inductance - Properties of inductance, Coupled inductors, Vector field theory derivations, Usage

A measure of a coil's ability to produce a voltage in another coil (mutual inductance, M) or in itself (self inductance, L) via changing magnetic fields; units H (henry). It is equal to the ratio of electromotive force produced to rate of change of current.

The inductance has the following relationship:

where

As we see here, the geometry and material properties (if material properties are same in surface S and the material is linear) of the current loop can be expressed with single scalar quantity L.

Properties of inductance

The equation relating inductance and flux linkages can be rearranged as follows:

Taking the time derivative of both sides of the equation 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. 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.

Phasor circuit analysis and impedance

Using phasors, the equivalent impedance of an inductance is given by:

where

is the inductive reactance, 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. Thus, there are three inductances defined for coupled inductors:

L11 - the self inductance of inductor 1 L22 - the self inductance of inductor 2 L12 = L21 - the mutual inductance associated with both inductors

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.

Vector field theory derivations

Mutual inductance

Mutual inductance is the concept that the current through one inductor can induce a voltage in another nearby inductor.

The mutual inductance also has the relationship:

where

M21 is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 to the current in coil 1. 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:

where

k is the coefficient of coupling and 0 ≤ k ≤ 1, L1 is the inductance of the first coil, and L2 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:

where

V is the voltage across the inductor of interest, L1 is the inductance of the inductor of interest, dI1 / dt is the derivative, with respect to time, of the current through the inductor of interest, M is the mutual inductance and dI2 / 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:

where

Vs is the voltage across the secondary inductor, Vp is the voltage across the primary inductor (the one connected to a power source), Ns is the number of turns in the secondary inductor, and Np is the number of turns in the primary inductor.

Conversely the current:

where

Is is the current through the secondary inductor, Ip is the current through the primary inductor (the one connected to a power source), Ns is the number of turns in the secondary inductor, and Np is the number of turns in the primary inductor.

Usage

The flux through the ith circuit in a set is given by:

so that the induced emf, , of a specific circuit, i, in any given set can be given directly by: