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Speaker
Speaker is an electromechanical device used to convert an electrical signal into sound. There are several existing designs, but the speakers consisting of dynamic drivers are most commonly used today. Here we will primarily focus on explaining how this type of speaker works.
A speaker can consist of one or more drivers. The main parts of a driver are a diaphragm, a basket (or frame), a voice coil and a permanent magnet. Diaphragm is on the wide end attached to the basket by a rim of flexible material called suspension or surround. On a narrow end it is connected to the voice coil. The voice coil is attached to the basket by a ring of flexible material called spider, which holds it in position, but also allows the coil to move back and forth freely.
A speaker produces the sound by vibration of the diaphragm, also called cone, which is usually made of paper, plastic or metal. The movement of the diaphragm is what sets the particles in air into motion, therefore creating a sound wave. The characteristics of the motion of the cone determine the characteristics of the sound produced. The frequency of the sound depends on how fast the diaphragm moves back and forth (frequency of its oscillation). The amplitude of the sound wave (loudness of the sound) is on the other hand much more complex, but one of the factors (the one that can be changed by changing the electrical signal) on which it depends is how much the diaphragm moves from its initial position.
As shown on the diagram, the cone is attached on one end to the voice coil and it is the movement of the voice coil what moves the cone. The voice coil consists of the coil of wire wrapped around a piece of metal, usually iron. It is important that the material used has ferromagnetic properties, so as the current flows trough the metal wire the magnetic field is created around the wire and the metal wrapped in the wire becomes magnetized. The magnetic field created by the coil and the piece of magnet is therefore
B=B(applied) + μ0M
(since magnetic field due to the current in the coil, Bappplied, alone and magnetic field due to the magnetised iron, μ0M, are in the same direction). μ0M can be easily calculated if the relative permeability of the material, Km, is known using
μ0M= Km B(applied) –B(applied)
As it will be explained further in the text, the direction of the magnetic field applied is being reversed constantly, so the metal used should be magnetically soft in order to avoid energy loss.
The magnetic field applied, Bappplied, depends on the current in the wire. Since the wire is a solenoid, the Bappplied in the middle of the coil can be found using
B(applied)=(1/2)(μ0nI)(l/(squareroot((l/2)^2 +r^2))
(where n is the number of turns in the coil per length, and l is the length of the coil and r the radius of the coil), or approximation
B= μ0nI
The magnetic flux (Φ) trough the coil would be related to the voltage drop (V) across the coil of self-inductance L and internal resistance r by
V= d (flux)/dt = d(LI)/dt + Ir
The field would therefore be related to the current by
B= Km μnI
The direction of the flow of the current determines direction of the magnetic field and therefore determines whether the voice coil will be attracted or repelled by the permanent magnet. As mentioned above, the voice coil is held in place by a ring of flexible material called spider, which allows the coil to move back or forth, as it is being attracted or repelled by the permanent magnet. Since the current flowing trough the wire is alternating current, the direction in which the current flows is being constantly reversed and, consequently, the direction (and magnitude) of the field changes constantly. This results in the quick changes in direction of the movement of the voice coil, as the direction of the magnetic force the permanent magnet is exerting on the voice coil is quickly changing.
The voice coil is therefore a driven oscillator, where the spider acts as a spring and air provides the frictional force (the reason why the motion is damped), so the equation that can be used to show how will it move is
M(d^2x/dt^2)= F – b(dx/dt) - kx
(where F is the driving force, M is the mass of the voice coil, k is the spring constant and k is the damping constant). Driving force varies with current, so it depends on frequency and time and can be calculated using
F= F0 cos ωt
where F0 is the force when current, and therefore the magnetic field, is maximum and ω is the angular frequency of the current. This is the consequence of the fact that force of attraction or repulsion between two magnets is proportional to magnetic fields, so (as B of the permanent magnet does not change) the force varies with the magnetic field, which, as already explained varies with current.
This equation can be used to show that, as the voice coil moves due to the force across the distance dx during time interval dt, the energy is conserved:
VIdt=Fdx + d ((1/2)LI^2)
On the left side of the equation is the total work done, which is equal to power, VI multiplied by the time interval dt. On the right side is the sum of the work done by the force while accelerating the voice coil over the distance dx, plus the energy stored in an inductor (the coil). This equation, combined with the previously mentioned equation that relates the voltage drop over the coil with the change in flux (LI) over the change in time can be combined into an alternative equation that describes the force:
F=(I^2/2)(dL/dx)
