Inductance  

Common symbols  L 
SI unit  henry (H) 
In SI base units  kg⋅m ^{2}⋅s ^{−2}⋅A ^{−2} 
Derivations from other quantities  
Dimension  M^{1}·L^{2}·T^{−2}·I^{−2} 
Articles about 
Electromagnetism 

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 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 (EMF) (voltage) in the conductors, a process known as electromagnetic induction. This induced voltage created by the changing current has the effect of opposing the change in current. This is stated by Lenz's law, and the voltage is called back EMF .
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 of nearby materials.^{ [1] } An electronic component designed to add inductance to a circuit is called an inductor. It typically consists of a coil or helix of wire.
The term inductance was coined by Oliver Heaviside in May 1884.^{ [2] }^{[ failed verification ]} It is customary to use the symbol for inductance, in honour of the physicist Heinrich Lenz.^{ [3] }^{ [4] } In the SI system, the unit of inductance is the henry (H), which is the amount of inductance that causes a voltage of one volt, when the current is changing at a rate of one ampere per second. It is named for Joseph Henry, who discovered inductance independently of Faraday.^{ [5] }
The history of electromagnetic induction, a facet of electromagnetism, began with observations of the ancients: electric charge or static electricity (rubbing silk on amber), electric current (lightning), and magnetic attraction (lodestone). 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 in 1831.^{ [6] }^{ [7] } 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, 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.^{ [8] } This current was induced by the change in magnetic flux that occurred when the battery was connected and disconnected.^{ [9] } 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").^{ [10] }
A current flowing through a conductor generates a magnetic field 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 (EMF) or voltage 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. The potential is therefore called a back EMF. 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. Selfinductance, 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 (H), named after American scientist Joseph Henry, which is the amount of inductance which generates a voltage of one volt when the current is changing at a rate of one ampere 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 of nearby materials; ferromagnetic materials with a higher permeability like iron 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^{ [11] }^{ [12] }^{ [13] }^{ [14] }
An inductor is an electrical component 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 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 linkages. 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 material in the hole in the center. The magnetic field of the coil magnetizes the material of the core, aligning its magnetic domains, 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 circuits 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 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 . The property describing the effect of one conductor on itself is more precisely called selfinductance, and the properties describing the effects of one conductor with changing current on nearby conductors is called mutual inductance.^{ [15] }
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^{ [16] }^{ [17] }^{ [18] }
From (1) above
When there is no current, there is no magnetic field and the stored energy is zero. Neglecting resistive losses, the energy (measured in joules, 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^{ [16] }^{ [17] }^{ [18] }
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 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 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.
When a sinusoidal alternating current (AC) is passing through a linear inductance, the induced backEMF is also sinusoidal. If the current through the inductance is , from (1) above the voltage across it is
where is the amplitude (peak value) of the sinusoidal current in amperes, is the angular frequency of the alternating current, with being its frequency in hertz, 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.^{ [19] } It is defined analogously to electrical resistance in a resistor, as the ratio of the amplitude (peak value) of the alternating voltage to current in the component
Reactance has units of ohms. It can be seen that inductive reactance 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; the voltage peaks occur earlier in each cycle than the current peaks. The phase difference between the current and the induced voltage is radians or 90 degrees, showing that in an ideal inductor the current lags the voltage by 90°.
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, the surface current densities and magnetic field may be obtained by solving the Laplace equation. Where the conductors are thin wires, selfinductance 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.
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 wholecircuit inductance which are omitted.
For derivation of the formulas below, see Rosa (1908).^{ [20] } The total low frequency inductance (interior plus exterior) of a straight wire is:
where
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 is much less than the length , 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, is then given by a very similar formula:
where the variables and 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 10 m long, made of 18 gauge wire, would only have an inductance of about 19 μH if stretched out straight.
There are two cases to consider:
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.
This is the generalized case of the paradigmatic twoloop 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 3dimensional 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 on a filamentary circuit is given by the double integral Neumann formula^{ [21] }
where
where
^{ [22] }
where
Stokes' theorem has been used for the 3rd equality step.
