The relationship between Xmax and SPL

[DevilDriver 28]

This is always an interesting and controversial topic. Every couple of weeks, a debate rages regarding the relevance of high Xmax and whether it is truly valuable to achieving SPL scores. Perhaps the greatest argument is that Digital Designs Audio drivers, who have a very impressive track record in SPL competition, have been routinely tested to show below average Xmax numbers. How is it possible that a driver with limited linear excursion can still be amongst the loudest? The answer is simple and complex all at once: Xmax is relevant to SPL and it's not. Now that I've confused you, let's proceed.

I'll touch on how a speaker makes sound and how it's interpreted at a later time, but let's skip to some ground works. Acceleration is a critical component of creating SPL. The higher the rate of acceleration, the more pressure that can be created. Let's figure out how to get there.

If you've been through any high school physics classes, you're probably very familiar with this equation.

F=m*a

Hey, that looks like Newton's Second Law of Motion. In short, it states that the net force of an object is equal to the product of it's mass and it's rate of acceleration. Of course, we can re-write the simple formula to better suit our needs.

F/m = a

After our spinning, the formula reads that the rate of acceleration is equal to the net force of an object divided by it's mass. Let's apply this to speakers for a moment, shall we? The force of an speaker is determined by the two factors:

1. BL - the combination of the magnetic field strength (B ) and the length of the voice coil in the gap (L)

2. i - the level of input current

Taking that information and our above equation, we can easily translate this to a speaker relevant equation.

(BL* i)/m = a

Let's make some logical conclusions from this formula.

1. Increasing the magnetic field strength (B ) or the length of the voice coil in the gap (L) will increase the rate of acceleration

2. Increasing the level of input current (i) will increase the rate of acceleration

3. Increasing the mass of the driver will decrease the rate of acceleration

Naturally, the inverse of these conclusions holds true as well. For example, if you decrease the moving mass of the driver, the rate of acceleration will increase. Simple stuff, me thinks. Now we have our foundation for understanding why Xmax is and is not relevant to SPL.

We know what we want to have a really loud driver, don't we? Huge BL, ability to take a lot of power, with extremely low moving mass. When looking at thiele/small parameters, we are looking for something with a high BL^2/Re, high power handling, and low Mms. It's odd that Xmax isn't factored into that equation, don't you think? Ahh, but it is, in a sense.

Once we apply current to the voice coil, it moves. Based on the Understanding Power Compression thread, we know that as the voice coil moves out of the gap, BL decreases. It stands to reason that as the voice coil moves out of the gap, BL decreases, and based on our previous conclusions, so does the rate of acceleration! In this sense, high Xmax is a good thing because it means the driver is capable of moving further without an alarming decrease in BL. In this sense, high Xmax is a great thing. A driver that can move great distances while keeping BL very linear can make for a very good, very loud driver. This is handy for competitions like Bass Race, where everyday music is played for longer durations.

With that said, there is another aspect of SPL competitions where high Xmax may not matter so much. In DBDrag style competition, high Xmax is not always critical. Typically, a competitor plays a short burst sine wave near the resonant frequency of the port(s) used in their enclosure. This excites the air mass in the port, causing it to become the primary source of our pressure. Meanwhile, the driver that is exciting the air mass is relatively stationary; it is not excurting itself very much at all. If your driver is only moving +/- 2mm, what is the difference in BL between a driver with 16mm of Xmax and a driver with 100mm of Xmax? Not much at all.

There is one more important consideration in the "high Xmax vs. SPL" equation. Again, looking back at that original equation, we know that high moving mass decreases our rate of acceleration. Let's pretend I'm engineering a driver for a client who wants high Xmax using conventional designs (more on this another day). How do I do so? Quite simple: I add length to the voice coil. There is, however, a caveat to adding length to your voice coil: more wire (be it copper, aluminum, etc) adds mass to the driver as well. This is not a good thing from an SPL standpoint.

As we can clearly see, high Xmax and SPL are not necessarily mutually exclusive; in fact, sometimes they go hand in hand. At the same time, some situations call for very little Xmax. This is why companies like Fi Car Audio deserve more credit than they receive: there is no one answer for every question, but we'll allow you to customize our answer to suit your question.

The point comes back to a daily engineering focal point: where can I afford to make sacrifices for the application? And that ultimately brings us to Newton's Third Law:

For every action, there is an equal and opposite reaction.

Cheers!

[Aaron Clinton 2,372]

Great write up.

A XMAX and SQ would be an interested addition to this topic, since designs like LMT and other monster excursion models.

[DevilDriver 28]

Great write up.

A XMAX and SQ would be an interested addition to this topic, since designs like LMT and other monster excursion models.

Good idea for tonight. Thanks Aaron!

[Aaron Clinton 2,372]

Great write up.

A XMAX and SQ would be an interested addition to this topic, since designs like LMT and other monster excursion models.

Good idea for tonight. Thanks Aaron!

Yeah, I still need to find the money for an RL-s. I know Kent loved it and anyone that had power for it love it too.

[JimJ 11]

Excellent article.

Of course, anyone that's done anything with high efficiency speakers already knows that Xmax as a singular spec is overrated, Lowthers can get plenty loud with a half millimeter of linear travel, just takes a six foot tall horn to do it

[topgun 262]

Excellent article.

