Electro-dynamic loudspeaker principle
Exemple : See the practical work on the speaker.
Fondamental : Mechanic and electric equations of the loudspeaker
The following figure represents a device used as a speaker or microphone.
is a permanent magnet of revolution symmetry around
axis.
In the air gap is a radial magnetic field.
In the area where the coil
moves, attached to the pavilion
, the magnetic field is written :
is a system of mass
, likely to move along
axis.
It can be set in translation movement if it is subject to an exterior force
.
It is also subject to dissipative forces of sum
, spring forces of sum
applied by a system of springs, as well as Laplace's forces of sum
applied on
.
The pavilion has a total length of wire equal to
and carries an intensity
.
It is admitted, by neglecting the helicity of
, that each element of the wire can be represented in cylindrical coordinates by :
is powered by a voltage source
through the circuit.
,
and
are the total resistance, inductance and capacitance, relatively to the whole of the circuit,
included.
can be expressed as a function of
,
and
.
Indeed, if we sum Laplace's forces on several elements of
:
The inertia center theorem applied to the pavilion
and projected on
axis gives the differential equation (M) verified by
.
It shows the mechanical behavior of the system :
The mechanic equation (M) of the system is obtained.
In order to obtain the electric equation,
,
and the induced electromotive force
in
must be expressed as functions of
.
The electromotive force is induced on an element
of
:
After integration along
in the positive direction of
:
(We also can obtain this expression for
using an analogy with the Laplace rails).
Kirchhoff's law gives the electric equation (E) of the circuit :
Fondamental : Energy review
Let's compute
:
This equation can be written as such :
With :
represents the mechanic energy of the system and its electromagnetic energy.
The previous equation expresses the energy conservation :
The derivative of
is equal to the sum of powers provided by the two energy sources of the system : the exterior force and the voltage source .
To this is subtracted the power dissipated via friction forces (or acoustic energy when the speaker emits a noise) or Joule effect.
Méthode : Study in forced sinusoidal regime
The system is now under a sinusoidal steady state at fixed frequency.
The equations (M') and (E') linking the complex representations of
,
,
and
can be written as such :
And :
The complex notation in
has been introduced and :
is the complex impedance of the
series dipole.
is the mechanic impedance of the mobile pavilion (ratio between Laplace's force and speed).
When it is used as a speaker,
is equal to zero, the energy of the system comes from the source of voltage
.
The answer to the system is characterized by relations
and
, given by (E') and (M') :
Hence :
The quantity
is introduced :
From an electro kinetic point of view, it is as though, because of the movement in the magnetic field, another electric impedance
is added to
.
This impedance, called motion impedance, characterizes the electro-mechanic coupling done by the assemblage.
If the frequency
is included in
, the vibrations of the pavilion will create an acoustic pressure wave which will generate audible sound.
When it is used as a microphone, then
is equal to zero.
The energy of the system comes from the sinusoidal force
which expresses the action of the pressure forces from a sound wave on
.
The answer of the system is then characterized by the function
:
The intensity is proportional to the applied force : the electric signal received is faithful to the force.
It will be recorded, treated and reproduced by another speaker.
