Theory about speaker+box resonance

[azira]

I'm having trouble finding hard information about how the speaker resonance and the box resonance interacts. Most people just plug in numbers into WinISD or whatever and magical figures appear.

My first problem is I don't understand how the impedance curve for a speaker gets translated into a high-pass filter kind of curve of a box (sealed enclosure for example).

My second problem is I can't find any information at all about what enclosure resonance curves look like. I assume that the resonance behavior would look like a LRC resonance circuit and would be caused by wave reflections inside the enclosure. If stuffing is used to dampen waves, does this "lower" the resonance frequency?.

And thirdly, how does the resonance of the speaker and the resonance of the box combine (like series resonance circuits? like parallel resonance circuits?)... and of course, how does all this get converted from a graph with a peak into a 2nd or 3rd or 4th order high-pass curve that is typically seen?

I am interested in the hard math and theory for the corelations, I already know it happens and that I can use WinISD to design using it...

For reference, here is an image of the impedance curve, resonance is obvious.

TIA.
--
Danny

[azira]

And here is the x-order response of a sealed enclosure that the above gets converted into.

(credits to Tang Band and the speaker workshop guys for these images.)

[sreten]

The simple answer is boxes do not have resonances,
and the internal volume is treated as an equivalent
air spring, or electrically a capacitor.

Unit suspension stiffness is quoted as an equivalent
box volume, this combined with the real box volume
determines the stiffness reacting with the mass of cone.

T/L's excepted, box resonances are at higher frequencies
and completely ignored by most modelling software, justifiably.

sreten.

[azira]

Originally posted by sreten
The simple answer is boxes do not have resonances,
and the internal volume is treated as an equivalent
air spring, or electrically a capacitor.

Unit suspension stiffness is quoted as an equivalent
box volume, this combined with the real box volume
determines the stiffness reacting with the mass of cone.

T/L's excepted, box resonances are at higher frequencies
and completely ignored by most modelling software, justifiably.

sreten.

Oh, cool, that makes a lot of sense. And since enclosed air is a positive value, it must add with the speaker compliance and push the resonance to a higher frequency because it's a larger apparent stiffness right?

That would take care of question 2&3 but still, how do you translate from the resonance curve (or I guess it would be an adjusted resonance curve with the enclosure) so the 2nd image?

[Ron E]

What are you trying to do?

If you understand transfer functions and circuit analysis: Get Richard Small's articles on enclosures from the early 1970's JAES.
and/or get :
"Theory and Design of Loudspeaker Encosures" by J.E. Benson
and/or:
"Acoustics" by L.L. Beranek

If you don't know this fairly advanced math, go to
www.diysubwoofers.org and play around with equations there.

[sreten]

Originally posted by azira



Oh, cool, that makes a lot of sense. And since enclosed air is a positive value, it must add with the speaker compliance and push the resonance to a higher frequency because it's a larger apparent stiffness right?

That would take care of question 2&3 but still, how do you translate from the resonance curve (or I guess it would be an adjusted resonance curve with the enclosure) so the 2nd image?

1) yes. true for sealed boxes and one of the frequencies for reflexes.

2) the curves are for a vented / reflex alignment.

Here you do have a "box" frequency, the tuning of the vent or port.
But its not related to box dimensions, but is related to
the port dimensions combined with the box dimensions.

Simply put there are various vented / reflex alignments that use
ratios of the port and sealed frequencies, and require specific
sealed box Q's ( for the same volume) for the alignment used.

Note that sealed boxes are second order high pass filters
and vented / reflex boxes are fourth order high pass filters.

sreten.

[azira]

Originally posted by Ron E
What are you trying to do?

If you understand transfer functions and circuit analysis: Get Richard Small's articles on enclosures from the early 1970's JAES.
and/or get :
"Theory and Design of Loudspeaker Encosures" by J.E. Benson
and/or:
"Acoustics" by L.L. Beranek

If you don't know this fairly advanced math, go to
www.diysubwoofers.org and play around with equations there.

The math is not the problem.

I'm a analog/digital electronics engineer. I've read some articles on modeling resonances as LRC circuits because they behave similarily so I was trying to use my circuit knowledge to better understand speaker behavior.

I'm trying to understand how to go from fig 1 -> fig 2. In a standard resonance circuit, the transfer function (and therefore freq response) slopes off on either side at the same rate (dB/dec). However, putting a speaker into a box produces some kind of high-pass like response. I can't find the theory/math behind this.

[azira]

Originally posted by sreten



1) yes. true for sealed boxes and one of the frequencies for reflexes.

2) the curves are for a vented / reflex alignment.


sreten.

Sorry, the black curve is a ported box, hence the extra bump near the cutoff frequency. The red curve is a sealed box however.

I'm just concerning myself with the sealed box vs the ported because it's the easiest one to start with and the most of the fancier boxes are just superpositioning of the behaviors.

[sreten]

Originally posted by azira


The math is not the problem.

I'm a analog/digital electronics engineer. I've read some articles on modeling resonances as LRC circuits because they behave similarily so I was trying to use my circuit knowledge to better understand speaker behavior.

