Inductor

Article Talk

For inductors whose magnetic properties rather than electrical

properties matter, see electromagnet.

An inductor, also called a coil, choke, or reactor, is a

passive two-terminal electrical component that stores
energy in a magnetic field when an electric current flows
through it.[1] An inductor typically consists of an
insulated wire wound into a coil.

Inductor

A selection of low-value inductors

Type Passive

Working principle Electromagnetic induction

Inventor Michael Faraday

Invention year 1831; 194 years ago

Number of terminals 2

Electronic symbol

When the current flowing through the coil changes, the

time-varying magnetic field induces an electromotive
force (emf) (voltage) in the conductor, described by
Faraday's law of induction. According to 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 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 (H) named for 19th century
American scientist Joseph Henry. In the measurement of
weber
magnetic circuits, it is equivalent to ​ ampere ​. Inductors
have values that typically range from 1 µH (10−6 H) to
20 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 electronic
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.

The term inductor seems to come from Heinrich Daniel

Ruhmkorff, who called the induction coil he invented in
1851 an inductorium.[2]

Description

An electric current I creates a magnetic

field B around it

An electric current flowing through a conductor

generates a magnetic field surrounding it. The magnetic
flux linkage generated by a given current depends
on the geometric shape of the circuit. Their ratio defines
the inductance .[3][4][5][6] Thus

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 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 terminals. 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
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 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 induced by any
change in magnetic flux through the circuit is given by[6]

Reformulating the definition of L above, we obtain[6]

It follows that

if 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.

Lenz's law
Main article: 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.[7] 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.[8][9][10] 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.

Positive form of current–voltage

relationship

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 at the current's entrance
terminal (as labeled in the schematic).

The derivative form of this current–voltage relationship

is then:

The integral form of this current–voltage relationship,

starting at time with some initial current , is
then:

The dual of the inductor is the capacitor, which stores

energy in an electric field rather than a magnetic field.
Its current–voltage relation replaces L with the
capacitance C and has current and voltage swapped
from these equations.

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 done per unit charge on the charges passing

through the inductor is . The negative sign indicates
that the work is done against the emf, and is not done by
the emf. The current is the charge per unit time
passing through the inductor. Therefore, the rate of
work done by the charges against the emf, that is the
rate of change of energy of the current, is given by

From the constitutive equation for the inductor,

so

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 . Neglecting losses, the energy
stored by an inductor with a current passing through
it is equal to the amount of work required to establish
the current through the inductor.

This is given by: , where is

the so-called "differential inductance" and is defined as:

. 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

For inductors with magnetic cores, the above equation is

only valid for 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
variable.

Voltage step response

When a 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 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

The constitutive equation describes the behavior of an

ideal inductor with inductance , and without
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 capacitance between turns of the wire.[11][12]

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.

Inductors with ferromagnetic cores experience

additional energy losses due to hysteresis and eddy
currents 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 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 of the resonant
circuit.

The Q factor of an inductor is defined as

where is the inductance, is the DC resistance, and

the product 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, and core losses increase R with
frequency; winding capacitance and variations in
permeability with frequency affect L.

At low frequencies and within limits, increasing the

number of turns N improves Q because L varies as N2
while R varies linearly with N. Similarly increasing the
radius r of an inductor improves (or increases) Q
because L varies with r2 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.

Applications

Example of signal filtering.

In this configuration, the
inductor blocks AC current,
while allowing DC current to
pass.

Example of signal filtering.

In this configuration, the
inductor 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 capacitors remove 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 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

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

capacitor, 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.[13] Transformers enable switched-
mode power supplies that 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
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 which can
synthesize inductance using capacitors.

Inductor construction

A ferrite core inductor A ferrite "bead" choke, consisting of an

with two 20 mH encircling ferrite cylinder, suppresses
windings. electronic noise in a computer power
cord.

Large 50 Mvar three-phase iron-core

loading inductor at a utility substation

An inductor usually consists of a coil of conducting

material, typically insulated copper wire, wrapped
around a core either of plastic (to create an air-core
inductor) 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. 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
interconnects. 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. 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.

Shielded inductors

Inductors used in power regulation systems, lighting,

and other systems that require low-noise operating
conditions, are often partially or fully shielded.[14][15] In
telecommunication circuits employing induction coils
and repeating transformers shielding of inductors in
close proximity reduces circuit cross-talk.

Types

Air-core inductor

High Q tank coil in tuned circuit of radio An antenna tuning coil at

transmitter an AM radio station.

These coils illustrate high power high Q construction: single layer

winding with turns spaced apart to reduce proximity effect losses, made
of silver-plated wire or tubing to reduce 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 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

Collection of RF inductors, showing techniques to reduce

losses. The three top left and the ferrite loopstick or rod
antenna,[16][17][18][19] bottom, have basket windings.

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 specialized construction
techniques are used to minimize these losses. The
losses are due to these effects:

Skin effect: The resistance of a wire to high frequency

current is higher than its resistance to direct current
because of skin effect.[20][21]: p.141 Due to induced eddy
currents, radio frequency alternating current does not
penetrate far into the body of a conductor but travels
along its surface. For example, at 6 MHz the skin depth of
copper wire is about 0.001 inches (25 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
proximity effect, which occurs in parallel wires that lie
close to each other.[22][21]: p.98 The individual magnetic
field of adjacent turns induces eddy currents 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 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 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 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.

(left) Spiderweb coil (right) Adjustable ferrite slug-tuned RF coil with

basketweave winding and litz wire

To reduce parasitic capacitance and proximity effect,

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 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: Magnetic core

A variety of types of ferrite core

inductors and transformers

Ferromagnetic-core or iron-core 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:

Core losses
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:
Eddy currents
From 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 resistance of
the core material. The amount of energy lost increases
with the area inside the loop of current.
Hysteresis
Changing or reversing the magnetic field in the core
also causes losses due to the motion of the tiny
magnetic domains 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 coercivity
have narrow hysteresis loops and so low hysteresis
losses.

