Inductor
Encyclopedia
An inductor is a passive
Passivity (engineering)
Passivity is a property of engineering systems, used in a variety of engineering disciplines, but most commonly found in analog electronics and control systems...

two-terminal
Terminal (electronics)
A terminal is the point at which a conductor from an electrical component, device or network comes to an end and provides a point of connection to external circuits. A terminal may simply be the end of a wire or it may be fitted with a connector or fastener...

electrical component
Electronic component
An electronic component is a basic electronic element and may be available in a discrete form having two or more electrical terminals . These are intended to be connected together, usually by soldering to a printed circuit board, in order to create an electronic circuit with a particular function...

used to store energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

in a magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

. An inductor's ability to store magnetic energy is measured by its inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

, in units of henries. Any conductor has inductance (see "Straight wire conductor" equation below) although the conductor is typically wound in loops to reinforce the magnetic field.

Due to the time-varying magnetic field inside the coil, a voltage is induced, according to Faraday's law of electromagnetic induction
Faraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...

, which by Lenz's Law
Lenz's law
Lenz's law is a common way of understanding how electromagnetic circuits must always obey Newton's third law and The Law of Conservation of Energy...

opposes the change in current that created it. Inductors are one of the basic components used in electronics where current and voltage change with time, due to the ability of inductors to delay and reshape alternating currents. Inductors called chokes
Choke (electronics)
A choke is a coil of insulated wire, often wound on a magnetic core, used as a passive inductor which blocks higher-frequency alternating current in an electrical circuit while passing signals of much lower frequency and direct current by having an impedance largely determined by reactance, which...

are used as parts of filters in power supplies or can be used to block AC signals from passing through a circuit.

## Overview

Inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

(L) results from the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

forming around a current-carrying conductor
Electrical conductor
In physics and electrical engineering, a conductor is a material which contains movable electric charges. In metallic conductors such as copper or aluminum, the movable charged particles are electrons...

which tends to resist changes in the current. Electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

through the conductor creates a magnetic flux
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...

proportional to the current. A change in this current creates a corresponding change in magnetic flux which, in turn, by Faraday's Law
Faraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...

generates an electromotive force
Electromotive force
In physics, electromotive force, emf , or electromotance refers to voltage generated by a battery or by the magnetic force according to Faraday's Law, which states that a time varying magnetic field will induce an electric current.It is important to note that the electromotive "force" is not a...

(EMF) that opposes this change in current. Inductance is a measure of the amount of EMF generated per 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
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

such as iron. This can increase the inductance by 2000 times.

### Ideal and real inductors

An "ideal inductor" has inductance, but no resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

or capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

, and does not dissipate or radiate energy. However real inductors have resistance (due to the resistance of the wire and losses in core material), and parasitic capacitance
Parasitic capacitance
In electrical circuits, parasitic capacitance, stray capacitance or, when relevant, self-capacitance , is an unavoidable and usually unwanted capacitance that exists between the parts of an electronic component or circuit simply because of their proximity to each other...

(due to the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

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 self-resonant. At frequencies above this the capacitive reactance becomes the dominant part of the impedance. Energy is dissipated by the resistance of the wire, and by any losses in the magnetic core due to hysteresis
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...

. At high currents, iron core inductors also show gradual departure from ideal behavior due to nonlinearity
Linear circuit
A linear circuit is an electronic circuit in which, for a sinusoidal input voltage of frequency f, any output of the circuit is also sinusoidal with frequency f...

caused by magnetic saturation. At higher frequencies, resistance and resistive losses in inductors grow due to skin effect
Skin effect
Skin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...

in the inductor's winding wires. Core losses also contribute to inductor losses at higher frequencies. Practical inductors work as antennas
An antenna is an electrical device which converts electric currents into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver...

