The Relationship Between Xmax and SPL

[KickinAudio]

Written by: Neil Middlemiss

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.