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 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.
Saturation
If the current through a magnetic core coil is high enough
that the core 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, audio signals can suffer intermodulation
distortion in saturated inductors. To prevent this, in linear
circuits 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.[23]
Curie point demagnetization
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 °C. However, for some ferrimagnetic materials
(ceramic iron compounds – ferrites) the Curie point can
be close to ambient temperatures (below
100 °C).[citation needed]

Laminated-core inductor

Laminated iron core ballast

inductor for a metal halide lamp

Low-frequency 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 low-conductivity
silicon steel to further reduce eddy current losses.

Ferrite-core inductor

Main article: 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 xxFe2O4 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

See also: Carbonyl iron

Another material is powdered iron cemented with a

binder. Medium frequency equipment almost exclusively
uses powdered iron cores, and inductors and
transformers built for the lower shortwaves are made
using either cemented powdered iron or
ferrites.[citation needed]

Toroidal-core inductor

Main article: Toroidal inductors and transformers

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
electromagnetic interference. 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
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.[24]

Variable inductor

(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 cm high. (right) A
variometer used in radio receivers in the 1920s

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 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 non-magnetic material
such as aluminum.[25] 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 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 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.

Choke

An MF or HF radio choke for tenths of an

ampere, and a ferrite bead VHF choke for
several amperes.

A 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 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 direct current; however, only superconducting
inductors have truly zero electrical resistance.

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:

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 ( ) of the current and the angular
frequency ( ) 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 (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:

Reactance

The ratio of the peak voltage to the peak current in an

inductor energised from an AC source is called the
reactance and is denoted XL.

Thus,

where ω is the 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:

Laplace circuit analysis (s-domain)

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

is the inductance, and

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:

where

is the inductance, and

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:

where
is the initial current in the inductor.
is the complex frequency.

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):

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.

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.

where M is the maximum mutual inductance possible

between 2 inductors and L1 and L2 are the two inductors.
In general

as only a fraction of self flux is linked with the other. This

fraction is called "Coefficient of flux linkage (K)" or
"Coefficient of coupling".

Inductance formulas

See also: Inductance § Self-inductance of thin wire shapes

The table below lists some common simplified formulas

for calculating the approximate inductance of several
inductor constructions.

Construction Formula

L = inductance in henries (H)

Cylindrical µ0 = permeability of free space = 4 × 10−7

air-core K = Nagaoka coefficient[26][a]

coil[26]
N = number of turns

A = area of cross-section of the coil in square metres (m

ℓ = length of coil in metres (m)

where:

L = inductance

ℓ = cylinder length

r = cylinder radius

µ0 = permeability of free space = 4 × 10−7

µ = conductor permeability
Straight wire ρ = resistivity
conductor[28]
ω = angular frequency

are Bessel functions.

= 0.2 µH/m, exactly.

(when d² f ≫ 1 mm² MHz

(when d² f ≪ 1 mm² MHz

L = inductance (nH)[30][31]

ℓ = length of conductor (mm)

d = diameter of conductor (mm)

f = frequency

= 0.2 µH/m, exactly.

L = inductance in the same units as µ0.

D = Diameter of the coil (conductor center-to-center)

d = diameter of the conductor

N = number of turns
Small loop or f = operating frequency (regular f, not ω)
very short
coil[33] σ = specific conductivity of the coil conductor

µr = relative permeability of the conductor

Total conductor length should be roughly

1⁄ wavelength or smaller.[34]
10

Proximity effects are not included: edge-to-edge gap

between turns should be 2×d or larger.

= 0.2 µH/m, exactly.

Medium or L = inductance (µH)

long air-core
cylindrical r = outer radius of coil (cm)
coil[35][36] ℓ = length of coil (cm)

N = number of turns

L = inductance (µH)
Multilayer
r = mean radius of coil (cm)
air-core
coil[37] ℓ = physical length of coil winding (cm)

N = number of turns

d = depth of coil (outer radius minus inner radius) (cm)

L = inductance (µH)

r = mean radius of coil (cm)

N = number of turns
Flat spiral
d = depth of coil (outer radius minus inner radius) (cm)
air-core
coil[38][39][40]

L = inductance (µH)

r = mean radius of coil (in)

N = number of turns

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

L = inductance (nH)

d = diameter of coil winding (cm)

Toroidal air- N = number of turns

core D = 2 * radius of revolution (cm)
(circular
cross-
section)[42]
L = inductance (nH)

d = diameter of coil winding (cm)

N = number of turns

D = 2 * radius of revolution (cm)

Toroidal air- L = inductance (nH)

core
d1 = inside diameter of toroid (cm)
(rectangular
cross- d2 = outside diameter of toroid (cm)
[41]
section)
N = number of turns

h = height of toroid (cm)

See also

Bellini–Tosi direction finder (radio goniometer)

Hanna curve

Induction coil

Induction cooking

Induction loop

LC circuit

RLC circuit

Saturable reactor – a type of adjustable inductor

Solenoid

Accumulator (energy)

Notes

a. ^ Nagaoka's coefficient (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).

References

External links

The Wikibook Electronics has a page on the topic of:

Inductors

Look up inductor in Wiktionary, the free dictionary.

Wikimedia Commons has media related to Inductors.

Last edited 2 months agoby Citation bot

R E L AT E D A R T I C L E S

Electromagnet
Magnet that creates a magnetic field through
an electric current

Inductance
Property of electrical conductors

Magnetic core
Object used to guide and confine magnetic
fields

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