Inductance is the property in an electrical circuit where a change in the electric current through that circuit induces an electromotive force (EMF) that opposes the change in current (See Induced EMF).
In electrical circuits, any electric current, i, produces a magnetic field and hence generates a total magnetic flux, Φ, acting on the circuit. This magnetic flux, due to Lenz's law, tends to act to oppose changes in the flux by generating a voltage (a back EMF) in the circuit that counters or tends to reduce the rate of change in the current. The ratio of the magnetic flux to the current is called the selfinductance, which is usually simply referred to as the inductance of the circuit. To add inductance to a circuit, electronic components called inductors are used, which consist of coils of wire to concentrate the magnetic field.
The term 'inductance' was coined by Oliver Heaviside in February 1886.^{[1]} It is customary to use the symbol L for inductance, possibly in honour of the physicist Heinrich Lenz.^{[2]}^{[3]}
The SI unit of inductance is the henry (H), named after American scientist and magnetic researcher Joseph Henry. 1 H = 1 Wb/A.
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The quantitative definition of the (self) inductance of a wire loop in SI units (webers per ampere, known as henries) is
where L is the inductance, Φ denotes the magnetic flux through the area spanned by the loop, N is the number of wire turns, and i is the current in amperes. The flux linkage thus is
There may, however, be contributions from other circuits. Consider for example two circuits K_{1}, K_{2}, carrying the currents i_{1}, i_{2}. The flux linkages of K_{1} and K_{2} are given by
According to the above definition, L_{11} and L_{22} are the selfinductances of K_{1} and K_{2}, respectively. It can be shown (see below) that the other two coefficients are equal: L_{12} = L_{21} = M, where M is called the mutual inductance of the pair of circuits.
The number of turns N_{1} and N_{2} occur somewhat asymmetrically in the definition above. However, L_{mn} is always proportional to the product N_{m}N_{n}, and thus the total currents N_{m}i_{m} contribute to the flux.
Self and mutual inductances also occur in the expression
for the energy of the magnetic field generated by K, electrical circuits where i_{n} is the current in the nth circuit. This equation is an alternative definition of inductance that also applies when the currents are not confined to thin wires so that it is not immediately clear what area is encompassed by the circuit nor how the magnetic flux through the circuit is to be defined.
The definition L = NΦ/i, in contrast, is more direct and more intuitive. It may be shown that the two definitions are equivalent by equating the time derivative of W and the electric power transferred to the system. It should be noted that this analysis assumes linearity, for nonlinear definitions and discussion see nonlinear inductance.
Taking the time derivative of both sides of the equation NΦ = Li yields:
In most physical cases, the inductance is constant with time and so
By Faraday's Law of Induction we have:
where is the Electromotive force (emf) and v is the induced voltage. Note that the emf is opposite to the induced voltage. Thus:
or
These equations together state that, for a steady applied voltage v, the current changes in a linear manner, at a rate proportional to the applied voltage, but inversely proportional to the inductance. Conversely, if the current through the inductor is changing at a constant rate, the induced voltage is constant.
The effect of inductance can be understood using a single loop of wire as an example. If a voltage is suddenly applied between the ends of the loop of wire, the current must change from zero to nonzero. However, a nonzero current induces a magnetic field by Ampère's law. This change in the magnetic field induces an emf that is in the opposite direction of the change in current. The strength of this emf is proportional to the change in current and the inductance. When these opposing forces are in balance, the result is a current that increases linearly with time where the rate of this change is determined by the applied voltage and the inductance.
An alternative explanation of this behaviour is possible in terms of energy conservation. Multiplying the equation for di/dt above with Li leads to
Since iv is the energy transferred to the system per time it follows that (L/2)i^{2} is the energy of the magnetic field generated by the current. A change in current thus implies a change in magnetic field energy, and this only is possible if there also is a voltage.
A mechanical analogy is a body with mass M, velocity v and kinetic energy (M/2)v^{2}. A change in velocity (current) requires or generates a force (an electrical voltage) proportional to mass (inductance).
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, 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:
where
The mutual inductance also has a relationship with the coupling coefficient. The coupling coefficient is always between 1 and 0, and is a convenient way to specify the relationship between a certain orientation of inductor with arbitrary inductance:
where
Once the mutual inductance, M, is determined from this factor, it can be used to predict the behavior of a circuit:
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.^{[4]}
When one inductor is closely coupled to another inductor through mutual inductance, such as in a transformer, the voltages, currents, and number of turns can be related in the following way:
where
Conversely the current:
where
Note that the power through one inductor is the same as the power through the other. Also note that these equations don't work if both transformers are forced (with power sources).
When either side of the transformer is a tuned circuit, the amount of mutual inductance between the two windings determines the shape of the frequency response curve. Although no boundaries are defined, this is often referred to as loose, critical, and overcoupling. When two tuned circuits are loosely coupled through mutual inductance, the bandwidth will be narrow. As the amount of mutual inductance increases, the bandwidth continues to grow. When the mutual inductance is increased beyond a critical point, the peak in the response curve begins to drop, and the center frequency will be attenuated more strongly than its direct sidebands. This is known as overcoupling.
The mutual inductance by a filamentary circuit i on a filamentary circuit j is given by the double integral Neumann formula
The symbol μ_{0} denotes the magnetic constant (4π × 10^{−7} H/m), C_{i} and C_{j} are the curves spanned by the wires, R_{ij} is the distance between two points. See a derivation of this equation.
Formally the selfinductance of a wire loop would be given by the above equation with i = j. However, 1/R becomes infinite and thus the finite radius a along with the distribution of the current in the wire must be taken into account. There remain the contribution from the integral over all points where R ≥ a/2 and a correction term,
Here a and l denote radius and length of the wire, and Y is a constant that depends on the distribution of the current in the wire: Y = 0 when the current flows in the surface of the wire (skin effect), Y = 1/4 when the current is homogenuous across the wire. This approximation is accurate when the wires are long compared to their crosssectional dimensions. Here is a derivation of this equation.
In some cases different current distributions generate the same magnetic field in some section of space. This fact may be used to relate self inductances (method of images). As an example consider the two systems:
The magnetic field of the two systems coincides (in a half space). The magnetic field energy and the inductance of the second system thus are twice as large as that of the first system.
Inductance per length L' and capacitance per length C' are related to each other in the special case of transmission lines consisting of two parallel perfect conductors of arbitrary but constant cross section,^{[5]}
Here ε and µ denote dielectric constant and magnetic permeability of the medium the conductors are embedded in. There is no electric and no magnetic field inside the conductors (complete skin effect, high frequency). Current flows down on one line and returns on the other. The signal propagation speed coincides with the propagation speed of electromagnetic waves in the bulk.
The selfinductance of many types of electrical circuits can be given in closed form. Examples are listed in the table.
Type  Inductance / μ_{0}  Comment 

