# Dictionary Definition

inductance

### Noun

1 (physics) a property of an electric circuit by
which an electromotive force is induced in it by a variation of
current [syn: induction]

2 an electrical device that introduces inductance
into a circuit [syn: inductor]

# User Contributed Dictionary

## English

### Noun

- The property of an electric
circuit by which a voltage is induced in it by a changing
magnetic
field.
- The power cable itself has enough inductance to disrupt the digital signal of the video output cable, due to poor sheilding.

- The quantity of the
resulting electromagnetic flux, measured in henries (SI symbol: H.)
- What is the inductance of that power supply's main inductor?

#### Derived terms

#### Translations

property

- Finnish: induktanssi
- German: Induktivität
- Italian: induttanza
- Portuguese: indutância

quantity

- Finnish: induktanssi
- Italian: induttanza

# Extensive Definition

An electric
current i flowing around a circuit
produces a magnetic field and hence a magnetic
flux \Phi through the circuit. The ratio of the magnetic flux
to the current is called the inductance, or more accurately
self-inductance of the circuit. The term was coined by Oliver
Heaviside in February 1886. It is customary
to use the symbol L for inductance, possibly in honour of the
physicist Heinrich
Lenz. The quantitative definition of the inductance in SI units
(webers per
ampere) is

- L= \frac.

In honour of Joseph
Henry, the unit of inductance has been given the name henry
(H): 1H = 1Wb/A.

In the above definition, the magnetic flux \Phi
is that caused by the current flowing through the circuit
concerned. There may, however, be contributions from other
circuits. Consider for example two circuits C_1, C_2, carrying the
currents i_1, i_2. The magnetic fluxes \Phi_1 and \Phi_2 in C_1 and
C_2, respectively, are given by

- \displaystyle \Phi_1 = L_i_1 + L_i_2,
- \displaystyle \Phi_2 = L_i_1 + L_i_2.

Self and mutual inductances also occur in the
expression

- \displaystyle W=\frac\sum_^L_i_i_

## Properties of inductance

The equation relating inductance and flux linkages can be rearranged as follows:- \Phi = Li \,

Taking the time derivative of both sides of the
equation yields:

- \frac = L \frac + i \frac \,

In most physical cases, the inductance is
constant with time and so

- \frac = L \frac

By Faraday's
Law of Induction we have:

- \frac = -\mathcal = v

where \mathcal is the Electromotive
force (emf) and v is the induced voltage. Note that the emf is
opposite to the induced voltage. Thus:

- \frac = \frac

- i(t) = \frac \int_0^tv(\tau) d\tau + i(0)

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 non-zero. However, a non-zero 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.

Multiplying the equation for di/dt above with Li
leads to

- Li\frac=\frac\fraci^=iv

Since iv is the energy transferred to the system
per time it follows that \left( L/2 \right)i^2 is the energy of the
magnetic field generated by the current.

### Phasor circuit analysis and impedance

- Z_L = V / I = j L \omega \,

- j is the imaginary unit,
- L is the inductance,
- \omega = 2 \pi f \, is the angular frequency,
- f is the frequency and
- L \omega \ = X_L is the inductive reactance.

### Induced emf

The flux \Phi_i\ \! through the i-th circuit in a set is given by:- \Phi_i = \sum_ M_I_j = L_i I_i + \sum_ M_I_j \,

- \mathcal = -\frac = -\frac \left (L_i I_i + \sum_ M_I_j \right ) = -\left(\fracI_i +\fracL_i \right) -\sum_ \left (\fracI_j + \fracM_ \right).

## Coupled inductors

Mutual inductance is the concept that the change in current in one inductor can induce 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:

- M_ = N_1 N_2 P_ \!

- M_ is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 to the current in coil 1.
- N_1 is the number of turns in coil 1,
- N_2 is the number of turns in coil 2,
- P_ is the permeance of the space occupied by the flux.

The mutual inductance also has a relationship
with the coefficient of coupling. The coefficient of coupling is
always between 1 and 0, and is a convenient way to specify the
relationship between a certain orientation of inductor with
arbitrary inductance:

- M = k \sqrt \!

- k is the coefficient of coupling and 0 ≤ k ≤ 1,
- L_1 is the inductance of the first coil, and
- L_2 is the inductance of the second coil.

Once this mutual inductance factor M is
determined, it can be used to predict the behavior of a circuit:

- V_1 = L_1 \frac + M \frac

- V is the voltage across the inductor of interest,
- L_1 is the inductance of the inductor of interest,
- dI_1 / dt is the derivative, with respect to time, of the current through the inductor of interest,
- M is the mutual inductance and
- dI_2 / dt is the derivative, with respect to time, of the current through the inductor that is coupled to the first inductor.

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:

- V_s = V_p \frac

- V_s is the voltage across the secondary inductor,
- V_p is the voltage across the primary inductor (the one connected to a power source),
- N_s is the number of turns in the secondary inductor, and
- N_p is the number of turns in the primary inductor.

