Speaker Equations: Amplitude & Frequency Explained

[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)

 

[antonantal]

Yes, the frequency of the sound played by the speaker is exactly the frequency of the current that flows through the coil (electromagnet).
The loudness of the speaker depends primarily on the intensity of the current through the coil (http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html#c2") and the strength of the permanent magnet. But actually it is given by the volume of air displaced by the membrane when it oscillates. This means that the loudness also depends on the dimensions of the speaker.
Another thing which increases the loudness of a speaker is it's enclosure. Because the sound waves produced by the back of the speaker are in antiphase with the waves produced by the front of the speaker, ther will be some destructive interference if the speaker is not enclosed. There are some useful equations for buiding a speaker enclosure. I used them when I built my bass-reflex subwoofer enclosure

 

[Velikikreten]

Could you give me any equations, with explanations?

I tought that amplitude of the sound must vary with frequency, but some people told me its not true...

therefore the factors that would affect the amplitude of the sound would be:

1. Current through the coil and since its resitance does not vary, it only depends on the voltage?

2. B of the permanent magnet

3. size of the speaker

Enclosure?

Do you by any chance know where could I find equations as simple as possible with explanations as understandable as possible?

When I typed Loudspeaker equations into google I got this
http://inst.eecs.berkeley.edu/~ee100/sp03/ee100lec33-01.pdf

Can you explain those?

Thanks a lot for your help.

 

[Xezlec]

Well here's my two cents.

I tought that amplitude of the sound must vary with frequency, but some people told me its not true...

They were wrong and you were right. Of course it depends upon frequency. The response of the coil depends strongly on frequency, the response of the mechanical system (springiness of the air and the paper, mass of air, paper, coil, etc. etc.) depends strongly on frequency, and so on.

When speaker manufacturers build speakers, they try very hard to make the output level not depend too much on frequency. They fail. Speakers always have some frequency repsonse that isn't a straight line. That's why some companies can charge you $10000 for a speaker, if it has a fairly flat response.

Building and modeling speakers is a very complex science, and acoustic engineers can make huge money trying to do it well. I think what you need is to study acoustics a little (it's an extensive science), if you want to know about how loud the output will be for a certain shape of speaker and a certain frequency. I know you want something simple, but then you shouldn't be asking about speakers

If you are interested in modeling the speaker electrically (i.e., you want to know how the speaker will affect the circuit, but not necessarily how loud the sound will be), that's a lot easier. A speaker is an inductor and a resistor in series. Of course, it's a little more than that if you want to model the effect of the mechanical system too. But for most electronics purposes, inductor + resistor is close enough.

therefore the factors that would affect the amplitude of the sound would be:

1. Current through the coil and since its resitance does not vary, it only depends on the voltage?

And frequency, and inductance of the coil. And that's only an approximation, because the mechanical motion of the speaker also affects the current through the coil. Sorry, but speakers are non-trivial devices.

2. B of the permanent magnet

Sure, that's involved in the loudness. So are all the mechanical characteristics of the surfaces that make up the speaker. Paper, plastic, wood, air, metal. Springiness, stiffness, density, thickness, and so on. All these things matter significantly in the equations.

3. size of the speaker

And shape! Shape is very important. My tiny, funny-shaped Harman-Kardon speakers can out-scream a lot of larger speakers.

Enclosure?

Aboslutely! This is critically important. A speaker with no enclosure can act like an acoustic dipole and radiate almost no sound, whereas the same speaker with a properly-shaped enclosure might sound more like a monopole (especially from some angles) and produce loud sound.

Also, you left out listening angle. Many speakers are highly directional. And don't forget sound speed, humidity, temperature, and all that good stuff too!

Do you by any chance know where could I find equations as simple as possible with explanations as understandable as possible?

No, but I'd start on http://en.wikipedia.org/wiki/Acoustics" . Don't let me scare you off, this is a fascinating field if you're willing to actually get your feet wet. I had more fun in my undergraduate acoustics class than most of my classes. And it's a great career path.

When I typed Loudspeaker equations into google I got this
http://inst.eecs.berkeley.edu/~ee100/sp03/ee100lec33-01.pdf

Can you explain those?

Not if you don't already understand them. I think that's intended for people with some electronics/physics background.

What I can say simply is that the electronics of the speaker aren't that important when you're talking about loudness and frequency response. In real systems, the frequency curve comes mainly from the mechanical system.

 

[Xezlec]

One more thing about that link you posted. Those equations don't appear to mention the loudness of the sound, they just tell you how to model the device electrically. I think.

 

[antonantal]

I tought that amplitude of the sound must vary with frequency, but some people told me its not true...

It depends. Some speakers are designed to have a flat frequency response from 20Hz to 20KHz, others are designed for certain bandwidths (for example a subwoofer has a range from 20Hz to about 200Hz).

Do you by any chance know where could I find equations as simple as possible with explanations as understandable as possible?

