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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 L 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 i 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 \Phi 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 \Phi 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, \mathcal) in the circuit, proportional to the rate of change of flux \mathcal(t) = -\frac\,\Phi(t) 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, L is the ratio between the induced voltage and the rate of change of the current v(t) = L\,\frac \qquad \qquad \qquad (1)\; 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 L = 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 ...
M_ of circuit k and circuit \ell as the ratio of voltage induced in circuit \ell to the rate of change of current in circuit k. 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 v(t) 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 t the power p(t) flowing into the magnetic field, which is equal to the rate of change of the stored energy U, is the product of the current i(t) and voltage v(t) across the conductor p(t) = \frac = v(t)\,i(t) From (1) above \begin \frac &= L(i)\,i\,\frac \\ pt \textU &= L(i)\,i\,\texti\, \end 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 ...
U (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 I 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: U = \int_^ L(i)\,i\,\text i\, If the inductance L(i) is constant over the current range, the stored energy is \begin U &= L\int_^\,i\,\text i \\ pt &= \tfrac L\,I^2 \end 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 i(t) = I_\text \sin\left(\omega t\right), from (1) above the voltage across it is \begin v(t) &= L \frac = L\,\frac\left _\text \sin\left(\omega t\right)\right\ &= \omega L\,I_\text\,\cos\left(\omega t\right) = \omega L\,I_\text\,\sin\left(\omega t + \right) \end where I_\text 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, \omega = 2\pi f 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 f 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 L is the inductance. Thus the amplitude (peak value) of the voltage across the inductance is V_p = \omega L\,I_p= 2\pi f\,L\,I_p 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 X_L = \frac = 2\pi f\,L 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 f, 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 \phi =\tfrac \pi
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: L_\text = 200\text\tfrac\, \ell \left ln\left(\frac\right) - 0.75 \right/math> where * L_\text is the "low-frequency" or DC inductance in nanohenry (nH or 10−9H), * \ell is the length of the wire in meters, * r is the radius of the wire in meters (hence a very small decimal number), * the constant 200\text\tfrac is the
permeability of free space The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
, commonly called \mu_\text, divided by 2 \pi; in the absence of magnetically reactive insulation the value 200 is exact when using the classical definition of ''μ''0 = , and correct to 7 decimal places when using the 2019-redefined SI value of ''μ''0 = . The constant 0.75 is just one parameter value among several; different frequency ranges, different shapes, or extremely long wire lengths require a slightly different constant ( see below). This result is based on the assumption that the radius r is much less than the length \ell, which is the common case for wires and rods. Disks or thick cylinders have slightly different formulas. For sufficiently high frequencies skin effects cause the interior currents to vanish, leaving only the currents on the surface of the conductor; the inductance for alternating current, L_\text is then given by a very similar formula: L_\text = 200\text\tfrac\, \ell \left ln\left(\frac\right) - 1 \right/math> where the variables \ell and r are the same as above; note the changed constant term now 1, from 0.75 above. In an example from everyday experience, just one of the conductors of a lamp cord long, made of 18  AWG wire, would only have an inductance of about if stretched out straight.


Mutual inductance of two parallel straight wires

There are two cases to consider: # Current travels in the same direction in each wire, and # current travels in opposing directions in the wires. Currents in the wires need not be equal, though they often are, as in the case of a complete circuit, where one wire is the source and the other the return.


Mutual inductance of two wire loops

This is the generalized case of the paradigmatic two-loop cylindrical coil carrying a uniform low frequency current; the loops are independent closed circuits that can have different lengths, any orientation in space, and carry different currents. Nonetheless, the error terms, which are not included in the integral are only small if the geometries of the loops are mostly smooth and convex: they do not have too many kinks, sharp corners, coils, crossovers, parallel segments, concave cavities or other topological "close" deformations. A necessary predicate for the reduction of the 3-dimensional manifold integration formula to a double curve integral is that the current paths be filamentary circuits, i.e. thin wires where the radius of the wire is negligible compared to its length. The mutual inductance by a filamentary circuit m on a filamentary circuit n is given by the double integral '' Neumann formula'' L_ = \frac \oint_\oint_ \frac where *C_m and C_n are the curves followed by the wires. *\mu_0 is the
permeability of free space The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
() *\mathrm\mathbf_m is a small increment of the wire in circuit Cm *\mathbf_m is the position of d\mathbf_m in space *\mathrm\mathbf_n is a small increment of the wire in circuit Cn *\mathbf_n is the position of d\mathbf_n in space


