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
.
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.
Attention : Self-induction of a wire circuit
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.
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
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.