Single layer solenoid^{[6]} 

N: Number of turns r: Radius l: Length w = r/l m = 4w^{2} E,K: Elliptic integrals 
Coaxial cable, high frequency 
a_{1}: Outer radius a: Inner radius l: Length 

Circular loop  r: Loop radius a: Wire radius 

Rectangle  b, d: Border length d >> a, b >> a a: Wire radius 

Pair of parallel wires 
a: Wire radius d: Distance, d ≥ 2a l: Length of pair 

Pair of parallel wires, high frequency 
a: Wire radius d: Distance, d ≥ 2a l: Length of pair 

Wire parallel to perfectly conducting wall 
a: Wire radius d: Distance, d ≥ a l: Length 

Wire parallel to conducting wall, high frequency 
a: Wire radius d: Distance, d ≥ a l: Length 
The symbol μ_{0} denotes the magnetic constant (4π × 10^{−7} H/m). For high frequencies the electrical current flows in the conductor surface (skin effect), and depending on the geometry it sometimes is necessary to distinguish low and high frequency inductances. This is the purpose of the constant Y: Y = 0 when the current is uniformly distributed over the surface of the wire (skin effect), Y = 1/4 when the current is uniformly distributed over the cross section of the wire. In the high frequency case, if conductors approach each other, an additional screening current flows in their surface, and expressions containing Y become invalid. Details for some circuit types are available on another page.
Using phasors, the equivalent impedance of an inductance is given by:
where
Many inductors make use of magnetic materials. These materials over a large enough range exhibit a nonlinear permeability with such effects as saturation. This inturn makes the resulting inductance a function of the applied current. Faraday's Law still holds but inductance is ambiguous and is different whether you are calculating circuit parameters or magnetic fluxes.
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.
There are similar definitions for nonlinear mutual inductances.
