Inductance is the tendency of an
electrical conductor
In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge (electric current) in one or more directions. Materials made of metal are common electrical conductors. Electric current is gener ...
to oppose a change in the
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
flowing through it. The flow of electric current creates a
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
around the conductor. The field strength depends on the magnitude of the current, and follows any changes in current. From
Faraday's law of induction, any change in magnetic field through a circuit induces an
electromotive force
In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transd ...
(EMF) (
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
) in the conductors, a process known as
electromagnetic induction
Electromagnetic or magnetic induction is the production of an electromotive force (emf) across an electrical conductor in a changing magnetic field.
Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk ...
. This induced voltage created by the changing current has the effect of opposing the change in current. This is stated by
Lenz's law
Lenz's law states that the direction of the electric current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes changes in the initial magnetic field. It is named after p ...
, and the voltage is called ''
back EMF
Counter-electromotive force (counter EMF, CEMF, back EMF),Graf, "counterelectromotive force", Dictionary of Electronics is the electromotive force (EMF) manifesting as a voltage that opposes the change in current which induced it. CEMF is the EMF c ...
''.
Inductance is defined as the ratio of the induced voltage to the rate of change of current causing it. It is a proportionality factor that depends on the geometry of circuit conductors and the
magnetic permeability
In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
of nearby materials. An
electronic component
An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are not ...
designed to add inductance to a circuit is called an
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
. It typically consists of a
coil or helix of wire.
The term ''inductance'' was coined by
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
in May 1884. It is customary to use the symbol
for inductance, in honour of the physicist
Heinrich Lenz
Heinrich Friedrich Emil Lenz (; ; also Emil Khristianovich Lenz, russian: Эмилий Христианович Ленц; 12 February 1804 – 10 February 1865), usually cited as Emil Lenz or Heinrich Lenz in some countries, was a Russian physic ...
. In the
SI system, the unit of inductance is the
henry
Henry may refer to:
People
*Henry (given name)
* Henry (surname)
* Henry Lau, Canadian singer and musician who performs under the mononym Henry
Royalty
* Portuguese royalty
** King-Cardinal Henry, King of Portugal
** Henry, Count of Portugal, ...
(H), which is the amount of inductance that causes a voltage of one
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defi ...
, when the current is changing at a rate of one
ampere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elect ...
per second. It is named for
Joseph Henry, who discovered inductance independently of Faraday.
History
The history of electromagnetic induction, a facet of electromagnetism, began with observations of the ancients: electric charge or static electricity (rubbing silk on
amber
Amber is fossilized tree resin that has been appreciated for its color and natural beauty since Neolithic times. Much valued from antiquity to the present as a gemstone, amber is made into a variety of decorative objects."Amber" (2004). In Ma ...
), electric current (
lightning
Lightning is a naturally occurring electrostatic discharge during which two electric charge, electrically charged regions, both in the atmosphere or with one on the land, ground, temporarily neutralize themselves, causing the instantaneous ...
), and magnetic attraction (
lodestone
Lodestones are naturally magnetized pieces of the mineral magnetite. They are naturally occurring magnets, which can attract iron. The property of magnetism was first discovered in antiquity through lodestones. Pieces of lodestone, suspen ...
). Understanding the unity of these forces of nature, and the scientific theory of electromagnetism began in the late 18th century.
Electromagnetic induction was first described by
Michael Faraday
Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
in 1831. In Faraday's experiment, he wrapped two wires around opposite sides of an iron ring. He expected that, when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. Using a
galvanometer
A galvanometer is an electromechanical measuring instrument for electric current. Early galvanometers were uncalibrated, but improved versions, called ammeters, were calibrated and could measure the flow of current more precisely.
A galvan ...
, he observed a transient current flow in the second coil of wire each time that a battery was connected or disconnected from the first coil. This current was induced by the change in
magnetic flux that occurred when the battery was connected and disconnected.
Faraday found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (
DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("
Faraday's disk
A homopolar generator is a DC electrical generator comprising an electrically conductive disc or cylinder rotating in a plane perpendicular to a uniform static magnetic field. A potential difference is created between the center of the disc and th ...
").
