Efficiency of speakers (and why towers and big cones play louder per watt)

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Spurred by another thread about tower floorstanding speakers versus bookshelf models https://www.audiosciencereview.com/...anders-really-bring-to-the-table.27360/page-3 I was spurred to finally dig out my copy of Richard Small's thesis and look up some things. (Once when I met him in Indiana and mentioned I had a copy he was astounded, he said only like 90 copies were printed. But my first employer had a copy in the corporate library and I photocopied the whole thing, over 2" thick. Still a great reference.)

The efficiency of a classical electrodynamic cone driver is set by a formula complicated to type here (thesis 1.28):
eta0 = (rho0/2*π*c) * (Bl)^2 *(Sd^2) / Re * (Mas^2)
eta0=reference efficiency. This is above resonance, but below cone breakup.
the second term is constants, which I'm forgetting and can't find in the previous umpteen thesis pages. rho0 = air density??? And c = air velocity? I don't think it's Einstein's speed of light ha ha.
Bl=magnetic strength, the product of the magnetic field + how much wire cuts the magnetic field.
Re=DC resistance of the voice coil. THIS is the biggest reason speakers tends towards 4 ohms-reducing the "ohms" directly increases efficiency. Because less power is burned up in resisitance, OK that's way oversimplified. Re is affected by the size of the wire and the length of it, and the wire in the magnetic field is affected by this.
Sd is the cone area, notice it is squared, this is a big factor, and why the most efficient direct radiator speakers have big woofers. Also why you don't see super-sensitive mids or especially tweeters. (Note: in Dick Small's thesis this is printed in the denominator however it is a typo and supposed to be in the numerator.)
Mas is the moving mass: cone + coil + part of the surround and spider + air loading.
- Translate this into Thiele-Small parameters and you get:
eta = (4*(π^2)) * fs^3 * Vas / (c^3 * Qes)
so notice here we see a foreshadowing that a "bigger volume" is more efficient (with a lot of other interrelationships).

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OP
H

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And now on to the efficiency of a speaker in a box (thesis 3.3):
eta0 = kn * (f3^3) * Vb
eta0=reference efficiency. Again above resonance, but below cone breakup.
kn is a big long constant which is too complex to type here. It can be somewhat higher for ported and some other box types.
f3 is the -3 dB point of the system, which Small agreed with me is not really so significant for speakers in rooms. It was just mathematically convenient since it is 50% and since it is used a lot in electrical filter theory which Benson and Thiele and Small were using to characterize/model direct radiators. f6 or f10 are really more significant.
Vb is the volume of the box.
--> Again we see the bigger box is more efficient.
But but BUT there are some big caveats:

- This presumes you could physically build a driver to make a certain f3 in a certain box, which is not always possible.
- It does NOT mean that the same driver in a bigger box (or in a ported box) becomes more efficient. That passband efficiency is set by the cone area, magnetic strength, and moving mass. The midrange sensitivity will be the same in whatever size box and ported or sealed.
- Does this mean a tower speaker using the same midrange and tweeter as a bookshelf will not be more efficient? Maybe. They could be padded down in the bookshelf speaker's crossover.
- And before anyone posts that Model XYZ-T tower is rated by the manufacturer as more sensitive than the comparable Model XYZ-B bookshelf, please understand all those manufacturer specs are sh!t, see next post.

OP
H

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Here are "Head_Unit’s Rules Of Protection" (speaking as a loudspeaker engineer who later worked in amplifier product planning and automotive testing):
1) If when things start to sound distorted or odd you TURN IT DOWN, you are unlikely to ever break anything.
2) If you constantly "turn it up to 11" you will break something.
3) The amp and speaker power ratings do not matter. Don’t bother “matching” the amp and speaker power. That is a seemingly sensible yet actually meaningless exercise, because:
- Speaker specifications are 92% useless, due to different manufacturers measuring the same specs in different ways on different setups. And then (shudder of horror!) the Marketing Department gets hold of them, yowZa!
- Specs for amps are not thorough since they are measured into resistors for pragmatic reasons and speakers are not resistors at all.
4) Amps' 4 ohm or even 2 ohm rating is the most meaningful even if your speakers are 8 ohms because again the speakers are not resistors Should be 20-20k Hz, distortion under 1% or it's baloney.
5) For amps "more" power means (IF specs are comparable) at least three times as much due to the logarithmic nature of hearing.
6) You are less likely to damage speakers with a big amp, since let’s face it everyone cranks it up sometime, and a small cheap amp is then more likely to clip and possibly put out DC and ultrasonics (This assumes the speakers are not tiny little pieces of poop).
7) Be merciful and get yourself one of these https://www.acinfinity.com/receiver-amp-cooling-fans/ because AVRs pack a LOT of stuff in one chassis, and every channel generates a chunk of waste heat even at low volume.

Jim Taylor

Major Contributor
Does this mean a tower speaker using the same midrange and tweeter as a bookshelf will not be more efficient?