For the last equality step, we used the Retarded potential expression for 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.
Formally, the selfinductance of a wire loop would be given by the above equation with . However, here becomes infinite, leading to a logarithmically divergent integral.^{ [loweralpha 1] } This necessitates taking the finite wire radius 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,^{ [23] }
where
A solenoid 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 within the coil is practically constant and is given by
where is the magnetic constant, the number of turns, the current and the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density by the crosssection area :
When this is combined with the definition of inductance , it follows that the inductance of a solenoid is given by:
Therefore, for aircore coils, inductance is a function of coil geometry and number of turns, and is independent of current.
Let the inner conductor have radius and permeability , let the dielectric between the inner and outer conductor have permeability , and let the outer conductor have inner radius , outer radius , and permeability . However, for a typical coaxial line application, we are interested in passing (nonDC) signals at frequencies for which the resistive skin effect cannot be neglected. In most cases, the inner and outer conductor terms are negligible, in which case one may approximate
Most practical aircore inductors are multilayer cylindrical coils with square crosssections to minimize average distance between turns (circular cross sections would be better but harder to form).
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. Saturation makes the resulting inductance a function of the applied current.
The secant or largesignal inductance is used in flux calculations. It is defined as:
The differential or smallsignal inductance, on the other hand, is used in calculating voltage. It is defined as:
The circuit voltage for a nonlinear inductor is obtained via the differential inductance as shown by Faraday's Law and the chain rule of calculus.
Similar definitions may be derived for nonlinear 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 M.
The inductance equations above are a consequence of Maxwell's equations. For the important case of electrical circuits consisting of thin wires, the derivation is straightforward.
In a system of wire loops, each with one or several wire turns, the flux linkage of loop , , is given by
Here denotes the number of turns in loop ; is the magnetic flux through loop ; and 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
where denotes the voltage induced in circuit . This agrees with the definition of inductance above if the coefficients are identified with the coefficients of inductance. Because the total currents contribute to it also follows that is proportional to the product of turns .
Multiplying the equation for v_{m} above with i_{m}dt and summing over m gives the energy transferred to the system in the time interval dt,
This must agree with the change of the magnetic field energy, W, caused by the currents.^{ [24] } The integrability condition
requires L_{m,n} = L_{n,m}. The inductance matrix, L_{m,n}, thus is symmetric. The integral of the energy transfer is the magnetic field energy as a function of the currents,
This equation also is a direct consequence of the linearity of Maxwell's equations. It is helpful to associate changing electric currents with a buildup 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 transformers work, but it can also cause unwanted coupling between conductors in a circuit.
The mutual inductance, , is also a measure of the coupling between two inductors. The mutual inductance by circuit on circuit is given by the double integral Neumann formula, see calculation techniques
The mutual inductance also has the relationship:
where
Once the mutual inductance, , is determined, it can be used to predict the behavior of a circuit:
where
The minus sign arises because of the sense the current has been defined in the diagram. With both currents defined going into the dots the sign of will be positive (the equation would read with a plus sign instead).^{ [25] }
The coupling coefficient is the ratio of the opencircuit actual voltage ratio to the ratio that would be obtained if all the flux coupled from one magnetic circuit 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 twoport matrix the opencircuit voltage ratio is found to be:
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
thus,
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^{ [26] } define it as . Allowing negative values of captures phase inversions of the coil connections and the direction of the windings.^{ [27] }
Mutually coupled inductors can be described by any of the twoport network parameter matrix representations. The most direct are the z parameters, which are given by
where is the complex frequency variable, and are the inductances of the primary and secondary coil, respectively, and is the mutual inductance between the coils.
Mutually coupled inductors can equivalently be represented by a Tcircuit of inductors as shown. If the coupling is strong and the inductors are of unequal values then the series inductor on the stepdown side may take on a negative value.