I'm trying to understand how to go from fig 1 -> fig 2. In a standard resonance circuit, the transfer function (and therefore freq response) slopes off on either side at the same rate (dB/dec). However, putting a speaker into a box produces some kind of high-pass like response. I can't find the theory/math behind this.

Sealed box is 2nd order high pass filter theory.

Reflexes are 4th order high pass filter theory.

sreten.

[Svante]

Originally posted by azira
I'm having trouble finding hard information about how the speaker resonance and the box resonance interacts. Most people just plug in numbers into WinISD or whatever and magical figures appear.

Indeed. Simulation programs are very powerful tools, but basic understanding of the mechanisms in the speaker is *very* helpful when using them.

Originally posted by azira

My first problem is I don't understand how the impedance curve for a speaker gets translated into a high-pass filter kind of curve of a box (sealed enclosure for example).

It doesn't. They are related, and have the same origin, but one does not cause the other.
Let's take the response curve first. Lets set up a few need-to-know things for this:
-Sound pressure is proportinal to the *acceleration* of the cone.
-The force from the coil acting on the cone is proprtional to the current running through the coil.
-The moving system of the driver consists of a mass, a spring, and some resistive losses.
-These mechanical impedances dominate in different frequency regions.
-At high frequencies the mass dominates, at low frequencies the spring dominates.

So, at high frequencies, the movement will be controlled by the mass, mainly, and since the force acting on a mass is proportional to the *acceleration* of the mass (F=ma), we will have an acceleration proportional to the driving current. Since sound pressure was proportional to acceleration, we get an essentially flat response for frequencies above the resonance frequency.

At low frequencies, the movement is controlled by the spring. Using the same reasoning we realise that the *cone excursion* will be proportional to the driving current. Now, acceleration (which was proportional to sound pressure) is the second derivative of the excursion. The second derivative of sin(wt) is -w^2 *sin(wt), we see that the response is proportional to the frequency squared. This is the same as a tilt of 12 dB/octave, and this we see below the resonance.

Now to the impedance curve. The impedance originates in two parts, one from the electric impedance of the voice coil, and one impedance corresponding to the mechanical load on the coil.

The electric part is the easy part. It is a resistance (the DC resistance) in series with a (lossy) inductance. This inductance is responsible for the raise of the impedance towards higher frequencies.

On top of this, there is a resonance which coincides with the mechanical resonance fs=1/(2*pi*sqrt(Mms*Cms)). This is harder to explain, but one way would be that at resonance the mechanical system is particularly easy to move. This will cause the coil to move a lot, and a lot of EMF is generated by the coil moving in the magnetic field. This EMF counteracts the driving voltage, and this makes the current drop. This is equivalent to a high electric impedance.

Originally posted by azira

My second problem is I can't find any information at all about what enclosure resonance curves look like. I assume that the resonance behavior would look like a LRC resonance circuit and would be caused by wave reflections inside the enclosure. If stuffing is used to dampen waves, does this "lower" the resonance frequency?.
[/B]

If we are talking about closed boxes, there are several types of resonances.
The main resonance is the one I described above, which really is a resonance in the driver. The thing is that the extra stiffness provided by the air enclosed in the box shifts the driver resonance upwards. So if you measure the electric impedance of a driver in a closed box, it will look the same as for the driver in free air, but with a higher resonance frequency.
There are also secondary resonances, for example standing waves inside the box, mechanical resonances in the box walls, pipe resonances in a bass reflex tube etc. All these are different and are rarely seen in the electrical impedance curve, but may have an impace on the response curve.

Stuffing inside the box mainly lowers the Q of the standing waves, and you should use this. It also has a *slight* effect on the resonance frequency of the driver/box, it is lowered. The reason for this is a thermal excange from the air to the stuffing. The air wont get as much heated during overpressure, and the air will not appear to be as stiff. So the raise of the driver resonance frequency as I described above will not be as large as for the unstuffed box. The acousticians call this "adiabatic" (without stuffing) and "isothermal" (with a lot of stuffing) compression, respectively.

Originally posted by azira

And thirdly, how does the resonance of the speaker and the resonance of the box combine (like series resonance circuits? like parallel resonance circuits?)... and of course, how does all this get converted from a graph with a peak into a 2nd or 3rd or 4th order high-pass curve that is typically seen?
[/B]

Let's see if I understand what you mean; the electrical impedance has an equivalent circuit, where the resonant part is a parallel cirquit. It is related to the response curve, but rarely "converted to" it.
A closed box has a 2:nd order HP response curve, a bass-reflex box has a 4:th order HP response curve. A 3:rd order responce curve would be very rare, the only type I can come to think of would be a bassreflex box where the port has been replaced with a flow resistance.

Originally posted by azira

I am interested in the hard math and theory for the corelations, I already know it happens and that I can use WinISD to design using it...
[/B]

If you really want to dig into this, you should read up on mechanic-electric analogies and the jw-method. If you wand to see a derivation of the reponse function for the bass-reflex box, have a look at
http://www.speech.kth.se/~svante/ela...s/image004.gif
...just don't be discouraged. The closed box is easier...

HTH
/Svante