, radiating a part of energy processed into surrounding space and circuits, and accepting electromagnetic emissions from other circuits, taking part in electromagnetic interference
Electromagnetic interference
Electromagnetic interference is disturbance that affects an electrical circuit due to either electromagnetic induction or electromagnetic radiation emitted from an external source. The disturbance may interrupt, obstruct, or otherwise degrade or limit the effective performance of the circuit...

. Circuits and materials close to the inductor will have near-field coupling to the inductor's magnetic field, which may cause additional energy loss. Real-world inductor applications may consider the parasitic
Parasitic element (electrical networks)
In electrical networks, a parasitic element is a circuit element that is possessed by an electrical component but which it is not desirable for it to have for its intended purpose. For instance, a resistor is designed to possess resistance, but will also possess unwanted parasitic...

parameters as important as the inductance.

## Applications

Inductors are used extensively in analog circuits and signal processing. Inductors in conjunction with capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s and other components form tuned circuits which can emphasize or filter
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...

out specific signal frequencies. Applications range from the use of large inductors in power supplies, which in conjunction with filter capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s remove residual hums known as the mains hum
Mains hum
Mains hum, electric hum, or power line hum is an audible oscillation of alternating current at the frequency of the mains electricity, which is usually 50 Hz or 60 Hz, depending on the local power line frequency...

or other fluctuations from the direct current output, to the small inductance of the ferrite bead
A ferrite bead is a passive electric component used to suppress high frequency noise in electronic circuits. It is a specific type of electronic choke. Ferrite beads employ the mechanism of high dissipation of high frequency currents in a ferrite to build high frequency noise suppression devices...

or torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

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 that have coupled magnetic flux form a transformer
Transformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...

, which is a fundamental component of every electric utility
Public utility
A public utility is an organization that maintains the infrastructure for a public service . Public utilities are subject to forms of public control and regulation ranging from local community-based groups to state-wide government monopolies...

power grid. The efficiency of a transformer may decrease as the frequency increases due to eddy currents in the core material and skin effect
Skin effect
Skin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...

on the windings. The 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.

An inductor is used as the energy storage device in some switched-mode power supplies
Switched-mode power supply
A switched-mode power supply is an electronic power supply that incorporates a switching regulator in order to be highly efficient in the conversion of electrical power...

. 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
Gyrator
A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal...

circuits.

## Inductor construction

An inductor is usually constructed as a coil
Coil
A coil is a series of loops. A coiled coil is a structure in which the coil itself is in turn also looping.-Electromagnetic coils:An electromagnetic coil is formed when a conductor is wound around a core or form to create an inductor or electromagnet...

of conducting
Electrical conductor
In physics and electrical engineering, a conductor is a material which contains movable electric charges. In metallic conductors such as copper or aluminum, the movable charged particles are electrons...

material, typically copper wire, wrapped around a core
Magnetic core
A magnetic core is a piece of magnetic material with a high permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, inductors and magnetic assemblies. It is made of ferromagnetic metal such...

either of air or of ferromagnetic or ferrimagnetic
Ferrimagnetism
In physics, a ferrimagnetic material is one in which the magnetic moments of the atoms on different sublattices are opposed, as in antiferromagnetism; however, in ferrimagnetic materials, the opposing moments are unequal and a spontaneous magnetization remains...

material. Core materials with a higher permeability
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

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
Electrical steel
Electrical steel, also called lamination steel, silicon electrical steel, silicon steel, relay steel or transformer steel, is specialty steel tailored to produce certain magnetic properties, such as a small hysteresis area and high permeability.The material is usually manufactured in the form of...

laminate
Laminate
A laminate is a material that can be constructed by uniting two or more layers of material together. The process of creating a laminate is lamination, which in common parlance refers to the placing of something between layers of plastic and gluing them with heat and/or pressure, usually with an...

d to prevent eddy current
Eddy current
Eddy currents are electric currents induced in conductors when a conductor is exposed to a changing magnetic field; due to relative motion of the field source and conductor or due to variations of the field with time. This can cause a circulating flow of electrons, or current, within the body of...

s. 'Soft' ferrite
Ferrite (magnet)
Ferrites are chemical compounds consisting of ceramic materials with iron oxide as their principal component. Many of them are magnetic materials and they are used to make permanent magnets, ferrite cores for transformers, and in various other applications.Many ferrites are spinels with the...

s 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
Enameled wire
Enamelled wire is wire coated with a very thin insulating layer. It is used in applications such as winding electric motor coils, speakers and transformers. It is also used in the construction of electromagnets and inductors....

