How does bandwidth affect a driver's max SPL?

[Cousin Mose]

Long time lurker, first time poster.

In an interview at Erin's Audio Corner, Purifi co-founder Lars Risbo mentioned that a driver playing a narrow passband can generally produce higher SPL than the same driver playing a wider part of the spectrum. If this is true, there must be some mathematical relationship between bandwidth and sensitivity or power handling. Despite my best and dogged googling, I can't find such as formula. In fact, I can't find much of any discussion on the topic.

In electrical engineering terms, I believe I'm looking for the acoustic equivalent of the "gain-bandwidth product".

Does such a formula exist for loudspeakers? My reason for asking is that I have been contemplating speaker designs with mid-range and mid-bass woofers that only cover about 2 octaves each (a la Revel's Salon2), and I would like to have a way to reasonably estimate the SPLmax of a driver over such a narrow bandwidth (I believe published power ratings are typically based on weighted noise signals that span at least a decade).

The basic premise seems intuitive enough. If SPLmax of a driver is limited by excursion at low frequencies, then lowering the top end will have no effect. However, eliminating those excursion-limiting frequencies on the low end by raising the high-pass filter will allow the driver to play higher frequencies with the same Xmax. Since it is well known that increasing frequency with constant excursion produces higher SPL, this is clearly a case where the driver can play louder by narrowing the passband.

In other cases, the driver may be thermally limited at SPLmax. In such cases, narrowing the passband at either end necessarily reduces the total power handled by the driver (the driver is producing less sound, so it must be using less energy). This, in turn, allows the driver to handle more power in the remaining passband. Again, SPLmax has increased by narrowing the passband.

So, how can I accurately calculate SPLmax in my target passband based on power handling/sensitivity ratings from standardized test bandwidths?

Thanks you in advance for your contributions.

 

[NTK]

The sensitivity of a driver also depends on its moving mass (see link below). Tweeters usually have light weight diaphragms and voice coils, and therefore have higher sensitivities than woofers, even with much smaller piston areas.

I guess one can also make do with lighter woofer/midrange diaphragms that are less stiff if they don't have to cover the upper frequencies.

Also, because in music there are much higher energy contents in the lower frequencies, one usually doesn't need the same SPL capabilities at the lower and upper ends.

A fast driver needs a light cone. Or does it? - PURIFI

Intuition can play us tricks. Take this simple example: how does a speaker cone move when you try to make it produce a “pulse”? I’m willing to bet quite a few of you will guess it will rapidly move forward and then equally rapidly return to its rest position. And from that it’d be quite... View...

purifi-audio.com

Linkwitz explained how to estimate max SPL from driver size and xmax as a function of frequency here.

Loudspeaker system design

 

[mhardy6647]

The sensitivity of a driver also depends on its moving mass (see link below).

well -- this and the electrical and magnetic properties of its motor.



ca. 104 dB SPL at 1 meter for 1 watt input (well... maybe 0.5 watts, 'cause it has a 16 ohm VC)

 

[tomtoo]

If you just have a small bandwide a driver has to handle, you can use the part where it plays loud. Dont think there is any easy formular for that.
If a driver has a huge bandwide and should be linear you have a lot of compromises that efficience has to pay for.

 

[Cousin Mose]

Thanks to everyone who responded so far. While I agree with everything that has been said, I do not believe that my specific question has been addressed. Perhaps it will be easier to understand if I use a specific example instead of speaking in general terms:

Suppose that a driver has a 100 hour noise rating of 60 watts when tested with a weighted pink noise signal and 2nd order Butterworth filters from 300Hz to 3kHz. Also suppose that this driver has a nominal impedance of 8 Ohm and sensitivity of 81.5 dB/W @ 1m across the entire test frequency range. From these values, one can determine the SPL at the rated power level is:

SPL = 81.5(dB) + 10*log[60(W)] = 99.3 (dB)

It has been stated (by a knowledgeable expert) that reducing the frequency range should allow the driver to play louder. Since frequency is nowhere to be found in the SPL formula, I can only assume that reducing the frequency range must affect sensitivity (seems unlikely), power handling (makes sense), or both. So, if this driver were re-tested from, say, 450Hz to 1.8kHz, what would be the new power handling and/or sensitivity rating? Is there a way to predict this mathematically, or is physical testing (and destroying several drivers in the process) the only option?

 

[DVDdoug]

It has been stated (by a knowledgeable expert) that reducing the frequency range should allow the driver to play louder.

I don't believe so...

But... If you restrict the bandwidth the driver can handle more power into the speaker system. i.e. You can use the tweeter in a higher-powered speaker if you use it as a "super tweeter" (say above 5 KHz) because all of the energy below 5KHz is directed toward other drivers.

And the tweeter may have a higher power rating with a limited bandwidth because speakers/drivers are rated as part of a speaker system. That's sort-of a trick with the specs... A "100W" tweeter can only the high-frequencies of a 100W amplifier playing regular program material, peaking at 100W.

 

[NTK]

Suppose that a driver has a 100 hour noise rating of 60 watts when tested with a weighted pink noise signal and 2nd order Butterworth filters from 300Hz to 3kHz. ...

In your example, you seem to be referring to the IEC 60268-5 power handling test. The "power" the method refers to is the total power of the full bandwidth IEC 286-1 pink noise, before any of the frequency limiting filtering.

Therefore, in your case where the signal is bandpass filtered to 300 - 3k Hz, the power the driver sees is NOT 60 W, it is only the 300 - 3k Hz portion of the 20 - 20k Hz (IEC 286-1) pink noise that totals to 60 W. It will thus be reasonable to see that if you have a narrower passband, the percentage of the total power of the test signal the driver will see will be less, and the "reported power handling capacity" can be raised, leading to a higher max SPL spec.

The 2 major factors limiting max SPL are mechanical excursion limits and voice coil temperature. Having a narrower frequency band will definitely help with voice coil temperature.

Reference:

https://www.scan-speak.dk/datasheet/tech/Scan-Speak_Technote02_Powertest.pdf

 

[Cousin Mose]

"Oh, man, my thinking about this case had become very uptight." - The Dude

Alright, I think I understand what Mr. Risbo meant. If I'm not mistaken, the SPLmax of the individual driver doesn't really change when you reduce the bandwidth, the SPLmax of the system increases when you use more drivers to cover the spectrum.

For example, if a driver that is rated at 81.5 dB sensitivity and 60W power handling can produce an SPL of 99.3 dB when playing its entire frequency range, then two identical drivers, each playing one half of the range, combine for 99.3 + 3 = 102.3 dB total.

So, when you are designing a system with a target SPLmax, each driver only needs to be able to reach:

SPLmax - 3 dB (2-way speaker)
SPLmax - 4.8 dB (3-way speaker)
SPLmax - 6 dB (4-way speaker)
etc...

assuming that the crossovers are evenly spaced. (I realize the the crossovers will not be evenly spaced, I just didn't want to post all the math for combining drivers at arbitrary crossover points here.) Am I on the right track, now?

 

[dasdoing]

how loud a driver can play is not an absolute value. it actualy means: how loud before it distorts too much.
So if the driver distorts heavily at let's say 2000Hz. I mean, the peak of distorsion is at that frequency. If you take this frequency out, you indeed can play it louder before it distorts too much.