Of course, anyone that's done anything with high efficiency speakers already knows that Xmax as a singular spec is overrated, Lowthers can get plenty loud with a half millimeter of linear travel, just takes a six foot tall horn to do it

My fostex's got pretty loud for low power and somewhere around .5mm of travel in their horn...

[JimJ 11]

Great write up.

A XMAX and SQ would be an interested addition to this topic, since designs like LMT and other monster excursion models.

Good idea for tonight. Thanks Aaron!

Yeah, I still need to find the money for an RL-s. I know Kent loved it and anyone that had power for it love it too.

I still want to hear one open baffle.

A sub with that much travel and a Q that high seems tailor-made for OB bass...

[topgun 262]

Great write up.

A XMAX and SQ would be an interested addition to this topic, since designs like LMT and other monster excursion models.

Good idea for tonight. Thanks Aaron!

Yeah, I still need to find the money for an RL-s. I know Kent loved it and anyone that had power for it love it too.

I still want to hear one open baffle.

A sub with that much travel and a Q that high seems tailor-made for OB bass...

You buy it and bring it down here to Kansas, I'll come up with the room and the power.

[JimJ 11]

I'd have the money this summer, but I can think of other audio-related things I could upgrade/buy before screwing with the substage...

Damn, now I know why I don't fit in anymore on CA.com

[helotaxi 46]

Why? Oh you mean that you realize that there is much more to an audio system than the substage and SPL? I never really fit in there from that perspective.

[bdawson72 0]

Good write up

And Jim i still wub you...even though you don't love the SPL'z

[mrray13 30]

awesome read! and i like aaron's idea of the xmax=sq article. the rl-s is indeed a sweet driver, and did very well at output too!

can't wait for the next one read.

wheeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

[PaulD 0]

interesting stuff,although I'm not a basshead either

On the SQ side, I have an interesting experiment for you to perform (or at least tink about). Take a length of transformer wire, it's coated so it won't short out against itself, and make a tightly wound coil with about a 1" or so tube down the middle - then attach the ends to an ammeter with an analog needle. Find a magnet that will fit in the gap tube) and slowy run it in and out, you will notice the ammeter (current meter) will start to move some. Now move it thru a little quicker, you will notice the needle will deflect a lot more, the faster you move it thru the higher the peak current reading will be. Be careful not to peg the meter to hard on this experiment.

Lessons for SQ ....... if you want powerful dynamics and sharp transient response, what you really need it LOTS of power, at least 2 - 3 times the max RMS of the speaker and just watch the gains. Note that this logic applies to mids and tweets too.

WARNING ... my power recommendations do NOT apply to bassheads who play electronic music with liitle or no headroom at full volume, the only thing you will get from it will be blown speakers.

[vladd 26]

i'm probably in left field here but does that not simply mean that when you're moving the coil faster that you're making more power. Therefore when you're making more power you're moving the coil further making more sound. Increasing the amplitude but not really affecting anything else?

[JimJ 11]

Lessons for SQ ....... if you want powerful dynamics and sharp transient response, what you really need it LOTS of power, at least 2 - 3 times the max RMS of the speaker and just watch the gains. Note that this logic applies to mids and tweets too.

Or lots of efficiency

[topgun 262]

Lessons for SQ ....... if you want powerful dynamics and sharp transient response, what you really need it LOTS of power, at least 2 - 3 times the max RMS of the speaker and just watch the gains. Note that this logic applies to mids and tweets too.

Or lots of efficiency

What about both...

[PaulD 0]

Actually if you take the experiment from the opposite direction, it means the extra power capability will increase the speed of the driver - not necessarily the amplitude, remember the max amplitude of the speaker in this case is really the Xmax and is fixed. Efficiency can also help, unfortunately it seems that all too many efficient drivers are not too good sounding.

I like these topics being discussed in this area, good stuff.

[helotaxi 46]

If you increase the speed of the cone, either the amplitude must change or the speaker hits a limit. The limit is really bad from both a distortion and a longevity standpoint. The increased amplitude is where your extra volume comes from.

F=ma as we all know and d= .5 at^2. t is a constant and is the period of the wave in question. If we increase power we increase a. An increase in a with a constant t means the driver moves farther during that time. Once you have enough power to hit the excursion limit of the driver, having more will not do anything for you.

This is where the interaction of the LPF and power handling comes in. In most cases Xmax (or Xlim) is what is governing your power handling with music. Thermal power isn't that important for the simple reason that you aren't applying high RMS power so it takes a while to temp out the coil. It is actually peak power that matters for music because it will dictate the excursion of the driver. The lower the freq, the longer the period and the farther the driver can potentially move with the same power. By limiting the low end freqs, you can apply increased power without hitting Xlim. This is the reason that you will see some manufacturers rate power handling for tweeters based on the crossover freq and slope. Higher freq and/or steeper slope means more power handling.

[holley2346 6]

Good article....

[ChristianMcKay 1]

This was a really good article. I love reading on these technical and how to forums because I learn new things like Xmax not being all that important.

[SpeakerBoy 64]

What I learned? Throw tons of power to a woofer at tuning, til you reach xmax. (match car and box peak frequency).