(magnet wire
Magnet wire
Magnet wire or enameled copper wire is a copper or aluminum wire covered with thin insulation. It is used in the construction of transformers, inductors, motors, headphones, loudspeakers, hard drive head positioners, potentiometers, and electromagnets, among other applications...

) wrapped around a ferrite
Ferrite (magnet)
Ferrites are chemical compounds consisting of ceramic materials with iron oxide as their principal component. Many of them are magnetic materials and they are used to make permanent magnets, ferrite cores for transformers, and in various other applications.Many ferrites are spinels with the...

bobbin
Bobbin
A bobbin is a spindle or cylinder, with or without flanges, on which wire, yarn, thread or film is wound. Bobbins are typically found in sewing machines, cameras, and within electronic equipment....

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 cylinder or bead on a wire.

Small inductors can be etched directly onto a printed circuit board
Printed circuit board
A printed circuit board, or PCB, is used to mechanically support and electrically connect electronic components using conductive pathways, tracks or signal traces etched from copper sheets laminated onto a non-conductive substrate. It is also referred to as printed wiring board or etched wiring...

by laying out the trace in a spiral
Spiral
In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...

pattern. Some such planar inductors use a planar core.

Small value inductors can also be built on integrated circuit
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...

s using the same processes that are used to make transistor
Transistor
A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...

s. 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
Gyrator
A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal...

" that uses a capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

and active components to behave similarly to an inductor.

### Air core inductor

The term air core coil describes an inductor that does not use a magnetic core
Magnetic core
A magnetic core is a piece of magnetic material with a high permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, inductors and magnetic assemblies. It is made of ferromagnetic metal such...

made of a ferromagnetic material. The term refers to coils wound on plastic, ceramic, or other nonmagnetic forms, as well as those that actually have 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
High frequency
High frequency radio frequencies are between 3 and 30 MHz. Also known as the decameter band or decameter wave as the wavelengths range from one to ten decameters . Frequencies immediately below HF are denoted Medium-frequency , and the next higher frequencies are known as Very high frequency...

Radio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals...

(RF), inductors have higher resistance and other losses. In addition to causing power loss, in resonant circuits this can reduce the Q factor
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....

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:
• Skin effect: The resistance of a wire to high frequency
High frequency
High frequency radio frequencies are between 3 and 30 MHz. Also known as the decameter band or decameter wave as the wavelengths range from one to ten decameters . Frequencies immediately below HF are denoted Medium-frequency , and the next higher frequencies are known as Very high frequency...

current is higher than its resistance to direct current
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...

because of skin effect
Skin effect
Skin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...

. Radio frequency alternating current does not penetrate far into the body of a conductor but travels along its surface. Therefore, in a solid wire, most of the cross sectional area of the wire is not used to conduct the current, which is in a narrow annulus on the surface. This effect increases the resistance of the wire in the coil, which may already have a relatively high resistance due to its length and small diameter.
• Proximity effect: Another similar effect that also increases the resistance of the wire at high frequencies is proximity effect
Proximity effect (electromagnetism)
In a conductor carrying alternating current, if currents are flowing through one or more other nearby conductors, such as within a closely wound coil of wire, the distribution of current within the first conductor will be constrained to smaller regions. The resulting current crowding is termed the...

