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Inductance
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Inductance (or '''Electric inductance''') is a measure of the amount of Magnetic Flux produced for a given Electric Current . The symbol ''L'' is used for inductance in honour of the physicist Heinrich Lenz . The term ''inductance'' was coined by Oliver Heaviside in February 1886 . The SI unit of inductance is the Henry (symbol: H).

The inductance has the following relationship:
:L= rac{\Phi}{i}
:::where
::::''L'' is the inductance in Henries ,
::::''i'' is the current in Ampere s,
::::''Φ'' is the magnetic flux in Weber s

Strictly speaking, the quantity just defined is called ''self-inductance'', because the magnetic field is created solely by the conductor that carries the current.

When a conductor is coiled upon itself N number of times around the same axis (forming a Solenoid ), the current required to produce a given amount of flux is reduced by a factor of N compared to a single turn of wire. Thus, the inductance of a coil of wire of N turns is given by:

::L= rac{\lambda}{i} = N rac{\Phi}{i}
where \lambda is the total 'flux linkage'.


INDUCTANCE OF A SOLENOID


The amount of magnetic flux produced by a current depends upon the Permeability of the medium surrounded by the current, the area inside the coil, and the number of turns. The greater the permeability, the greater the magnetic flux generated by a given current. Certain ( Ferromagnetic ) materials have much higher permeability than air. If a conductor (wire) is wound around such a material, the magnetic flux becomes much greater and the inductance becomes much greater than the inductance of an identical coil wound in air. The self-inductance ''L'' of such a solenoid can be calculated from

: L = {\mu_0 \mu_r N^2 A \over l} = rac{N \Phi}{i}
:::where
::::''μ''0 is the Permeability of free space (4π × 10-7 Henries per Metre )
::::''μr'' is the relative permeability of the core (dimensionless)
::::''N'' is the number of turns.
::::''A'' is the cross sectional area of the coil in Square Metre s.
::::''l'' is the length of the coil in Metre s.
::::''\Phi = BA'' is the flux in webers (''B'' is the flux density, ''A'' is the area).
::::''i'' is the current in amperes

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. However, 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.


PROPERTIES OF INDUCTANCE

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

Taking the time derivative of both sides of the equation yields:
: rac{d\lambda}{dt} = L rac{di}{dt} + i rac{dL}{dt} \,

In most physical cases, the inductance is constant with time and so
: rac{d\lambda}{dt} = L rac{di}{dt}

By Faraday's Law of Induction we have:
: rac{d\lambda}{dt} = -\mathcal{E} = v

where \mathcal{E} is the Electromotive Force (emf) and v is the induced voltage. Note that the emf is opposite to the induced voltage. Thus:
: rac{di}{dt} = rac{v}{L}
or
:i(t) = rac{1}{L} \int_0^tv( au) d au + 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 proportionally 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 Ampere'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.


Phasor circuit analysis and impedance


Using Phasors , the equivalent Impedance of an inductance is given by:

:Z_L = V / I = j L \omega \,
:::where
:::: X_L = L \omega \, is the inductive Reactance ,
:::: \omega = 2 \pi f \, is the angular frequency,
:::: ''L'' is the Inductance ,
:::: ''f'' is the frequency, and
:::: ''j'' is the Imaginary Unit .


COUPLED INDUCTORS

When the magnetic flux produced by an inductor links another inductor, these inductors are said to be coupled. Coupling is often undesired but in many cases, this coupling is intentional and is the basis of the Transformer . When inductors are coupled, there exists a mutual inductance that relates the current in one inductor to the flux linkage in the other inductor. Thus, there are three inductances defined for coupled inductors:

:L_{11} - the self inductance of inductor 1
:L_{22} - the self inductance of inductor 2
:L_{12} = L_{21} - the mutual inductance associated with both inductors


VECTOR FIELD THEORY DERIVATIONS


Mutual inductance







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 rac{N_s}{N_p}
:::where
::::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.


:I_s = I_p rac{N_p}{N_s}
:::where
::::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).


Self-inductance

Self-inductance, denoted ''L'', is the usual inductance one talks about with an Inductor . It is a special case of mutual inductance where, in the above equation, ''i'' =''j''. Thus,




SEE ALSO



REFERENCES