Theory about speaker+box resonance

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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
 

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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.
 
sreten said:
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?
 
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.
 
azira said:



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.
 
Ron E said:
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.
 
sreten said:



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.
 
azira said:


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.
 
azira said:
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.

azira said:

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.

azira said:
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.

azira said:
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.
azira said:
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/elak/harledningar_files/image004.gif
...just don't be discouraged. The closed box is easier... ;)

HTH
/Svante
 
azira said:


However, putting a speaker into a box produces some kind of high-pass like response. I can't find the theory/math behind this.

Ahh... That is easy. Sound pressure is proportional to cone acceleration. In mechanical analogies the cone velocity is the equivalent to the current. So a mechanical series resonance circuit would be bandpass for the *velocity*. Differentiate this and you have the acceleration. Differentiation corresponsds to a tilt of +6dB/oct. So the bandpass will be a highpass. Voilá!
 
azira said:


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.



Dynamic simulation is a major interest of mine. The theory developed in the books I cited (I own them, too) is the hardest data I know of, and that is what you said you were looking for. I gave you the Resources you need. I was not willing to waste time spouting equations or theories you might not understand. A true understanding of this will not be had from a forum discussion, although this might be a fair place to come if you had questions. You'll find someday there's nothing like going to the source when you want results.

If you want a web resource, check out the Benson style Thiele/Small model that is developed by Robert Bullock in a windows help file format....Google for it.
 
Ron E said:


I was not willing to waste time spouting equations or theories you might not understand. A true understanding of this will not be had from a forum discussion, although this might be a fair place to come if you had questions.


Fair enough. On the other hand, I'm not willing to dump $100 on the first book that someone on this same forum says is the definitive guide to this or that. Same coin.
I agree I won't find the definitive answer, but atleast I might get a start with a direction to start exploring.
Anyway, only way to raise the educational level at which this forum is discussed is to contribute at that level.
 
Both books mentioned by Ron are a must-have for anyone interested in loudspeaker theory. I have a copy of the Acoustics book, it helped a great deal when I was making the current versions of my speaker programs; particularly SubSim.

There are so many texts form the Journal of the Audio Engineering Society. You may also be interested in the book "Introduction to Electroacoustics and Audio Amplifier Design" by W.M. Leach Jr. whose JAES contributions I found most informative.
 
azira said:

I just did a sim...

Try to simulate the circuit below instead. All parameters are T/S parameters, RML is the mechanical radiation resistance:

RML=(Ss*w)^2 * rho0 /(2*pi*c)

Note that it is dependent on the frequency squared.

Simulate this and you should see the driver impedance as the impedance connected to the voltage generator, and if you can find the *power* dissipated in RML, you have the HP function!
 

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Hi,

If it helps to understand a loudspeaker unit, long time ago I made an electromechanical model to simulate in Pspice. Basically it is a second order mass-spring system. The acoustical output is proportional to the acceleration.

For a closed box, the springiness of the air in the box acts in parallel to the springiness (compliance) of the speaker itself. For a BR the speaker is coupled to a Helmholz resonator formed by the springiness of the air in the box acting on the BR-port and the air-mass in the BR port.

Cheers ;)
 

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Nice simulation! This is the Physisists way of doing it.

The electrical engineer might do it like this, using electric-mechanic-acoustic analogies. To the left currents are flowing through the cables and voltages appear over the components, in thew middle velocities "flow" through the components and forces appear over them, to the left, volume flows flow through the components and pressures appear over the components.
The diagram is the equivalent of a loudspeaker. We feed it a voltage (us) and a current (is) in the electric domain, and out of it we get a volume flow (Qs) and a pressure ((Ps) in the acoustic domain.

T=Bl and the symbol below the T is a gyrator, Ss is the cone area and the symbol below is an ideal transformer.
 

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Guys, Guys, can I get my 2 cents in here ??? (OK, 3 cents)

In live band renforcement (live sound) in smaller venue's we would use a thing called an "iso-box" for the lead & rythm gutairs an iso-box was this, a micropone & a speaker in an foam lined box (yes the speaker was baffled) but no "outside sound", as in nothing could be heard outside the box (totally inclosed).

Why you ask ? well, the thing is , that it gave us a "controled space" to aquire the "screaming" lead solo's from, AND a consistant source for basing the rest of the "room sound" from (bar style venue's, low ceiling (12ft) and generally a "large living room", once it filled with partons).

What does this have to do with the original question ??? read on..


If one "sings in the shower" geez... would that basicly be a fairly standard type of enclosure (shiny on 3 sides, dull on the 4th(curtain), and the persons voice is offset to the port end of the box (standing up, the area above the curtain is the port) and the steel tub, could be construde as the back of the tweeter(s) ???

Now, sit back and think back... When you where in a different bathroom.(hotel, distant friends place, etc.)... a tub is a tub... (generally speaking) how come when you sang there, it sounded different then when you sing at home ????

Plot that ! Why are they different ??? it's your voice, your head cavitys, a curtain, and a steel structure at you feet, and generally ceramic tile on 3 sides, are ya gona tell me the curtain make that much difference ????

:D
 
Svante said:
Nice simulation! This is the Physisists way of doing it.

The electrical engineer might do it like this, using electric-mechanic-acoustic analogies. To the left currents are flowing through the cables and voltages appear over the components, in thew middle velocities "flow" through the components and forces appear over them, to the left, volume flows flow through the components and pressures appear over the components.
The diagram is the equivalent of a loudspeaker. We feed it a voltage (us) and a current (is) in the electric domain, and out of it we get a volume flow (Qs) and a pressure ((Ps) in the acoustic domain.

T=Bl and the symbol below the T is a gyrator, Ss is the cone area and the symbol below is an ideal transformer.

Svante,
thanks for that info. Just out of curiosity, Qms would be the Q of the mechanical RLC circuit (Rms, Mms, Cms), right? Then what would Qes be? There is no capacitor on the electrical side (left of the gyrator). Or have I got it all wrong?

Thanks
- Ashwin
 
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