, which occurs in parallel wires that lie close to each other. The individual magnetic field of adjacent turns induces eddy current
Eddy current
Eddy currents are electric currents induced in conductors when a conductor is exposed to a changing magnetic field; due to relative motion of the field source and conductor or due to variations of the field with time. This can cause a circulating flow of electrons, or current, within the body of...

s in the wire of the coil, which causes the current in the conductor to be concentrated in a thin strip on the side near the adjacent wire. Like skin effect, this reduces the effective cross-sectional area of the wire conducting current, increasing its resistance.
• Parasitic capacitance: The capacitance between individual wire turns of the coil, called parasitic capacitance
Parasitic capacitance
In electrical circuits, parasitic capacitance, stray capacitance or, when relevant, self-capacitance , is an unavoidable and usually unwanted capacitance that exists between the parts of an electronic component or circuit simply because of their proximity to each other...

, 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
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

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.

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 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.
• Honeycomb 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.
• 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
Litz wire
Litz wire is a type of cable used in electronics to carry alternating current. The wire is designed to reduce the skin effect and proximity effect losses in conductors used at frequencies up to about 1 MHz...

. Instead of a single solid conductor, litz wire consists of several 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 braided together. The braid pattern ensures that each wire strand spends the same amount of its length on the outside of the braid, so skin effect distributes the current equally between the strands, resulting in a larger cross-sectional conduction area than an equivalent single wire.

### Ferromagnetic core inductor

Ferromagnetic-core or iron-core inductors use a magnetic core
Magnetic core
A magnetic core is a piece of magnetic material with a high permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, inductors and magnetic assemblies. It is made of ferromagnetic metal such...

made of a ferromagnetic or ferrimagnetic material such as iron or ferrite
Ferrite
Ferrite may refer to:* Ferrite , iron or iron alloys with a body centred cubic crystal structure.* Ferrite , ferrimagnetic ceramic materials used in magnetic applications....

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 current
Eddy current
Eddy currents are electric currents induced in conductors when a conductor is exposed to a changing magnetic field; due to relative motion of the field source and conductor or due to variations of the field with time. This can cause a circulating flow of electrons, or current, within the body of...

s: From Faraday's law of induction
Faraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...

, 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
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

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
Coercivity
In materials science, the coercivity, also called the coercive field or coercive force, of a ferromagnetic material is the intensity of the applied magnetic field required to reduce the magnetization of that material to zero after the magnetization of the sample has been driven to saturation...

have narrow hysteresis loops and so low hysteresis losses.
For both of these processes, the energy loss per cycle of alternating current is constant, so core losses increase linearly with frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

. Online core loss calculators are available to calculate the energy loss. Using inputs such as input voltage, output voltage, output current, frequency, ambient temperature, and inductance these calculators can predict the losses of the inductors core and AC/DC based on the operating condition of the circuit being used.
• Nonlinearity
Linear circuit
A linear circuit is an electronic circuit in which, for a sinusoidal input voltage of frequency f, any output of the circuit is also sinusoidal with frequency f...

: If the current through a ferromagnetic core coil is high enough that the magnetic core saturates
Saturation (magnetic)
Seen in some magnetic materials, saturation is the state reached when an increase in applied external magnetizing field H cannot increase the magnetization of the material further, so the total magnetic field B levels off...

, the inductance will not remain constant but will change with the current through the device. This is called nonlinearity
Linear circuit
A linear circuit is an electronic circuit in which, for a sinusoidal input voltage of frequency f, any output of the circuit is also sinusoidal with frequency f...

and results in distortion of the signal. For example, audio signal
Audio signal
An audio signal is an analog representation of sound, typically as an electrical voltage. Audio signals may be synthesized directly, or may originate at a transducer such as a microphone, musical instrument pickup, phonograph cartridge, or tape head. Loudspeakers or headphones convert an electrical...

s can suffer intermodulation distortion in saturated inductors. To prevent this, in linear circuit
Linear circuit
A linear circuit is an electronic circuit in which, for a sinusoidal input voltage of frequency f, any output of the circuit is also sinusoidal with frequency f...

s the current through iron core inductors must be limited below the saturation level. Using a powdered iron core with a distributed air gap allows higher levels of magnetic flux which in turn allows a higher level of direct current through the inductor before it saturates.