When I typed Loudspeaker equations into google I got this
http://inst.eecs.berkeley.edu/~ee100...00lec33-01.pdf

I don't know of any equations made specifically for calculating the intensity of the sound played by a speaker. Instead you can use general physics equations (like those in the pdf you mentioned) to find, for example, the relation between the intensity of the input current and the distance traveled by the speaker's cone, and then relate this distance to the intensity of the sound.

 

[Velikikreten]

Thanks for replies.

Could you give me an example how would the shape of the speaker affect the amplitude of the sound? Is it just the volume of the air displaced by the speaker or is there something more?

I think I understand first two equations well, but please correct me if I am wrong:

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 don't 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?

Yes, I see that the equations don't describe the loudness, as it depends on many other things including the properties of the air.

I just want equations relating the current and the distance traveled by the cone.

Thanks!

 

[Xezlec]

Thanks for replies.

Could you give me an example how would the shape of the speaker affect the amplitude of the sound? Is it just the volume of the air displaced by the speaker or is there something more?

There is a LOT more. The way air vibrates in 3 dimensions is pretty complicated, and any resonances of the system will dramatically affect the loudness of the sound at certain frequencies. That said, I guess if you have a good enclosure around your speaker, and you are listening from directly in front of it, and you're neglecting the effects of the room you're in, you could approximate it by just thinking about the area of air it displaces. But understand that is only a rough approximation.

I think I understand first two equations well, but please correct me if I am wrong:

Oh, you do have some physics background. Sorry I didn't assume that.

Force equation:

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

They've combined the previous two equations to get an equation that gives you the force of the speaker as a function of (current and change in coil inductance as a function of displacement).

I believe they don't need B here because they've explicitly included "change in coil inductance as a function of displacement", which will probably involve B and frequency somehow.

Loudspeaker equations:

I don't understand anything here, so please help me (they are all just guesses):

what is x? The displacement of the speakers cone?

Yes.

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

Well, mass of the speaker I guess.

omega - angular frequency ?

Yes. They're using the standard phasor (complex number) representations of things that engineers use a lot. This is the frequency (in rad/sec rather than Hz) of the sound you're playing through them.

j - ?

The square root of -1. This is standard in engineering.

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)

I'm not sure, but if k is a spring constant, I might guess that the acoustic impedance of the air seen by the speaker would make a bigger difference than the spring you're talking about. Though I'm really unsure of that.

B - B of the permanent magnet?

It appears so.

l - ?

This is some length parameter relating to the coil, I believe. I think it might be the length of all the wire wrapped around the coil.

i - ?

Current in the coil.

Yes, I see that the equations don't describe the loudness, as it depends on many other things including the properties of the air.

I just want equations relating the current and the distance traveled by the cone.

That depends on the acoustic impedance seen by the speaker, which involves every single bit of the complexity of the problem. These equations seem to be based on some very basic approximations, and don't involve acoustics at all. Though I think acoustics will be most of the work in the problem.

May I ask why you want to know all this? I mean what are you trying to do? There might be an easy way out.

 

[antonantal]

Force equation:

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

It says at the top of the page "changing L". That's where the B of the magnet comes in. It affects the inductance of the coil.

Loudspeaker equations:

I don't 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 - ?

Here the coil, which is attached to the speaker cone, is treated as a driven oscillator (a http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html#c1", where you will also find most of the notations explained. I don't think you will find these:
j = square root of -1.
i = the current (sinusoidal form)
l=length of a circle of wire in the coil
B*l*i=the magnetic force of a field B exerted on a wire which carries the current i and has length l
B*l*x=the flux of the field B through a surface l*x
V0=the voltage at the output of the amplifier (i.e. at the input of the speaker) - it also should be in the sinusoidal form.
I guess you know the impedance of an inductor is j\omega L
One thing that you should try is to differentiate in time the equation x = A\cdot e^{-\gamma t} \cdot e^{j\omega{t}+\varphi} and replace \dot{x} and the second derivative of x (I don't know how to latex it ) in the motion equation of the driven oscillator. See if it looks similar to any of those loudspeaker equations.

 

[rcgldr]

I think of a speaker as a servo device. It's commanded to position itself based on the +/- voltage received. The accuracy of it's response determines the quality. It's power handling capability detemines it's "quantity".

Similar to a good servo, a good speaker system will need some sort of feedback to prevent overshoot and a means to dampen the overshoot tendency. A drag inducing device or negative feedback circuitry will do this. The result is increased accuracy, but reduced loudness (efficiency in terms of sound power output versus input power is reduced).

In terms of quality, true studio monitor systems are the most accrate, followed closely (sometime the same accuracy, but nicer looking) by "reference" speaker systems targeted towards music listening. Home theater systems, are a big step down from this.

 

[Velikikreten]

Thanks, you have all been very 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...

Thanks

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)

 

[antonantal]

It looks OK to me, if you wrote the formulas correctly. You should learn how to use LaTex.

 

[Velikikreten]

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

 

[Velikikreten]

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

 

[antonantal]

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

Yes, because it has a core, so the field inside the solenoid is much stronger than the field outside it.