Derivation

M_ \mathrel\stackrel \frac where * I_j is the current through the jth wire, this current creates the magnetic flux \Phi_\ \,through the ith surface * \Phi_ is the magnetic flux through the ''i''th surface due to the
electrical 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, ...
outlined by C_j: \Phi_ = \int_ \mathbf_j\cdot\mathrm\mathbf = \int_ (\nabla\times\mathbf)\cdot\mathrm\mathbf = \oint_ \mathbf_j\cdot\mathrm\mathbf_i = \oint_ \left(\frac \oint_ \frac\right) \cdot \mathrm\mathbf_i where
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
has been used for the 3rd equality step. For the last equality step, we used the
retarded potential In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so the ...
expression for A_j and we ignore the effect of the retarded time (assuming the geometry of the circuits is small enough compared to the wavelength of the current they carry). It is actually an approximation step, and is valid only for local circuits made of thin wires.


Self-inductance of a wire loop

Formally, the self-inductance of a wire loop would be given by the above equation with m = n. However, here 1/, \mathbf - \mathbf', becomes infinite, leading to a logarithmically divergent integral.since \int \fracdx = \ln(x) for x>0 This necessitates taking the finite wire radius a and the distribution of the current in the wire into account. There remains the contribution from the integral over all points and a correction term, L = \frac \left oint_\oint_ \frac\right+ \frac\,\ell\,Y + O \quad \text \; \left, \mathbf - \mathbf'\ > \fraca where *\mathbf s and \mathbf' are distances along the curves C and C' respectively *a is the radius of the wire *\ell is the length of the wire *Y is a constant that depends on the distribution of the current in the wire: Y = 0 when the current flows on the surface of the wire (total
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 ...
), Y = \frac when the current is evenly over the cross-section of the wire. *O is an error term O(\mu_0 a) when the loop has sharp corners, and O\mathord\left( \frac \right)when it is a smooth curve. These are small when the wire is long compared to its radius.


Inductance of a solenoid

A
solenoid upright=1.20, An illustration of a solenoid upright=1.20, Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines A solenoid () is a type of electromagnet formed by a helix, helical coil of wire whose ...
is a long, thin coil; i.e., a coil whose length is much greater than its diameter. Under these conditions, and without any magnetic material used, the
magnetic flux density 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 ...
B within the coil is practically constant and is given by B = \frac where \mu_0 is the
magnetic constant The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
, N the number of turns, i the current and l the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density B by the cross-section area A: \Phi = \frac, When this is combined with the definition of inductance L = \frac, it follows that the inductance of a solenoid is given by: L = \frac. Therefore, for air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current.


Inductance of a coaxial cable

Let the inner conductor have radius r_i and permeability \mu_i, let the dielectric between the inner and outer conductor have permeability \mu_d, and let the outer conductor have inner radius r_, outer radius r_, and permeability \mu_0. However, for a typical coaxial line application, we are interested in passing (non-DC) signals at frequencies for which the resistive
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 ...
cannot be neglected. In most cases, the inner and outer conductor terms are negligible, in which case one may approximate L' = \frac \approx \frac \ln \frac


Inductance of multilayer coils

Most practical air-core inductors are multilayer cylindrical coils with square cross-sections to minimize average distance between turns (circular cross -sections would be better but harder to form).