#### Laminated core inductor

Low-frequency inductors are often made with laminated cores to prevent eddy currents, using construction similar to transformer
Transformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...

s. 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 the energy loss greatly. The laminations are made of low-coercivity
Coercivity
In materials science, the coercivity, also called the coercive field or coercive force, of a ferromagnetic material is the intensity of the applied magnetic field required to reduce the magnetization of that material to zero after the magnetization of the sample has been driven to saturation...

silicon steel, to reduce hysteresis losses.

#### Ferrite-core inductor

For higher frequencies, inductors are made with cores of ferrite
Ferrite (magnet)
Ferrites are chemical compounds consisting of ceramic materials with iron oxide as their principal component. Many of them are magnetic materials and they are used to make permanent magnets, ferrite cores for transformers, and in various other applications.Many ferrites are spinels with the...

. 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
Coercivity
In materials science, the coercivity, also called the coercive field or coercive force, of a ferromagnetic material is the intensity of the applied magnetic field required to reduce the magnetization of that material to zero after the magnetization of the sample has been driven to saturation...

and thus low hysteresis losses. Another similar material is powdered iron cemented with a binder.

#### Toroidal core inductor

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 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
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

core material. A higher magnetic field and inductance can be achieved by forming the core in a closed magnetic circuit
Magnetic circuit
A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other...

. The magnetic field lines form closed loops within the core without leaving the core material. The shape often used is a toroidal
Toroidal
Toroidal describes something which resembles or relates to a torus or toroid:*Torus*Toroid, a doughnut-shaped object which resembles a torus*Toroidal inductors and transformers, a type of electrical device...

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
Electromagnetic interference
Electromagnetic interference is disturbance that affects an electrical circuit due to either electromagnetic induction or electromagnetic radiation emitted from an external source. The disturbance may interrupt, obstruct, or otherwise degrade or limit the effective performance of the circuit...

than other shapes. Toroidal core coils are manufactured of various materials, primarily ferrite, Kool Mu MPP, powdered iron and laminated cores.

### Variable inductor

A variable inductor can be constructed by making one of the terminals of the device a sliding spring contact that can move along the surface of the coil, increasing or decreasing the number of turns of the coil included in the circuit. An alternative construction method is to use a moveable magnetic core, which can be slid in or out of the coil. Moving the core farther into the coil increases the permeability
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

, increasing 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. They decrease the inductance because the magnetic field must bypass them.

Another method to control the inductance without any moving parts requires an additional DC or LF current biased winding which controls the permeability of an easy saturable core material. See Magnetic amplifier
Magnetic amplifier
The magnetic amplifier is an electromagnetic device for amplifying electrical signals. The magnetic amplifier was invented early in the 20th century, and was used as an alternative to vacuum tube amplifiers where robustness and high current capacity were required...

.

## In electric circuits

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
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...

; however, only superconducting inductors have truly zero electrical resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

.

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
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

:

When there is a sinusoidal alternating current
Alternating current
In alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....

(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.

In this situation, the phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

of the current lags that of the voltage by π/2.

If an inductor is connected to a direct current source with value I via a resistance R, 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:

### 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
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:

(Note that 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
L is the inductance, and
is the initial current in the inductor.

### Inductor networks

Inductors in a parallel
Series and parallel circuits
Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur very frequently. Components connected in series are connected along a single path, so the same current flows through all of the...

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

The current through inductors in series
Series and parallel circuits
Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur very frequently. Components connected in series are connected along a single path, so the same current flows through all of the...

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.

### Stored energy

The energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

(measured in joule
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

s, in SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

) 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 (non-saturated) regions of the magnetic flux linkage and current relationship.

## 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 electric current through the coils into heat, thus causing a loss of inductive quality. The quality factor
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....