This can be analyzed as a two port network. With the output terminated with some arbitrary impedance, , the voltage gain, , is given by,
where is the coupling constant and is the complex frequency variable, as above. For tightly coupled inductors where this reduces to
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,
For this reduces to
Thus, current gain, is not independent of load unless the further condition
is met, in which case,
and
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 Tcircuit, it can be generalized^{ [28] } to circuits consisting of more than two coupled inductors. Equivalent circuit elements , have physical meaning, modelling respectively magnetic reluctances of coupling paths and magnetic reluctances of leakage paths. For example, electric currents flowing through these elements correspond to coupling and leakage magnetic fluxes. Ideal transformers normalize all selfinductances to 1 Henry to simplify mathematical formulas.
Equivalent circuit element values can be calculated from coupling coefficients with
where coupling coefficient matrix and its cofactors are defined as
and
For two coupled inductors, these formulas simplify to
and
and for three coupled inductors (for brevity shown only for and )
and
When a capacitor is connected across one winding of a transformer, making the winding a tuned circuit (resonant circuit) it is called a singletuned 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 and thus function as a bandpass filter, allowing frequencies near their resonant frequency 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 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 doubletuned circuits is described as loose, critical, or overcoupled depending on the value of the coupling coefficient . 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.
Stonglycoupled selfresonant coils can be used for wireless power transfer between devices in the mid range distances (up to two metres).^{ [29] } Strong coupling is required for a high percentage of power transferred, which results in peak splitting of the frequency response.^{ [30] }^{ [31] }
When , the inductor is referred to as being closely coupled. If in addition, the selfinductances go to infinity, the inductor becomes an ideal transformer. In this case the voltages, currents, and number of turns can be related in the following way:
where
Conversely the current:
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.
The table below lists formulas for the selfinductance of various simple shapes made of thin cylindrical conductors (wires). In general these are only accurate if the wire radius is much smaller than the dimensions of the shape, and if no ferromagnetic materials are nearby (no magnetic core).
Type  Inductance  Comment 

Single layer solenoid  The wellknown Wheeler's approximation formula for currentsheet model aircore coil:^{ [32] }^{ [33] } (English) (cgs) 

Coaxial cable (HF)  : Outer cond.'s inside radius : Inner conductor's radius : Length : see table footnote.  
Circular loop^{ [34] }  : Loop radius : Wire radius : see table footnotes.  
Rectangle made of round wire^{ [35] }  : Side lengths : Wire radius : see table footnotes.  
Pair of parallel wires  : Wire radius : Separation distance, : Length of pair : see table footnotes.  
Pair of parallel wires (HF)  : Wire radius : Separation distance, : Length (each) of pair : see table footnote. 
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), when the current is evenly spread over the crosssection of the wire (direct current). For round wires, Rosa (1908) gives a formula equivalent to:^{ [20] }
where
is represents small term(s) that have been dropped from the formula, to make it simpler. Read the symbol "" as "plus small corrections on the order of" . See also Big O notation.
An inductor, also called a coil, choke, or reactor, is a passive twoterminal 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 coil.
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 the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a magnetic field that varies with location will exert a force on a range of nonmagnetic materials by affecting the motion of their outer atomic electrons. Magnetic fields surround magnetized materials, and are created by electric currents such as those used in electromagnets, and by electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, called a vector field.
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:
In physics, the Poynting vector represents the directional energy flux or power flow of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m^{2}); kg/s^{3} in base SI units. It is named after its discoverer John Henry Poynting who first derived it in 1884. Nikolay Umov is also credited with formulating the concept. Oliver Heaviside also discovered it independently in the more general form that recognises the freedom of adding the curl of an arbitrary vector field to the definition. The Poynting vector is used throughout electromagnetics in conjunction with Poynting's theorem, the continuity equation expressing conservation of electromagnetic energy, to calculate the power flow in electromagnetic fields.
Electromagnetic or magnetic induction is the production of an electromotive force across an electrical conductor in a changing magnetic field.