Magnetic cores

Many inductors include a magnetic core at the center of or partly surrounding the winding. Over a large enough range these exhibit a nonlinear permeability with effects such as
magnetic saturation Seen in some magnetic materials, saturation is the state reached when an increase in applied external magnetic field ''H'' cannot increase the magnetization of the material further, so the total magnetic flux density ''B'' more or less levels off ...
. Saturation makes the resulting inductance a function of the applied current. The secant or large-signal inductance is used in flux calculations. It is defined as: L_s(i) \mathrel\overset \frac = \frac The differential or small-signal inductance, on the other hand, is used in calculating voltage. It is defined as: L_d(i) \mathrel\overset \frac = \frac The circuit voltage for a nonlinear inductor is obtained via the differential inductance as shown by Faraday's Law and the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
of calculus. v(t) = \frac = \frac\frac = L_d(i)\frac Similar definitions may be derived for nonlinear mutual inductance.


Mutual inductance

Mutual inductance is defined as the ratio between the EMF induced in one loop or coil by the rate of change of current in another loop or coil. Mutual inductance is given the symbol .


Derivation of mutual inductance

The inductance equations above are a consequence of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
. For the important case of electrical circuits consisting of thin wires, the derivation is straightforward. In a system of K wire loops, each with one or several wire turns, the
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 ...
of loop m, \lambda_m, is given by \displaystyle \lambda_m = N_m \Phi_m = \sum\limits_^K L_\ i_n\,. Here N_m denotes the number of turns in loop m; \Phi_m is the magnetic flux through loop m; and L_ are some constants described below. This equation follows from Ampere's law: ''magnetic fields and fluxes are linear functions of the currents''. By Faraday's law of induction, we have \displaystyle v_m = \frac = N_m \frac = \sum\limits_^K L_\frac, where v_m denotes the voltage induced in circuit m. This agrees with the definition of inductance above if the coefficients L_ are identified with the coefficients of inductance. Because the total currents N_n\ i_n contribute to \Phi_m it also follows that L_ is proportional to the product of turns N_m\ N_n.


Mutual inductance and magnetic field energy

Multiplying the equation for ''vm'' above with ''imdt'' and summing over ''m'' gives the energy transferred to the system in the time interval ''dt'', \sum \limits_m^K i_m v_m \textt = \sum\limits_^K i_m L_ \texti_n \mathrel\overset \sum\limits_^K \frac \texti_n. This must agree with the change of the magnetic field energy, ''W'', caused by the currents. The
integrability condition In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
\displaystyle\frac = \frac requires ''Lm,n = Ln,m''. The inductance matrix, ''Lm,n'', thus is symmetric. The integral of the energy transfer is the magnetic field energy as a function of the currents, \displaystyle W\left(i\right) = \frac \sum \limits_^K i_m L_ i_n. This equation also is a direct consequence of the linearity of Maxwell's equations. It is helpful to associate changing electric currents with a build-up or decrease of magnetic field energy. The corresponding energy transfer requires or generates a voltage. A mechanical analogy in the ''K'' = 1 case with magnetic field energy (1/2)''Li''2 is a body with mass ''M'', velocity ''u'' and kinetic energy (1/2)''Mu''2. The rate of change of velocity (current) multiplied with mass (inductance) requires or generates a force (an electrical voltage). Mutual inductance occurs when the change in current in one inductor induces a voltage in another nearby inductor. It is important as the mechanism by which
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' ...
s work, but it can also cause unwanted coupling between conductors in a circuit. The mutual inductance, M_, is also a measure of the coupling between two inductors. The mutual inductance by circuit i on circuit j is given by the double integral '' Neumann formula'', see calculation techniques The mutual inductance also has the relationship: M_ = N_1\ N_2\ P_ \! where Once the mutual inductance, M, is determined, it can be used to predict the behavior of a circuit: v_1 = L_1\ \frac - M\ \frac where The minus sign arises because of the sense the current i_2 has been defined in the diagram. With both currents defined going into the dots the sign of M will be positive (the equation would read with a plus sign instead).