Source of inductance
A current
flowing through a conductor generates a
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
around the conductor, which is described by
Ampere's circuital law. The total
magnetic flux through a circuit is equal to the product of the perpendicular component of the magnetic flux density and the area of the surface spanning the current path. If the current varies, the
magnetic flux through the circuit changes. By
Faraday's law of induction, any change in flux through a circuit induces an
electromotive force
In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transd ...
(EMF,
) in the circuit, proportional to the rate of change of flux
The negative sign in the equation indicates that the induced voltage is in a direction which opposes the change in current that created it; this is called
Lenz's law
Lenz's law states that the direction of the electric current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes changes in the initial magnetic field. It is named after p ...
. The potential is therefore called a
back EMF
Counter-electromotive force (counter EMF, CEMF, back EMF),Graf, "counterelectromotive force", Dictionary of Electronics is the electromotive force (EMF) manifesting as a voltage that opposes the change in current which induced it. CEMF is the EMF c ...
. If the current is increasing, the voltage is positive at the end of the conductor through which the current enters and negative at the end through which it leaves, tending to reduce the current. If the current is decreasing, the voltage is positive at the end through which the current leaves the conductor, tending to maintain the current. Self-inductance, usually just called inductance,
is the ratio between the induced voltage and the rate of change of the current
Thus, inductance is a property of a conductor or circuit, due to its magnetic field, which tends to oppose changes in current through the circuit. The unit of inductance in the
SI system is the
henry
Henry may refer to:
People
*Henry (given name)
* Henry (surname)
* Henry Lau, Canadian singer and musician who performs under the mononym Henry
Royalty
* Portuguese royalty
** King-Cardinal Henry, King of Portugal
** Henry, Count of Portugal, ...
(H), named after
Joseph Henry, which is the amount of inductance which generates a voltage of one
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defi ...
when the current is changing at a rate of one
ampere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elect ...
per second.
All conductors have some inductance, which may have either desirable or detrimental effects in practical electrical devices. The inductance of a circuit depends on the geometry of the current path, and on the
magnetic permeability
In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
of nearby materials;
ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
materials with a higher permeability like
iron
Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in f ...
near a conductor tend to increase the magnetic field and inductance. Any alteration to a circuit which increases the flux (total magnetic field) through the circuit produced by a given current increases the inductance, because inductance is also equal to the ratio of
magnetic flux to current
An
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
is an
electrical component
An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are not ...
consisting of a conductor shaped to increase the magnetic flux, to add inductance to a circuit. Typically it consists of a wire wound into a
coil or
helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
. A coiled wire has a higher inductance than a straight wire of the same length, because the magnetic field lines pass through the circuit multiple times, it has multiple
flux linkage
In circuit theory, flux linkage is a property of a two-terminal element. It is an extension rather than an equivalent of magnetic flux and is defined as a time integral
:\lambda = \int \mathcal \,dt,
where \mathcal is the voltage across the de ...
s. The inductance is proportional to the square of the
number of turns in the coil, assuming full flux linkage.
The inductance of a coil can be increased by placing a
magnetic core of
ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
material in the hole in the center. The magnetic field of the coil magnetizes the material of the core, aligning its
magnetic domain
A magnetic domain is a region within a magnetic material in which the magnetization is in a uniform direction. This means that the individual magnetic moments of the atoms are aligned with one another and they point in the same direction. When c ...
s, and the magnetic field of the core adds to that of the coil, increasing the flux through the coil. This is called a
ferromagnetic core inductor. A magnetic core can increase the inductance of a coil by thousands of times.
If multiple
electric circuit
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, ...
s are located close to each other, the magnetic field of one can pass through the other; in this case the circuits are said to be ''
inductively coupled''. Due to
Faraday's law of induction, a change in current in one circuit can cause a change in magnetic flux in another circuit and thus induce a voltage in another circuit. The concept of inductance can be generalized in this case by defining the
mutual inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of th ...
of circuit
and circuit
as the ratio of voltage induced in circuit
to the rate of change of current in circuit
. This is the principle behind a ''
transformer
A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...
