Inductance 

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When current flows within a wire, a magnetic field is created. The potion of this magnetic field perpendicular to the wire is called the magnetic flux (measured in weber, Wb). Inductance is the ratio of magnetic flux to current in a circuit. The unit of inductance is the henry, H (Wb/A) and is normally represented by the symbol L.

Self Inductance

Whenever current in a coil of wire changes,  the magnetic field]it produces will change. That will change the magnetic flux through the coil and hence produce a voltage across the coil. This is called self-inductance, and the coil is referred to as an inductor.

Example – Self Inductance of a Solenoid

Consider a coil of wire around a magnetic core, whose length is much greater than it's diameter. 

Given the permeability of free space μo , relative permeability of the magnetic core μr , number of turns N and length of solenoid l, with a current i flowing, the magnetic flux density, B within the coil is is given by:

image00

The magnetic flux, Φ is obtained by multiplying flux density by the cross sectional area A:

image01

Given that the inductance is the ratio of magnetic flux to current, we have:

image02

Note: if the solenoid is wound around a non-magnetic core, then μr = 1.

Mutual Inductance

When the magnetic field of one coil links with that of another coil, a change in current in the first coil will produce a linking magnetic field with the second coil.  The magnetic field linking with the second coil will produce a voltage within that coil.  This is called mutual inductance.

For two inductors (coils) we have:

  • L11 – self inductance of inductor 1
  • L22 – self inductance of inductor 2
  • L12 = L21 – mutual inductance between the two

Inductance Formulae

The table below lists some common formulae for calculating the theoretical inductance of several inductor constructions.

  • L = inductance (H)
  • μ0 = permeability of free space = 4 π× 10-7 (H/m)
  • μr = relative permeability of core material

Cylindrical coil:

myElectrical Equation N = number of turns
A = area of cross-section of the coil in square metres (m2)
l = length of coil in metres (m)

 

Straight wire conductor:

myElectrical Equation l = length of conductor (m)
d = diameter of conductor (m)

 

Flat spiral air-core coil:

myElectrical Equation r = mean radius of coil (m)
N = number of turns
d = depth of coil (outer radius minus inner radius) (m)

Toroidal core, circular cross-section:

myElectrical Equation N = number of turns
r = radius of coil winding (m)
D = overall diameter of toroid (m)

Application to Circuits

The quantitative measure of applying an inductance L, to a circuit defined by:

myElectrical Equation

An inductor can store energy. The power (= energy / time) being stored in an inductor is:

myElectrical Equation

This implies (by a little calculus) that the energy stored in an inductor is :

myElectrical Equation



Steven McFadyen's avatar Steven McFadyen

Steven has over twenty five years experience working on some of the largest construction projects. He has a deep technical understanding of electrical engineering and is keen to share this knowledge. About the author

myElectrical Engineering

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  1. Chris's avatar Chris says:
    7/12/2012 3:52 AM

    You list the relative permeability of non-magnetic materials as 0. Shouldn't it be 1?

    • Steven's avatar Steven says:
      7/14/2012 9:45 AM

      Thanks for spotting the error Chris. I have corrected it.


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