Coupling coefficient

The coupling coefficient is the ratio of the open-circuit actual voltage ratio to the ratio that would be obtained if all the flux coupled from one
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 l ...
to the other. The coupling coefficient is related to mutual inductance and self inductances in the following way. From the two simultaneous equations expressed in the two-port matrix the open-circuit voltage ratio is found to be: _\text = where while the ratio if all the flux is coupled is the ratio of the turns, hence the ratio of the square root of the inductances _\text = \sqrt thus, M = k \sqrt where The coupling coefficient is a convenient way to specify the relationship between a certain orientation of inductors with arbitrary inductance. Most authors define the range as , but some define it as Allowing negative values of k captures phase inversions of the coil connections and the direction of the windings.


Matrix representation

Mutually coupled inductors can be described by any of the
two-port network A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
parameter matrix representations. The most direct are the z parameters, which are given by mathbf z= s \begin L_1 \ M \\ M \ L_2 \end where s is the
complex frequency In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the compl ...
variable, L_1 and L_2 are the inductances of the primary and secondary coil, respectively, and M is the mutual inductance between the coils.


Equivalent circuits


T-circuit

Mutually coupled inductors can equivalently be represented by a T-circuit of inductors as shown. If the coupling is strong and the inductors are of unequal values then the series inductor on the step-down side may take on a negative value. This can be analyzed as a two port network. With the output terminated with some arbitrary impedance, Z, the voltage gain, A_v, is given by, A_\mathrm = \frac = \frac where k is the coupling constant and s is the
complex frequency In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the compl ...
variable, as above. For tightly coupled inductors where this reduces to A_\mathrm v = \sqrt which is independent of the load impedance. If the inductors are wound on the same core and with the same geometry, then this expression is equal to the turns ratio of the two inductors because inductance is proportional to the square of turns ratio. The input impedance of the network is given by, Z_\text = \frac = \frac\, Z\, \left( \frac \right) \left( 1 + \frac \right) For this reduces to Z_\text = \frac = \frac\, Z\, \left( \frac \right) Thus, current gain, A_i is ''not'' independent of load unless the further condition , sL_2, \gg , Z, is met, in which case, Z_\text \approx Z and A_\text \approx \sqrt =


π-circuit

Alternatively, two coupled inductors can be modelled using a ''π'' equivalent circuit with optional ideal transformers at each port. While the circuit is more complicated than a T-circuit, it can be generalized to circuits consisting of more than two coupled inductors. Equivalent circuit elements L_\text, L_\text have physical meaning, modelling respectively
magnetic reluctance Magnetic reluctance, or magnetic resistance, is a concept used in the analysis of magnetic circuits. It is defined as the ratio of magnetomotive force (mmf) to magnetic flux. It represents the opposition to magnetic flux, and depends on the geo ...
s of coupling paths and
magnetic reluctance Magnetic reluctance, or magnetic resistance, is a concept used in the analysis of magnetic circuits. It is defined as the ratio of magnetomotive force (mmf) to magnetic flux. It represents the opposition to magnetic flux, and depends on the geo ...
s of leakage paths. For example, electric currents flowing through these elements correspond to coupling and leakage magnetic fluxes. Ideal transformers normalize all self-inductances to 1 Henry to simplify mathematical formulas. Equivalent circuit element values can be calculated from coupling coefficients with \begin L_ &= \frac \\ pt L_ &= \frac \end where coupling coefficient matrix and its cofactors are defined as : \mathbf = \begin 1 & k_ & \cdots & k_ \\ k_ & 1 & \cdots & k_ \\ \vdots & \vdots & \ddots & \vdots \\ k_ & k_ & \cdots & 1 \end\quad and \quad \mathbf_ = (-1)^\,\mathbf_. For two coupled inductors, these formulas simplify to : L_ = \frac\quad and \quad L_ = L_ \!=\! k_ + 1, and for three coupled inductors (for brevity shown only for L_\text and L_\text) : L_ = \frac \quad and \quad L_ = \frac .


