Inductor

Inductor

A selection of low-value inductors
Type	Passive
Working principle‍	Electromagnetic induction
First production	Michael Faraday (1831)
Electronic symbol

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

.

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

, 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 flowing through a conductor generates a magnetic field surrounding it. The magnetic flux linkage ${\displaystyle \Phi _{\mathbf {B} }}$ generated by a given current ${\displaystyle I}$ depends on the geometric shape of the circuit. Their ratio defines the inductance ${\displaystyle L}$.^[3][4][5][6] Thus

${\displaystyle L:={\frac {\Phi _{\mathbf {B} }}{I}}}$.

The inductance of a circuit depends on the geometry of the current path as well as the

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 ${\displaystyle {\mathcal {E}}}$ induced by any change in magnetic flux through the circuit is given by^[6]

${\displaystyle {\mathcal {E}}=-{\frac {d\Phi _{\mathbf {B} }}{dt}}}$.

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

${\displaystyle \Phi _{\mathbf {B} }=LI}$.

It follows that

${\displaystyle {\mathcal {E}}=-{\frac {d\Phi _{\mathbf {B} }}{dt}}=-{\frac {d}{dt}}(LI)}$

if L is independent of time, current and magnetic flux linkage.

Thus, inductance is also a measure of the amount of

(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 current–voltage relation is obtained by exchanging current and voltage in the inductor equations and replacing L with the capacitance C.

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 voltage 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.

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 the inductor is ${\displaystyle -{\mathcal {E}}}$. The negative sign indicates that the work is done against the emf, and is not done by the emf. The current ${\displaystyle I}$ is the charge per unit time passing through the inductor. Therefore, the rate of work ${\displaystyle W}$ done by the charges against the emf, that is the rate of change of energy of the current, is given by

${\displaystyle {\frac {dW}{dt}}=-{\mathcal {E}}I}$

From the constitutive equation for the inductor, ${\displaystyle -{\mathcal {E}}=L{\frac {dI}{dt}}}$ so

${\displaystyle {\frac {dW}{dt}}=L{\frac {dI}{dt}}\cdot I=LI\cdot {\frac {dI}{dt}}}$

${\displaystyle dW=LI\cdot dI}$

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 ${\displaystyle L(I)}$. Neglecting losses, the energy ${\displaystyle W}$ stored by an inductor with a current ${\displaystyle I_{0}}$ passing through it is equal to the amount of work required to establish the current through the inductor.

This is given by: ${\displaystyle W=\int _{0}^{I_{0}}L_{d}(I)\,I\,dI}$, where ${\displaystyle L_{d}(I)}$ is the so-called "differential inductance" and is defined as: ${\displaystyle L_{d}={\frac {d\Phi _{\mathbf {B} }}{dI}}}$. 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

${\displaystyle W=L\int _{0}^{I_{0}}I\,dI}$

For inductors with magnetic cores, the above equation is only valid for

level of the inductor, where the inductance is approximately constant. Where this is not the case, the integral form must be used with ${\displaystyle L_{d}}$ variable.

Voltage step response - short and long term limit

When a voltage step is applied to an inductor, its short and long-term response are easy to calculate:

• 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
  .
• 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 ${\displaystyle L}$, and without

A real inductor's

.

Inductors with ferromagnetic cores experience additional energy losses due to

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

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

where ${\displaystyle L}$ is the inductance, ${\displaystyle R}$ is the DC resistance, and the product ${\displaystyle \omega L}$ 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

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

Inductors are used extensively in

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

to generate sinusoidal signals.

Two (or more) inductors in proximity that have coupled magnetic flux (

Transformers enable switched-mode power supplies that 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

synthesize inductance

using capacitors.

Inductor construction

An inductor usually consists of a coil of conducting material, typically insulated

, 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

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.

telecommunication

circuits employing induction coils and repeating transformers shielding of inductors in close proximity reduces circuit cross-talk.

Types

Air-core inductor

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

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.^[20]

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

metal halide lamp

Low-frequency inductors are often made with

to further reduce eddy current losses.

Ferrite-core inductor

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

are used, which have low coercivity and thus low hysteresis losses.

Powdered-iron-core inductor

Another material is powdered iron cemented with a binder.

]

Toroidal-core inductor

In an inductor wound on a straight rod-shaped core, the

Variable inductor

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.^[22] 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

; 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

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 resistricts 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

.

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:

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

When there is a

(AC) through an inductor, a sinusoidal voltage is induced. The amplitude of the voltage is proportional to the product of the amplitude (${\displaystyle I_{P}}$) of the current and the angular frequency (${\displaystyle \omega }$) of the current.

${\displaystyle {\begin{aligned}i(t)&=I_{\mathrm {P} }\sin(\omega t)\\{\frac {di(t)}{dt}}&=I_{\mathrm {P} }\omega \cos(\omega t)\\v(t)&=LI_{\mathrm {P} }\omega \cos(\omega t)\end{aligned}}}$

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:

${\displaystyle i(t)=Ie^{-{\frac {R}{L}}t}}$

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 X_L.

${\displaystyle X_{\mathrm {L} }={\frac {V_{\mathrm {P} }}{I_{\mathrm {P} }}}={\frac {\omega LI_{\mathrm {P} }}{I_{\mathrm {P} }}}}$

Thus,

${\displaystyle X_{\mathrm {L} }=\omega L}$

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

defined as:

${\displaystyle f_{\mathrm {3\,dB} }={\frac {R}{2\pi L}}}$

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:

${\displaystyle Z(s)=Ls\,}$

where

${\displaystyle L}$ is the inductance, and

${\displaystyle s}$ is the complex frequency.

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

Inductor networks

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

${\displaystyle {\frac {1}{L_{\mathrm {eq} }}}={\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+\cdots +{\frac {1}{L_{n}}}}$

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:

${\displaystyle L_{\mathrm {eq} }=L_{1}+L_{2}+\cdots +L_{n}\,\!}$

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

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.

${\displaystyle M={\sqrt {L_{1}L_{2}}}}$

where M is the maximum mutual inductance possible between 2 inductors and L₁ and L₂ are the two inductors. In general

${\displaystyle M\leq {\sqrt {L_{1}L_{2}}}}$

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".

