# A technical question - dB in relation to Hz and Vd



## mvw2 (Oct 2, 2005)

Ok, this is an answer I haven't been able to find. I'm trying to figure out what dB level would correlate to a given frequency and air volume movement(Vd = cone area(Sd)xexcursion(Xmax/Xmag)). I've come across relations of dB to pressure changes and power, but I have yet to find anybody providing a clear cut equation saying you need to move X amount of air at Y frequncy to get Z dB output.


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## Guest (Feb 15, 2007)

*
SPL = 100.3 + 20log(xmax) + 40log(d) + 40log(f) - 20log(r)
*
where: xmax=piston excursion, d=piston diameter, f=frequency, r=distance to driver
units: dB, meters, Hertz

alternatively:
*
SPL = 102.4 + 20log(Vp) + 40log(f) - 20log(r)
*
where : Vp=volume displacement (cubic meters)


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## Guest (Feb 15, 2007)

In those expressions, xmax is a one-way measurement (as is volume, or piston, displacement of course).

And a rough estimate of generic cabin gain is 12dB per octave boost, as frequency decreases, below about 80Hz ... nicely offsetting the frequency dependency in the above relationships below 80Hz


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## mvw2 (Oct 2, 2005)

Ok, thanks.

As far as SPL versus wattage, can I assume dB raises linearly based from the driver's sensitivity? (assuming I am still in the driver's linear excrusion range)


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## mvw2 (Oct 2, 2005)

Ok, I came across this at:
http://www.hps4000.com/pages/spksamps/powering_surround_speakers.pdf

Peak SPL = ((speaker’s 1 meter sensitivity -2) +l0log amplifier power -20log(D/4)

I used my Excel woofer as the test. I used Seas listed sensitivity numbers and applied the above equation at frequency intervals of 10Hz. I used rated rms wattage of the speaker for the amp power, assuming this is the safe thermal limit of the driver.

I plotted a SPL curve for both the Xmax and Xmag levels of excursion and overlayed the raise in SPL for cabin gain as well. I've attached my MS Excel spreadsheet and chart of my results.

Do these numbers look appropriate?

I know one thing I'm not including is the non-linear excursion of the actual driver and actual SPL at various wattage and frequency outputs. This doesn't change the Xmax/Xmag plot, but it should affect the SPL level of the woofer at any given wattage and frequency. I figure it would limit the thermal output further as excursion increases and more resistance is incured. If so, I would assume thermal limits remain the dominate factor much farther down. As it is now, assuming linearity, thermal limitation becomes dominate at 150Hz on up. 

As it is, I would assume I would need to cross at 150Hz to keep the woofer below its excursion limits at rms wattage(sine waves  ). Depending on the non-linear excursion/sensitivity relationship, this value may be much lower. I don't know...

I added a part where you can enter the desired SPL level and it would give you the excursion required to achieve that dB level. I included the cabin gain for frequencies below 80Hz.


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## Guest (Feb 15, 2007)

a point of clarification ... sensitivity or efficiency does NOT impact the above equations. The SPL achievable from a given driver depends _only_ on its volume displacement. Yes, sensitivity will tell you how much amp power is required to achieve that volume displacement ... but it does not impact the achievable SPL.

mvw2 ... i don't think you are confused on this point, but it deserves it's own post for emphasis and clarification.

make sense?


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## mvw2 (Oct 2, 2005)

Yes, I'm aware of the difference.

Can I assume that I am still limited by rated power? For example 80w rms gives me an output of around 115dB(somewhat flat curve). Below about 180Hz, the physical excursion of the driver seems to be the limiting factor. Above this 180Hz point, it seems I will put more than 80w rms into the driver before I would reach maximum excursion. In this sense, could I say the woofer is mechanically limited below 180Hz and thermally limited above 180Hz?

I'll make a note I only use sensitivity to estimate max thermal SPL at the various frequencies. All the other equations SPL is simply arbetrary, something calculated through Xmax or a given SPL.


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## Guest (Feb 16, 2007)

dude it would take me a while to digest that equation you found ... for example, a quick browse through your link indicates that "D" in your equation is the distance from the screen speaker to the center of the theater ... the link is all about powering surround speakers in a home theater.

I understand the derivation of the equations I posted, and they are "general purpose" 

*The achievable SPL from any driver is a function of : volume displacement, frequency, and distance. Nothing more (excepting cabin gain or reflected energy, of course).*

For example : if you drive the cone outside the accepted "linear range", beyond xmax, can you get more SPL? Of course ... because you are creating more volume displacement. The SPL equations above tell us nothing about linearity, harmonic distortion, intermod distortion, etc.

The "xmax" in the first equation should really be considered, simply, "x" ... the driver displacement. When "x" reaches "xmax", you achieve what is reasonably considered to be the maximum linear SPL from the driver.

So i'm afraid I've got little more to offer


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## Guest (Feb 16, 2007)

... by the way, the second equation i posted is easily derived from the first  so there's no "new information" in equation number two.

The (trivial) derivation is left as an exercise for the reader


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## mvw2 (Oct 2, 2005)

I whipped this up for something else, but I should add it here as it's simply an extension of what I've done above.

This is a chart indicating excursion and thermal limits of a particular 3-way setup, in this case Seas popular 1" neo aluminum tweeter, Seas Excel (W18NX) woofer, and Peerless' XXLS aluminum coned 12" woofer. It shows a rough relation between the capabilities of the drivers showing strong/weak points in the system as well possible appropriate x-over points for the setup.

However, it's a bit simplified. It doesn't take into account enclosures, if exists, for the subwoofer and midwoofer which can improve low end response, both for sensitivity and excursion needs. As well, it doesn't show frequency response nor effect on this response due to crossovers. It's a rough glimpse if you will.


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## Jhemi80 (Jul 18, 2008)

If taken out of context, the title of this post is just too funny.....

I think we all want to know how dB relates to VD! Hahaha

Sorry, that tickled my funny bone for some reason.


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