I did some maths

I thought, what if you have a sealed-box type cabinet. What if the moving mass and cone area of the driver (or drivers) are fixed. Suppose I care about a single frequency in the low midrange that is (a) too high for the driver's fundamental resonance to have any effect and (b) too low for beaming. Suppose I can change the BL of the driver until I get the most efficiency at that frequency.

This is interesting because the extremes of BL won't work. Too low and the motor isn't pushing the cone hard enough, mass dominates and efficiency is low (as with eg car subwoofer drivers). If BL is too high, electrical damping dominates and the efficiency is again very low (like a compression driver diaphragm without its horn).

Turns out if I care mostly about a single frequency, I'm mostly better off setting BL such that the efficiency badwidth product (EBP) is near that frequency. EBP is Fs/Qts and is the point at which the driver transisions from electrical damping to mass-limited mode. So, not surprising really (remember, the experiment requires keeping mass constant - you can always do better if you can resuce your mass, but copper and aluminium weigh what they weigh...)

Then I hit the math. Efficiency at EPB is about half (-3dB) that of the reference efficiency, which is

Efficiency (N0) = 9.64 * 10^(-10) * Fs^3 * Vas / Qes

for half-space in SI units. Substituting the usual formuas that link speaker specs, we get approx

Nebp = Sd * Febp / Dsd / 245

Where Dsd = Mms/Sd. I call Dsd the surface density, and it is intersting because it is roughly constant across different drivers. Midrange drivers have values of 0.5 to 1.0 and bass drivers have values of 1.0-2.0. This formula is good for multiple drivers too, just scale up Sd.

What does it tell us? Well, direct drivers are better at higher frequencies (always assuming we can adjust the BL to suit whatever frequency we care about), until beaming sets in. More area is always good. But if you are limited in area (eg because of cabinet size restrictions) and in terms of frequency response (because you have to meet a sub) then there's only so far you can go.

Example: I can afford 1.5 square feet front panel, minus a tweeter and I have to cross to a sub at 150Hz, so I need max output at 200ish Hz. I can fit maybe 2*10" drivers, and I'll assume they are good midrange drivers, Dst=0.7. I choose a BL that gives Febp=200Hz and then using the formula I get (2*0.034) * 200 / 0.7 / 245 = 0.079 which is 7.9% or 101dB. Of course that is a half-space figure and the baffle-step will reduce the sensitivity somewhat in whole-space.

More maths!

Recalling Nebp = Sd * Febp / Dsd / 245

it looks like we can get impossibly efficient by just making Sd or Febp really big (Nebp=1 would be 100% efficient, more than that is unpossible). What happens?

Well, I did state that Sd and Febp were such that no beaming occurs. The limit of beaming is approximately

Sd = 42700/F^2

and beyond this point the efficiency will stop rising (on-axis might rise, depending on array shape, but only over an increasingly narrow coverage area). Putting this back into the sensitivity formula becomes

Nebp = 174/Febp/Dsd [1]

which an upper limit for the power response for direct drivers of *any* area. If Dsd=0.5 is the lowest Dsd you can find, then direct drivers are *always* falling in power response above about 350Hz.

What if we go lower in frequency - does this latest formula also indicate Nebp>1? Yes, and there's a second limiting mechanism that prevents the impossible. This is the airload impedence. Eventually, the airload takes over from cone mass as the main limiting factor (apart from electrical damping).

The frequency range this can work to is limited by [1]. With Dsd=0.5 and Febp in the range of 350Hz, sensitivity would peak at about 25% I think in theory.

If you use high power drivers, Dsd rises and if the arraying is imperfect (total Sd > n*driver Sd) it rises even further. You may not be able to acheive Dsd less than 2.0 in practice. This reduces the practical freq limit to about 85Hz for 25%, requiring a 4 square metre array. If you take Febp even lower for the same Dsd (and allow the array to get really huge) you could approach 50% - perfect impedance matching - but only at very low freqs below 50Hz.

Direct cone drivers represent a band of capabilities then, with horn-like 50% levels (109dB) acheivable with high power drivers below 50Hz with very big arrays, a respectable 25%ish (106dB) at 200-350Hz with pretty big arrays of light-coned drivers, and then a ceiling that falls at 3dB per octave above 350Hz, so 101dB at 1K, 96dB at 3k5 and 91dB at 10K (always power response in half-space).

That last one resembles old 3" cone tweeters from yesteryear - good to know they were optimal!

jsg wrote:
Turns out if I care mostly about a single frequency, I'm mostly better off setting BL such that the efficiency badwidth product (EBP) is near that frequency. EBP is Fs/Qts and is the point at which the driver transisions from electrical damping to mass-limited mode. So, not surprising really (remember, the experiment requires keeping mass constant - you can always do better if you can resuce your mass, but copper and aluminium weigh what they weigh...)

EBP is Fs/Qes not Qts

You are right. All this stuff is assuming low mechanical damping, so Qms is high and Qts roughly = Qes.

There are other assumptions in there too.