(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 is capacitive or inductive reactance at resonance:

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
Saturation (magnetic)
Seen in some magnetic materials, saturation is the state reached when an increase in applied external magnetizing field H cannot increase the magnetization of the material further, so the total magnetic field B levels off...

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
Alloy
An alloy is a mixture or metallic solid solution composed of two or more elements. Complete solid solution alloys give single solid phase microstructure, while partial solutions give two or more phases that may or may not be homogeneous in distribution, depending on thermal history...

in liquid helium
Liquid helium
Helium exists in liquid form only at extremely low temperatures. The boiling point and critical point depend on the isotope of the helium; see the table below for values. The density of liquid helium-4 at its boiling point and 1 atmosphere is approximately 0.125 g/mL Helium-4 was first liquefied...

or liquid nitrogen
Liquid nitrogen
Liquid nitrogen is nitrogen in a liquid state at a very low temperature. It is produced industrially by fractional distillation of liquid air. Liquid nitrogen is a colourless clear liquid with density of 0.807 g/mL at its boiling point and a dielectric constant of 1.4...

. 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
Superconducting magnetic energy storage
Superconducting Magnetic Energy Storage systems store energy in the magnetic field created by the flow of direct current in a superconducting coil which has been cryogenically cooled to a temperature below its superconducting critical temperature....

). Bear in mind that for inductors with cores, core losses still exist.

## Inductance formulae

The table below lists some common simplified formulas for calculating the approximate inductance of several inductor constructions.
! Construction
! Formula
! Dimensions
| Notes
|-
! Cylindrical air-core coil
|
|
• L = inductance in henries (H)
• μ0 = permeability of free space = 4 × 10−7 H/m
• K = Nagaoka coefficient
• N = number of turns
• A = area of cross-section of the coil in square metres (m2)
• l = length of coil in metres (m)

|
|-
! rowspan="3"|Straight wire conductor
|

|
• L = inductance
• l = cylinder length
• μ0 = vacuum permeability = nH/cm
• μ = conductor permeability
• p = resistivity
• ω = phase rate

| exact if ω = 0 or ω = ∞
|-
| +0-3%
|
• L = inductance (µH)
• l = length of conductor (mm)
• d = diameter of conductor (mm)
• f = frequency

|
• Cu or Al
• l > 100 d
• d2 f > 1 mm2 MHz

|-
| +0-3%
|
• L = inductance (µH)
• l = length of conductor (mm)
• d = diameter of conductor (mm)
• f = frequency

|
• Cu or Al
• l > 100 d
• d2 f < 1 mm2 MHz

|-
! Short air-core cylindrical coil
|
|
• L = inductance (µH)
• r = outer radius of coil (in)
• l = length of coil (in)
• N = number of turns

|-
! Multilayer air-core coil
|
|
• 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)

|-
! rowspan="2"|Flat spiral air-core coil
|
|
• L = inductance (µH)
• r = mean radius of coil (cm)
• N = number of turns
• d = depth of coil (outer radius minus inner radius) (cm)

|-
|
|
• 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.

|-
! rowspan="2"|Toroidal core (circular cross-section)
|
|
• L = inductance (µH)
• μ0 = permeability
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

of free space
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

= 4 × 10−7 H/m
• μr = relative permeability of core material
• d = diameter of coil winding (in)
• N = number of turns
• D = 2 * radius of revolution (in)

|-
|
|
• L = inductance (µH)
• μ0 = permeability
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

of free space
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

= 4 × 10−7 H/m
• μr = relative permeability of core material
• d = diameter of coil winding (in)
• N = number of turns
• D = 2 * radius of revolution (in)
• approximation when d < 0.1 D

|-
! Toroidal core (rectangular cross-section)
|
|
• L = inductance (µH)
• μ0 = permeability
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

of free space
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

= 4 × 10−7 H/m
• μr = relative permeability of core material
• d2 = inside diameter of toroid (in)
• d2 = outside diameter of toroid (in)
• N = number of turns
• h = height of toroid (in)

|}>