Conversely the current:

- I_s = I_p \frac

- I_s is the current through the secondary inductor,
- I_p is the current through the primary inductor (the one connected to a power source),
- N_s is the number of turns in the secondary inductor, and
- N_p is the number of turns in the primary inductor.

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 over-coupling. 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.

## Calculation techniques

### Mutual inductance

The mutual inductance by circuit i on circuit j
is given by the double integral Neumann
formula

- M_ = \frac \oint_\oint_ \frac

### Self-inductance

Formally the self-inductance of a wire loop would be given by the above equation with i =j. However, 1/R now gets singular and the finite radius a and 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| \ge a/2 and a correction term,- L_ = L = \left (\frac \oint_\oint_ \frac\right )_

## Inductance of simple electrical circuits

The self-inductance of many types of electrical circuits can be given in closed form. Examples are listed in the table.The constant \mu_0 is the
permeability of free space (4\pi × 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. If conductors approach each other then in the high frequency
case an additional screening current flows in their surface and the
expressions containing Y get invalid.

### Inductance of a solenoid

A solenoid is a long, thin coil,
i.e. a coil whose length is much greater than the diameter. Under
these conditions, and without any magnetic material used, the
magnetic flux
density B within the coil is practically constant and is given
by

- \displaystyle B = \mu_0 Ni/l

- \displaystyle \Phi = \mu_0N^2iA/l,

- \displaystyle L = \mu_0N^2A/l.

This, and the inductance of more complicated
shapes, can be derived from Maxwell's
equations. For rigid air-core coils, inductance is a function
of coil geometry and number of turns, and is independent of
current.

Similar analysis applies to a solenoid with a
magnetic core, but only if the length of the coil is much greater
than the product of the relative permeability of the magnetic core
and the diameter. That limits the simple analysis to
low-permeability cores, or extremely long thin solenoids. Although
rarely useful, the equations are,

- \displaystyle B = \mu_0\mu_r Ni/l

- \displaystyle \Phi = \mu_0\mu_rN^2iA/l,

- \displaystyle L = \mu_0\mu_rN^2A/l.

Note that since the permeability of ferromagnetic
materials changes with applied magnetic flux, the inductance of a
coil with a ferromagnetic core will generally vary with
current.

### Inductance of a coaxial line

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_o. Assume that a DC current I flows in opposite directions in
the two conductors, with uniform current density. The magnetic
field generated by these currents points in the azimuthal direction
and is a function of radius r; it can be computed using Ampère's
Law:

- 0 \leq r \leq r_i: B(r) = \frac
- r_i \leq r \leq r_: B(r) = \frac
- r_ \leq r \leq r_: B(r) = \frac \left( \frac \right)

The flux per length l in the region between the
conductors can be computed by drawing a surface containing the
axis:

- \frac = \int_^ B(r) dr = \frac \ln\frac

Inside the conductors, L can be computed by
equating the energy stored in an inductor, \fracLI^2, with the
energy stored in the magnetic field:

- \fracLI^2 = \int_V \frac dV

For a cylindrical geometry with no l dependence,
the energy per unit length is

- \fracL'I^2 = \int_^ \frac 2 \pi r~dr

where L' is the inductance per unit length. For
the inner conductor, the integral on the right-hand-side is \frac;
for the outer conductor it is \frac \left( \frac \right)^2 \ln\frac
- \frac \left( \frac \right) - \frac

Solving for L' and summing the terms for each
region together gives a total inductance per unit length of:

- L' = \frac + \frac \ln\frac + \frac \left( \frac \right)^2 \ln\frac - \frac \left( \frac \right) - \frac

However, for a typical coaxial line application
we are interested in passing (non-DC) 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

- \frac \approx \frac \ln\frac

## See also

## References

- Inductance Calculations
- Introduction to Electrodynamics (3rd ed.)
- Electromagnetic Fields
- Electrical & Electronic Technology (8th ed.)
- Küpfmüller K., Einführung in die theoretische Elektrotechnik, Springer-Verlag, 1959.
- Heaviside O., Electrical Papers. Vol.1. – L.; N.Y.: Macmillan, 1892, p. 429-560.

inductance in Afrikaans: Induktansie

inductance in Arabic: محاثة تبادلية

inductance in Catalan: Inductància

inductance in Czech: Indukčnost

inductance in German: Induktivität

inductance in Spanish: Inductancia

inductance in Esperanto: Induktanco

inductance in French: Inductance

inductance in Korean: 인덕턴스

inductance in Croatian: Induktivitet

inductance in Italian: Induttanza

inductance in Japanese: インダクタンス

inductance in Hebrew: השראות

inductance in Latvian: Induktivitāte

inductance in Malay (macrolanguage):
Induktans

inductance in Dutch: Zelfinductie

inductance in Norwegian: Induktans

inductance in Polish: Indukcyjność

inductance in Portuguese: Indutância

inductance in Russian: Индуктивность

inductance in Slovenian: Induktivnost

inductance in Finnish: Induktanssi

inductance in Swedish: Induktans

inductance in Tamil: தூண்டம்

inductance in Ukrainian: Індуктивність

inductance in Chinese: 电感