An electromagnet is a type of magnet in which the magnetic field is produced by an electric current. Electromagnets usually consist of wire wound into a coil. A current through the wire creates a magnetic field which is concentrated in the hole, denoting the center of the coil. The magnetic field disappears when the current is turned off. The wire turns are often wound around a magnetic core made from a ferromagnetic or ferrimagnetic material such as iron; the magnetic core concentrates the magnetic flux and makes a more powerful magnet.
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 differs in that reactance does not lead to dissipation of electrical energy as heat. Instead, energy is momentarily stored in the reactance, and a quartercycle later returned to the circuit, whereas a resistance continuously loses energy.
A solenoid is a type of electromagnet formed by a helical coil of wire whose length is substantially greater than its diameter, which generates a controlled magnetic field. The coil can produce a uniform magnetic field in a volume of space when an electric current is passed through it. The term solenoid was coined in 1823 by AndréMarie Ampère.
A Rogowski coil, named after Walter Rogowski, is an electrical device for measuring alternating current (AC) or highspeed current pulses. It sometimes consists of a helical coil of wire with the lead from one end returning through the centre of the coil to the other end so that both terminals are at the same end of the coil. This approach is sometimes referred to as a counterwound Rogowski. Other approaches use a full toroid geometry that has the advantage of a central excitation not exciting standing waves in the coil. The whole assembly is then wrapped around the straight conductor whose current is to be measured. There is no metal (iron) core. The winding density, the diameter of the coil and the rigidity of the winding are critical for preserving immunity to external fields and low sensitivity to the positioning of the measured conductor.
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 conductor. The electric current flows mainly at the "skin" of the conductor, between the outer surface and a level called the skin depth. Skin depth depends on the frequency of the alternating current; as frequency increases, current flow moves to the surface, resulting in less skin depth. Skin effect reduces the effective crosssection of the conductor and thus increases its effective resistance. Skin effect is caused by opposing eddy currents induced by the changing magnetic field resulting from the alternating current. At 60 Hz in copper, the skin depth is about 8.5 mm. At high frequencies the skin depth becomes much smaller.
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of James Clerk Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws. These laws can be applied in time and frequency domains and form the basis for network analysis.
Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.
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 materials in the path. Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as electric motors, generators, transformers, relays, lifting electromagnets, SQUIDs, galvanometers, and magnetic recording heads.
Magnetostatics is the study of magnetic fields in systems where the currents are steady. It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static — as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory.
The search coil magnetometer or induction magnetometer, based on an inductive sensor, is a magnetometer which measures the varying magnetic flux. An inductive sensor connected to a conditioning electronic circuit constitutes a search coil magnetometer. It is a vector magnetometer which can measure one or more components of the magnetic field. A classical configuration uses three orthogonal inductive sensors. The searchcoil magnetometer can measure magnetic field from mHz up to hundreds of MHz.
A pinch is the compression of an electrically conducting filament by magnetic forces, or a device that does such. The conductor is usually a plasma, but could also be a solid or liquid metal. Pinches were the first type of device used for experiments in controlled nuclear fusion power.
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 proximity effect. This crowding gives an increase in the effective resistance of the circuit, which increases with frequency.
Toroidal inductors and transformers are inductors and transformers which use magnetic cores with a toroidal shape. They are passive electronic components, consisting of a circular ring or donut shaped magnetic core of ferromagnetic material such as laminated iron, iron powder, or ferrite, around which wire is wound.
The gyrator–capacitor model  sometimes also the capacitorpermeance model  is a lumpedelement model for magnetic circuits, that can be used in place of the more common resistance–reluctance model. The model makes permeance elements analogous to electrical capacitance rather than electrical resistance. Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.
An electropermanent magnet or EPM is a type of permanent magnet in which the external magnetic field can be switched on or off by a pulse of electric current in a wire winding around part of the magnet. The magnet consists of two sections, one of "hard" magnetic material and one of "soft" material. The direction of magnetization in the latter piece can be switched by a pulse of current in a wire winding about the former. When the magnetically soft and hard materials have opposing magnetizations, the magnet produces no net external field across its poles, while when their direction of magnetization is aligned the magnet produces an external magnetic field.