Resonant transformer

When a capacitor is connected across one winding of a transformer, making the winding a
tuned circuit An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
(resonant circuit) it is called a single-tuned transformer. When a capacitor is connected across each winding, it is called a double tuned transformer. These '' resonant transformers'' can store oscillating electrical energy similar to a
resonant circuit An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
and thus function as a
bandpass filter A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. Description In electronics and signal processing, a filter is usually a two-port ...
, allowing frequencies near their
resonant frequency Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
to pass from the primary to secondary winding, but blocking other frequencies. The amount of mutual inductance between the two windings, together with the
Q factor In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
of the circuit, determine the shape of the frequency response curve. The advantage of the double tuned transformer is that it can have a wider bandwidth than a simple tuned circuit. The coupling of double-tuned circuits is described as loose-, critical-, or over-coupled depending on the value of the coupling coefficient k. When two tuned circuits are loosely coupled through mutual inductance, the bandwidth is narrow. As the amount of mutual inductance increases, the bandwidth continues to grow. When the mutual inductance is increased beyond the critical coupling, the peak in the frequency response curve splits into two peaks, and as the coupling is increased the two peaks move further apart. This is known as overcoupling. Stongly-coupled self-resonant coils can be used for
wireless power transfer Wireless power transfer (WPT), wireless power transmission, wireless energy transmission (WET), or electromagnetic power transfer is the transmission of electrical energy without wires as a physical link. In a wireless power transmission system ...
between devices in the mid range distances (up to two metres). Strong coupling is required for a high percentage of power transferred, which results in peak splitting of the frequency response.


Ideal transformers

When k = 1, the inductor is referred to as being closely coupled. If in addition, the self-inductances go to infinity, the inductor becomes an ideal
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' ...
. In this case the voltages, currents, and number of turns can be related in the following way: V_\text = \frac V_\text where Conversely the current: I_\text = \frac I_\text where The power through one inductor is the same as the power through the other. These equations neglect any forcing by current sources or voltage sources.


Self-inductance of thin wire shapes

The table below lists formulas for the self-inductance of various simple shapes made of thin cylindrical conductors (wires). In general these are only accurate if the wire radius a is much smaller than the dimensions of the shape, and if no ferromagnetic materials are nearby (no magnetic core). Y is an approximately constant value between 0 and 1 that depends on the distribution of the current in the wire: when the current flows only on the surface of the wire (complete
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 ...
), when the current is evenly spread over the cross-section of the wire (
direct current Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even ...
). For round wires, Rosa (1908) gives a formula equivalent to: Y \approx \frac where \mathcal(x) is represents small term(s) that have been dropped from the formula, to make it simpler. Read the term as "plus small corrections that vary on the order of x" (see
big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
).


See also

*
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 ...
*
Gyrator A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventio ...
*
Hydraulic analogy The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Lodge) is the most widely used analogy for "electron fluid" in a metal conductor. Since electric current is invisible and the processes in play in ...
*
Leakage inductance Leakage inductance derives from the electrical property of an imperfectly-coupled transformer whereby each winding behaves as a self-inductance in series with the winding's respective ohmic resistance constant. These four winding constants also in ...
*
LC circuit An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
,
RLC circuit An RLC circuit is an electrical circuit consisting of a electrical resistance, resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the ...
,
RL circuit A resistor–inductor circuit (RL circuit), or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor, eithe ...
* Kinetic inductance


Footnotes


References


General references

* * * * * Küpfmüller K., ''Einführung in die theoretische Elektrotechnik,'' Springer-Verlag, 1959. * Heaviside O., ''Electrical Papers.'' Vol.1. – L.; N.Y.: Macmillan, 1892, p. 429-560. * Fritz Langford-Smith, editor (1953).
Radiotron Designer's Handbook
', 4th Edition, Amalgamated Wireless Valve Company Pty., Ltd. Chapter 10, "Calculation of Inductance" (pp. 429–448), includes a wealth of formulas and nomographs for coils, solenoids, and mutual inductance. * F. W. Sears and M. W. Zemansky 1964 ''University Physics: Third Edition (Complete Volume)'', Addison-Wesley Publishing Company, Inc. Reading MA, LCCC 63-15265 (no ISBN).


External links



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