I was a curious young whippersnapper in the late 60s/early 70s. I experimented a lot, but had no real knowledge of what I was doing. It was sort of a "poke it with a stick and see what happens" type of thing.
One thing that I found out (at that time and with drivers from that era) is that a bigger box simply produced lower bass. In this crude experiment, I did not observe any other changes, including max drive level, which surprised me. Jim

Blumlein 88

Grand Contributor
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Worth reading your post just for the AC Infinity fans. I've done similar putting very quiet computer case fans on AVR's. The ones you link to look nicely made, provide additional functionality to an always on fan without being outrageous in cost.

Blumlein 88

Grand Contributor
Forum Donor
I was a curious young whippersnapper in the late 60s/early 70s. I experimented a lot, but had no real knowledge of what I was doing. It was sort of a "poke it with a stick and see what happens" type of thing.
One thing that I found out (at that time and with drivers from that era) is that a bigger box simply produced lower bass. In this crude experiment, I did not observe any other changes, including max drive level, which surprised me. Jim
Within reason you can trade-off efficiency with bass extension.

rho0 = air density???
Yes.
And c = air velocity?
Speed of sound in air (depends on air temperature), I believe. Usually denoted by c or a.

OP
H

Forum Donor
I was a curious young whippersnapper in the late 60s/early 70s. I experimented a lot, but had no real knowledge of what I was doing. It was sort of a "poke it with a stick and see what happens" type of thing.
Hey, that is exactly how science works, seriously. Someone starts poking around, and eventually things get figured out. Such as Henning Brandt deciding to boil all the urine he had collected from his neighbors, and discovering the element phosphorus (true story!). I worked for another famous speaker guy, Jim Novak, and some of his work was basically "poke it with a stick" as was later magnetics work I did.

mhardy6647

Master Contributor
... some of his work was basically "poke it with a stick" as was later magnetics work I did.
Magnetic stick?

Spkrdctr

Major Contributor
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After reading this I have only one thing to say. Are you sure? LOL.

Great posts. The most important point you brought out is that if you turn it up to 11, you have a good chance of destroying something. That is a HUGE issue. Also, the upper spl levels of many speakers is not all that important. If it can play 90db at 1 meter cleanly, then for most people they will be good speakers. Too many people do not realize that buying speakers for the few minutes you are going to turn it up to 11 to show a friend your speakers is a big waste of money. 90 to 95db is fine for showing off how good something sounds. Any louder and your brain/ear interface starts getting heavily involved and says Hey! turn it down or I will!

"Hey! turn it down or I will!" is a common reaction to loud distortion and compression whereas the same audio at the same SPL but undistorted and uncompressed will not elicit the same reaction. Dirty loud audio is much more painful and objectionable than clean loud audio.

KSTR

Major Contributor
Re=DC resistance of the voice coil. THIS is the biggest reason speakers tends towards 4 ohms-reducing the "ohms" directly increases efficiency. Because less power is burned up in resistance, OK that's way oversimplified. Re is affected by the size of the wire and the length of it, and the wire in the magnetic field is affected by this.
I may add that we should note (BL)²/Re is a constant with regard to how the VC is wound, it doesn't matter for power efficiency. It does matter for voltage sensitivity, though.
Consider a dual 4ohms voice coil woofer, VCs in parallel. Then rewire to series and Re quadruples but L only doubles, keeping (BL)²/Re constant. So, basically it's the mass of the voice coil copper (or whatever) in the gap which governs that constant, besides B of course. The only way to increase efficiency is to put more total copper volume on the VC or theoretically, use a much higher conductivity material ;-)

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OP
H

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"Hey! turn it down or I will!" is a common reaction to loud distortion and compression whereas the same audio at the same SPL but undistorted and uncompressed will not elicit the same reaction.
Actually I find a lot of people in a lot of situations (not so much concerts but clubs and even weddings) unfortunately don't react like that and just take being blasted as par for the course. I was in one club for a birthday party, the DJ had it louder than almost all the concerts I've been to, we could not even scream at each other. It was painfully loud even after I went to the car and got earplugs.

JSmith

Major Contributor
we could not even scream at each other.
As it should be in a club or at a good party.

I remember a party I went to in the late 80's... was so loud some people got sick from the bass hitting their stomach over and over. The setup was just insane, it was heard suburbs away... one couldn't even talk outside the venue properly, whole place was shaking/vibrating. There was a literal wall of subs with huge stacks of speakers on top. No wonder I can't hear over 16KHz these days.

JSmith

OP
H

Forum Donor
As it should be in a club or at a good party.
Eh, I think something is wrong when rock concerts are not as loud as a wedding! I miss the days of for instance Iron Maiden's massive Turbosound PA on the World Slavery Tour. Maybe I gotta go to less weddings and clubs, should go to more concerts to have conversations-like the boors next to us at a Lord Huron show at Hollywood Forever, or behind us at The Eagles.