Chapter 3

# Self-inductance, mutual inductance

### Définition : Self-inductance

A closed wire loop is travelled by an intensity .

Its own magnetic field , which is given by Biot-Savart law , is proportional to .

Definition of the self-inductance of a circuit

The flux of the proper magnetic field through an outline which is oriented in the positive direction of the chosen current, is also called “own flux”.

It is proportional to  :

The coefficient  depends on the geometric characteristics of the circuit only.

It is called the self-inductance of the circuit .

Sign of :

• If , the magnetic field has the direction represented on the figure.

Its flux is positive so .

• Si , direction of the magnetic field changes and the flux is negative.

Hence, is a positive coefficient.

The unit of the flux is Weber and the unit of the inductance is Henry.

### Exemple : Self-inductance of a solenoid

The magnetic field inside an infinite solenoid is :

The proper flux through wire loops occupying a given length is : ( )

Hence the self-inductance :

Order of magnitude :

For a coil of length , formed of whorls and which has a diameter of , the self-inductance is around : Henry is a pretty big unit.

Bigger Self-inductances can be obtained by iron-core coils.

But the equation that gives is more complicated ( not only depends on the geometry of the circuit but also on the intensity).

### Exemple : Self-inductance of a torus coil of rectangular section

A torus coil has a rectangular section of height and radii and .

It contains joined whorls travelled by intensity .

Any meridian plane is a plane of symmetry.

In any point of this plane, in cylindrical coordinates, the proper field is ortho-radial and depends on and :

The field lines are circles around axis.

Magnetic field of a torus

Ampere's circuital law applied to a field line of radius  :

The field actually does not depend on .

Its own field through the whorls is :

Hence the inductance :

### Définition : Mutual Inductance

Interacting circuits

Two wire circuits and are travelled by intensities and .

The flux of the magnetic field through the closed outline oriented by the positive direction of current is proportional to  :

Likewise, the flux of the magnetic field through the closed outline oriented by the positive direction of is proportional to :

is the mutual inductance of these two circuits.

Unlike the inductance, which is always positive, can be positive or negative, depending on the orientation of the circuits.

Remark :

If can be calculated, and can be deduced.

Sometimes, one of the two fluxes is hard to calculate whereas the other one is easier.

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