${\displaystyle M=K{\sqrt {L_{1}L_{2}}}}$

Inductance formulas

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

Construction	Formula	Notes
Cylindrical air-core coil[23]	${\displaystyle L=\mu _{0}KN^{2}{\frac {A}{\ell }}}$ L = inductance in henries (H) μ0 = permeability of free space = 4${\displaystyle \pi }$ × 10−7 H/m K = Nagaoka coefficient[23][a] N = number of turns A = area of cross-section of the coil in square metres (m2) ℓ = length of coil in metres (m)	${\displaystyle K\approx 1}$ Calculation of Nagaoka's coefficient (K) is complicated; normally it must be looked up from a table.[24]
Straight wire conductor[25]	${\displaystyle L={\frac {\mu _{0}}{2\pi }}\ \ell \left(A\;-\;B\;+C\right)}$, where: ${\displaystyle {\begin{aligned}A&=\ln \left({\frac {\ell }{r}}+{\sqrt {\left({\frac {\ell }{r}}\right)^{2}+1}}\right)\\B&={\frac {1}{{\frac {r}{\ell }}+{\sqrt {1+\left({\frac {r}{\ell }}\right)^{2}}}}}\\C&={\frac {1}{4+r{\sqrt {{\frac {2}{\rho }}\omega \mu }}}}\end{aligned}}}$ L = inductance ℓ = cylinder length r = cylinder radius μ0 = permeability of free space = 4${\displaystyle \pi }$ × 10−7 H/m μ = conductor permeability ρ = resistivity ω = phase rate ${\displaystyle {\tfrac {\mu _{0}}{2\pi }}}$ = 0.2 µH/m, exactly.	Exact if ω = 0, or if ω = ∞. The term B subtracts rather than adds.
	${\displaystyle L={\frac {\mu _{0}}{2\pi }}\ \ell \left[\ln \left({\frac {4\ell }{d}}\right)-1\right]}$ (when d² f ≫ 1 mm² MHz) ${\displaystyle L={\frac {\mu _{0}}{2\pi }}\ \ell \left[\ln \left({\frac {4\ell }{d}}\right)-{\frac {3}{4}}\right]}$ (when d² f ≪ 1 mm² MHz) L = inductance (nH)[26][27] ℓ = length of conductor (mm) d = diameter of conductor (mm) f = frequency ${\displaystyle {\tfrac {\mu _{0}}{2\pi }}}$ = 0.2 µH/m, exactly.	Requires ℓ > 100 d[28] For relative permeability μr = 1 (e.g., Al ).
Small loop or very short coil[29]	${\displaystyle L\approx {\frac {\mu _{0}}{2\pi }}N^{2}\pi D\left[\ln \left({\frac {D}{d}}\right)+\left(\ln 8-2\right)\right]+{\sqrt {\frac {\mu _{0}}{2\pi }}}\;{\frac {ND}{d}}{\sqrt {\frac {\mu _{\text{r}}}{2f\sigma }}}}$ 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 f = operating frequency (regular f, not ω) σ = specific conductivity of the coil conductor μr = relative permeability of the conductor Total conductor length ${\displaystyle \ell _{\text{c}}\approx N\pi D}$ should be roughly 1⁄10 wavelength or smaller.[30] Proximity effects are not included: edge-to-edge gap between turns should be 2×d or larger. ${\displaystyle {\tfrac {\mu _{0}}{2\pi }}}$ = 0.2 µH/m, exactly.	Conductor μr should be as close to 1 as possible – aluminum rather than a magnetic or paramagnetic metal.
Medium or long air-core cylindrical coil[31][32]	${\displaystyle L={\frac {r^{2}N^{2}}{23r+25\ell }}}$ L = inductance (µH) r = outer radius of coil (cm) ℓ = length of coil (cm) N = number of turns	Requires cylinder length ℓ > 0.4 r: Length must be at least 1⁄5 of the diameter. Not applicable to single-loop antennas or very short, stubby coils.
Multilayer air-core coil[33]	${\displaystyle L={\frac {r^{2}N^{2}}{19r+29\ell +32d}}}$ L = inductance (µH) r = mean radius of coil (cm) ℓ = physical length of coil winding (cm) N = number of turns d = depth of coil (outer radius minus inner radius) (cm)
Flat spiral air-core coil[34][35][36]	${\displaystyle L={\frac {r^{2}N^{2}}{20r+28d}}}$ L = inductance (µH) r = mean radius of coil (cm) N = number of turns d = depth of coil (outer radius minus inner radius) (cm)
	${\displaystyle L={\frac {r^{2}N^{2}}{8r+11d}}}$ L = inductance (µH) r = mean radius of coil (in) N = number of turns d = depth of coil (outer radius minus inner radius) (in)	Accurate to within 5 percent for d > 0.2 r.[37]
Toroidal air-core (circular cross-section)[38]	${\displaystyle L=2\pi N^{2}\left(D-{\sqrt {D^{2}-d^{2}}}\right)}$ L = inductance (nH) d = diameter of coil winding (cm) N = number of turns D = 2 * radius of revolution (cm)
	${\displaystyle L\approx \pi {d^{2}N^{2} \over D}}$ L = inductance (nH) d = diameter of coil winding (cm) N = number of turns D = 2 * radius of revolution (cm)	Approximation when d < 0.1 D
Toroidal air-core (rectangular cross-section)[37]	${\displaystyle L=2N^{2}h\ln \left({\frac {d_{2}}{d_{1}}}\right)}$ L = inductance (nH) d1 = inside diameter of toroid (cm) d2 = outside diameter of toroid (cm) 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

References

Source

• Terman, Frederick (1943). Radio Engineers' Handbook. McGraw-Hill.

External links

The Wikibook Electronics has a page on the topic of: Inductors

Look up inductor in Wiktionary, the free dictionary.

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