Speaker equations

[Velikikreten]

I am looking for equations which would describe sound that a speaker produces.

 

If I understood the principle how a speaker works correctly, there are a permanent magnet and an electromagnet.

 

As alternate current flows trough a wire, the poles of electromagent constantly swich places. When current flows in one direction, the magnet repels the electromagnet, when the current flows in another direction, the magnet atttracts the electromagnet, so the electromagnet oscillates.

 

The frequency at which it oscillates probably depends only on the frequency of the signal, but what the amplitude of the sound (loudness) depends on? Does anyone know any equations with explanations?

 

(if I am wrong about anything here please tell me so)

[Klaynos]

I believe the amplitude is related to how far the speakers diafram moves for each oscillation. As it works by just forcing the air infront of it to move.

[swansont]

Right, so the volume is going to depend on the amount of current, since a stronger repulsion/attraction will result in a larger displacement of the diaphragm.

[Velikikreten]

The current is going to depend on the voltage and resistance (Ohms law).

 

The resitance cannot be changed, so the the only thing that changes is the voltage... right?

 

The voltage therefore determines the B of the electromagnet and the force of repulsion or attraction to the permanent magnet.

 

Now, if the frequency changes, the amplitude must change as well! (the smaller the frequency, everything else held constant, the longer will the electromagnet travel in one direction, so the amplitude will be larger)

 

Also the greater the B of the permanent magnet the greater the force will be...

 

but, DOES ANYONE KNOW THE EXACT EQUATIONS? I cant find them anywhere and I need them quite soon.

 

Please help. Thanks

[Klaynos]

It's a bit more complicated than just resistance.

 

Because most speakers will be on some kind of LCR circuit, so this introduces a significant frequency dependence on the impedance of the system.

 

The equations will depend on your specific speakers, and the electronics inside them. You would need circuit diagrams and to analyse them (probably best done using something like spice),

[YT2095]

the amplitude does NOT have to alter with the frequency.

the X and Y are entirely seperate.

 

X is your Frequency

Y is the Power (which is taken as Voltage X Current)

 

over the very narrow spectrum of the Audio band Impedance will play a very minor to trivial role and can be ingnored.

[Klaynos]

the amplitude does NOT have to alter with the frequency.

the X and Y are entirely seperate.

 

X is your Frequency

Y is the Power (which is taken as Voltage X Current)

 

over the very narrow spectrum of the Audio band Impedance will play a very minor to trivial role and can be ingnored.

 

Why then do we use LCR circuits to pull out base signals to one diafram and higher to another?

[YT2095]

that`s Filtering (often passive) like twin `T` or PI filters.

 

direct audio into a speaker it will see All frequencies, and you`ll get Harmonic distortion, so you strip them off to what the speaker Can operate in to keep the THD to a min, and ideally aim for flat response.

 

it`s a Physical/Mechanical problem, not an electronic one.

that`s why most Full Range speakers are Ovoid and some even have a "tweeter" cone in the center.

[swansont]

that`s Filtering (often passive) like twin `T` or PI filters.

 

direct audio into a speaker it will see All frequencies, and you`ll get Harmonic distortion, so you strip them off to what the speaker Can operate in to keep the THD to a min, and ideally aim for flat response.

 

it`s a Physical/Mechanical problem, not an electronic one.

that`s why most Full Range speakers are Ovoid and some even have a "tweeter" cone in the center.

 

To amplify this, as it were, the speaker is going to respond to the drive signal, which will be a very complex waveform with many frequencies from 10's of Hz to ~20kHz, and a single speaker composition, size and shape will be limited to how well it can respond (flat surfaces would be Bessel functions). So you use filters to send the low frequencies to the woofer and the high frequencies to the tweeter, which are designed for a narrower frequency range.

[Velikikreten]

Actually I am looking for equations which would describe the distance travelled by the cone (relating it to the current in the coil), since finding actual loudness depends on too many factors, such as properties of air, size and shape of the cone, enclosure...

 

I dont understand why the loudness is not related to the frequency: if the frequency is lower the current is flowing in one direction longer, therefore the the coil will travel in one direction longer?

 

Here is something I found by typing loudspeaker equations into google:

http://inst.eecs.berkeley.edu/~ee100/sp03/ee100lec33-01.pdf

 

 

Here is how I attempted to interpred the equations, please correct me where I am wrong:

 

(I found the equations labeled loudspeaker equations most confusing, others I understand more or less):

 

V Idt = Fdx + d((1/2)LI^2)

 

Left side is the power times dt, therefore the energy.

Right side is the the work the force does while moving a coil + the energy in stored in the magnetic filed of the coil?

 

V=dfi/dt=dLI/dt

 

Relates voltage to the magnetic flux and flux to the current and L of the coil.

 

Force equation:

 

I guess it describes the force acting on the magnet, but why is there no B of the permanent magnet there?

 

Loudspeaker equations:

 

I dont understand anything here, so please help me (they are all just guesses):

 

what is x? The displacement of the speakers cone?

 

m - mass of the coil and the piece of iron???

omega - angular frequency ???

j - ???

gamma - ???

k - I am really guessing here: the elsticity constant of the thing that holds the voice coil in place (I think it is called a spider)

B - B of the permanent magnet?

l - ?????????????

i - ?????????????

 

Why are the left and right side equal?

 

Second equation:

 

Vo - peak voltage?

 

What does the second equation describe?

[swansont]

I dont understand why the loudness is not related to the frequency: if the frequency is lower the current is flowing in one direction longer, therefore the the coil will travel in one direction longer?

 

But if the frequency is lower it is moving more slowly. The amplitude depends on the current, not the frequency.

[Velikikreten]

Thank you, you have all been very helpful.

 

Here is text I wrote, and I would like you to tell me did I understand everything correctly. If I misunderstood something or made a mistake somewhere, please tell me so that I can correct it...

 

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)

[Velikikreten]

Is it ok to use approximation for long solenoid here, B= μ0nI?