lags that of the voltage by 90 degrees. #

 

If an inductor is connected to a DC current source, with
value I via a resistance, R, and then the current source short
circuited, the differential relationship above shows that the current through
the inductor will discharge with an exponential decay:

 

\ i(t) = I (e^{\frac{-tR}{L}})

 

 

[edit]
Laplace circuit analysis (s-domain)

 

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

 

Z(s) = Ls\,

where
L is the inductance, and
s is the complex frequency

 

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

 


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

 

L I_0 \,

 

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

 


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

 

\frac{I_0}{s}

where
L is the inductance, and
I0 is the initial current in the
inductor.

 

 

[edit]
Inductor networks

 

Main article: Series and
parallel circuits

 

Inductors in a parallel
configuration each have the same potential difference (voltage). To find their
total equivalent inductance (Leq):

 

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

 

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

 

The current through inductors in series
stays the same, but the voltage across each inductor can be different. The sum
of the potential differences (voltage) is equal to the total voltage. To find
their total inductance:

 

A diagram of several inductors, connected end to end, with the same amount of current going through each

 

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

 

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

 

 

[edit]
Stored energy

 

The energy (measured in joules, in SI) stored by an inductor is equal to
the amount of work required to establish the current through the inductor, and
therefore the magnetic field. This is given by:

 

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

 

where L is inductance and I is the current through the
inductor(****).

 

 

[edit]
Q factor

 

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

 

The Q factor of an inductor can be found through the following formula, where
R is its internal electrical resistance and ωL is capacitive or
inductive reactance at resonance:

 

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

 

By using a ferromagnetic core, the
inductance is greatly increased for the same amount of copper, multiplying up
the Q. Cores however also introduce losses that increase with frequency. A grade
of core material is chosen for best results for the frequency band. At VHF or higher frequencies an air
core is likely to be used.

 

Inductors wound around a ferromagnetic core may saturate at
high currents, causing a dramatic decrease in inductance (and Q). This
phenomenon can be avoided by using a (physically larger) air core inductor. A
well designed air core inductor may have a Q of several hundred.

 

An almost ideal inductor (Q approaching infinity) can be created by immersing
a coil made from a superconducting alloy in liquid helium or liquid nitrogen. This
supercools the wire, causing its winding resistance to disappear. Because a
superconducting inductor is virtually lossless, it can store a large amount of
electrical energy within the surrounding magnetic field (see superconducting
magnetic energy storage).

 

 

[edit]
Inductance formulae

 

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

 

Construction
Formula
Dimensions

 

Cylindrical coil[2]
L=\frac{\mu_0KN^2A}{l}

 

 

Straight wire conductor
L = l\left(\ln\frac{4l}{d}-1\right) \cdot 200 \times 10^{-9}

 


  • L = inductance (H)

  • l = length of conductor (m)

  • d = diameter of conductor (m)

 

L = 5.08 \cdot l\left(\ln\frac{4l}{d}-1\right)

 


  • L = inductance (nH)

  • l = length of conductor (in)

  • d = diameter of conductor (in)

 

Short air-core cylindrical coil
L=\frac{r^2N^2}{9r+10l}

 


  • L = inductance (µH)

  • r = outer radius of coil (in)

  • l = length of coil (in)

  • N = number of turns

 

Multilayer air-core coil
L = \frac{0.8r^2N^2}{6r+9l+10d}

 


  • L = inductance (µH)

  • r = mean radius of coil (in)

  • l = physical length of coil winding (in)

  • N = number of turns

  • d = depth of coil (outer radius minus inner radius) (in)

 

Flat spiral air-core coil
L=\frac{r^2N^2}{(2r+2.8d) \times 10^5}

 


  • L = inductance (H)

  • r = mean radius of coil (m)

  • N = number of turns

  • d = depth of coil (outer radius minus inner radius) (m)

 

L=\frac{r^2N^2}{8r+11d}

 


  • L = inductance (µH)

  • r = mean radius of coil (in)

  • N = number of turns

  • d = depth of coil (outer radius minus inner radius) (in)

 

Toroidal core (circular cross-section)
L=\mu_0\mu_r\frac{N^2r^2}{D}

 


  • L = inductance (H)

  • μ0 = permeability
    of free space = 4π ×
    10-7 H/m

  • μr = relative permeability of core material

  • N = number of turns

  • r = radius of coil winding (m)

  • D = overall diameter of toroid (m)

 

 

[edit]
See also

 

 

 

[edit]
Synonyms

 

 

 

[edit]
Notes

 


  1. ^ http://www.wonderquest.com/expounding-aircraft-electrical-systems.htm

  2. ^ a
    b
    Nagaoka,
    Hantaro    . The Inductance Coefficients of Solenoids[1].
    27. Journal of the College of Science, Imperial University, Tokyo, Japan.
    p. 18.

 

 

[edit]
External links

 

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