''. The property describing the effect of one conductor on itself is more precisely called ''self-inductance'', and the properties describing the effects of one conductor with changing current on nearby conductors is called ''mutual inductance''.
[Sears and Zemansky 1964:743]
Self-inductance and magnetic energy
If the current through a conductor with inductance is increasing, a voltage
is induced across the conductor with a polarity that opposes the current—in addition to any voltage drop caused by the conductor's resistance. The charges flowing through the circuit lose potential energy. The energy from the external circuit required to overcome this "potential hill" is stored in the increased magnetic field around the conductor. Therefore, an inductor stores energy in its magnetic field. At any given time
the power
flowing into the magnetic field, which is equal to the rate of change of the stored energy
, is the product of the current
and voltage
across the conductor
From (1) above
When there is no current, there is no magnetic field and the stored energy is zero. Neglecting resistive losses, the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
(measured in
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...
s, in
SI) stored by an inductance with a current
through it is equal to the amount of work required to establish the current through the inductance from zero, and therefore the magnetic field. This is given by:
If the inductance
is constant over the current range, the stored energy is
Inductance is therefore also proportional to the energy stored in the magnetic field for a given current. This energy is stored as long as the current remains constant. If the current decreases, the magnetic field decreases, inducing a voltage in the conductor in the opposite direction, negative at the end through which current enters and positive at the end through which it leaves. This returns stored magnetic energy to the external circuit.
If
ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
materials are located near the conductor, such as in an inductor with a
magnetic core, the constant inductance equation above is only valid for
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
regions of the magnetic flux, at currents below the level at which the ferromagnetic material
saturates, where the inductance is approximately constant. If the magnetic field in the inductor approaches the level at which the core saturates, the inductance begins to change with current, and the integral equation must be used.
Inductive reactance
When a
sinusoidal
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
alternating current
Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
(AC) is passing through a linear inductance, the induced
back- is also sinusoidal. If the current through the inductance is
, from (1) above the voltage across it is
where
is the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
(peak value) of the sinusoidal current in amperes,
is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the alternating current, with
being its
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
in
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
, and
is the inductance.
Thus the amplitude (peak value) of the voltage across the inductance is
Inductive
reactance is the opposition of an inductor to an alternating current.
It is defined analogously to
electrical resistance
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels ...
in a resistor, as the ratio of the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
(peak value) of the alternating voltage to current in the component
Reactance has units of
ohm
Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm.
Ohm or OHM may also refer to:
People
* Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm''
* Germán Ohm (born 1936), Mexican boxer
* Jörg Ohm (b ...
s. It can be seen that
inductive reactance
In electrical circuits, reactance is the opposition presented to alternating current by inductance or capacitance. Greater reactance gives smaller current for the same applied voltage. Reactance is similar to resistance in this respect, but does ...
of an inductor increases proportionally with frequency
, so an inductor conducts less current for a given applied AC voltage as the frequency increases. Because the induced voltage is greatest when the current is increasing, the voltage and current waveforms are
out of phase
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it ...
; the voltage peaks occur earlier in each cycle than the current peaks. The phase difference between the current and the induced voltage is
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s or 90 degrees, showing that in an ideal inductor ''the current lags the voltage by 90°''.
Calculating inductance
In the most general case, inductance can be calculated from Maxwell's equations. Many important cases can be solved using simplifications. Where high frequency currents are considered, with
skin effect
Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the co ...
, the surface current densities and magnetic field may be obtained by solving the
Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
. Where the conductors are thin wires, self-inductance still depends on the wire radius and the distribution of the current in the wire. This current distribution is approximately constant (on the surface or in the volume of the wire) for a wire radius much smaller than other length scales.
Inductance of a straight single wire
As a practical matter, longer wires have more inductance, and thicker wires have less, analogous to their electrical resistance (although the relationships aren't linear, and are different in kind from the relationships that length and diameter bear to resistance).
Separating the wire from the other parts of the circuit introduces some unavoidable error in any formulas’ results. These inductances are often referred to as “partial inductances”, in part to encourage consideration of the other contributions to whole-circuit inductance which are omitted.
Practical formulas
For derivation of the formulas below, see Rosa (1908).
The total low frequency inductance (interior plus exterior) of a straight wire is: