I added an example video clip in my above post #37...
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How much comparable power do we need for different frequencies?
Chris A
Active Member
There isn't a simple formula that relates efficiency of loudspeakers (in SPL) below 100 Hz because the difference in efficiencies between types of bass bins vary by more than a factor of a hundred. Horn bass bins, like corner horns, show over 10% efficiency (i.e., conversion of electrical power into acoustic power, in watts) down to their cutoff frequencies, while closed box subwoofers typically have less than 0.1% efficiency.
There are formulas for equal loudness curves available from ISO 226:2023 which relate loudness level in phons (perceived human loudness) to actual decibels (dB):
MODELING THE ISO 226:2023 EQUAL-LOUDNESS-LEVEL CONTOURS BY STANDARDIZED LOUDNESS METHODS
This allows you to get a handle on the human hearing part.
The combination of these two efficiencies would yield a formula for electrical power input to achieve acoustic watt output. But then room acoustics begin to take over the picture, with in-room reflections and cancellations introducing extremely large variations in SPL vs. location that the listener experiences simply by moving his/her head a few inches to a new location.
So what you are asking for has basically no utility for calculation purposes since the "noise" variables completely swamp any attempt at calculation of SPL output at the listener's ears as a function of electrical power input--for low frequencies. Even higher frequencies experience rather large deviations in SPL vs. position and even time (the so-called "dense mode region" of the room--generally above 200 Hz for typical home hi-fi sized listening rooms.
In the sparse mode region below 200 Hz (in my room, the calculated Schroeder frequency based on RT30 reverberation time is about 100 Hz), deviations in SPL due to interior acoustic reflections can be greater than 100:1 in terms of SPL (linear scale).
Note that I used to model problems in like manner for a living (operations analysis), and build operational simulations that made use of these type of constructed relationships mathematically, so I do feel comfortable laying out the basic relationships in mathematical formulae. But in this case, what you want and what you are going to be able to achieve are really too far apart.
It's better to calculate the rule-of-thumb efficiencies of the human hearing system separately from the aggregated electrical-acoustic efficiencies of the various bass loudspeaker types: closed box, bass reflex, and horn-loaded. Then calculate a room gain vs. frequency based on the positioning of the bass bins relative to the room wall and corner positions (i.e., half-space, quarter space, eighth space room acoustic loadings vs. frequency), since these details in the problem space yield a difference in SPL in-room of over 18 dB.
Chris
There are formulas for equal loudness curves available from ISO 226:2023 which relate loudness level in phons (perceived human loudness) to actual decibels (dB):
MODELING THE ISO 226:2023 EQUAL-LOUDNESS-LEVEL CONTOURS BY STANDARDIZED LOUDNESS METHODS
This allows you to get a handle on the human hearing part.
The combination of these two efficiencies would yield a formula for electrical power input to achieve acoustic watt output. But then room acoustics begin to take over the picture, with in-room reflections and cancellations introducing extremely large variations in SPL vs. location that the listener experiences simply by moving his/her head a few inches to a new location.
So what you are asking for has basically no utility for calculation purposes since the "noise" variables completely swamp any attempt at calculation of SPL output at the listener's ears as a function of electrical power input--for low frequencies. Even higher frequencies experience rather large deviations in SPL vs. position and even time (the so-called "dense mode region" of the room--generally above 200 Hz for typical home hi-fi sized listening rooms.
In the sparse mode region below 200 Hz (in my room, the calculated Schroeder frequency based on RT30 reverberation time is about 100 Hz), deviations in SPL due to interior acoustic reflections can be greater than 100:1 in terms of SPL (linear scale).
Note that I used to model problems in like manner for a living (operations analysis), and build operational simulations that made use of these type of constructed relationships mathematically, so I do feel comfortable laying out the basic relationships in mathematical formulae. But in this case, what you want and what you are going to be able to achieve are really too far apart.
It's better to calculate the rule-of-thumb efficiencies of the human hearing system separately from the aggregated electrical-acoustic efficiencies of the various bass loudspeaker types: closed box, bass reflex, and horn-loaded. Then calculate a room gain vs. frequency based on the positioning of the bass bins relative to the room wall and corner positions (i.e., half-space, quarter space, eighth space room acoustic loadings vs. frequency), since these details in the problem space yield a difference in SPL in-room of over 18 dB.
Chris
MusicScope 2.1.0 is also powerful/useful tool, I believe.
You can find all the tracks of "SONY Super Audio Check CD 48DG3" analyzed by the color spectrum (by Adobe Audition 3.0.1) and analysis data by MusicScope 2.1.0 in my post #651 on my project thread.
I also shared five video clips of "transient-sound" tracks within this "Audio Check CD 48DG3" played and analyzed by MusicScope 2.1.0 in my post here #760.
You can find all the tracks of "SONY Super Audio Check CD 48DG3" analyzed by the color spectrum (by Adobe Audition 3.0.1) and analysis data by MusicScope 2.1.0 in my post #651 on my project thread.
I also shared five video clips of "transient-sound" tracks within this "Audio Check CD 48DG3" played and analyzed by MusicScope 2.1.0 in my post here #760.
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Thank you for sharing this diagram.
Maybe, my post here#63 on remote thread would be showing the similar aspects of "whole" the music tracks;
- A nice smooth-jazz album for bass (low Fq) and higher Fq tonality check and tuning
Edit: let me copy-paste YouTube links and two diagrams from that post.
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Peaks show up in spectra as multiple frequencies, look at an impulse response in the freq domain, so your question dosnt relate to the OPs question.
It needs to handle both, the accumulated plot shows how hot the amp/voice coils will get.
You want the answer to the OPs question look at the 3 drivers in a 3 way speaker (that goes down to 20hz). Which driver is built for the most power? Another hint, why are built in subwoofer amps so large?
Its Flethcher Munson. Low freqs dont sound as loud as mids so they are mixed louder.
Your woofers are 25db more sensitive than the tweeter! And your filtering the woofers below 60hz, so no surprise and not realistic.
Thats low freq heavey. Very little music (of course theres EDM and the odd other thing) has anything below 30hz.
- Thread Starter
- #50
Thank you Chris. But you dont need to add "room sound" to calculation. Absolutely you cant do it, since everybody has diffirent rooms.
What we are searching is a general formula which can be apllied to all. So it should be anechoic space calculations.
- Thread Starter
- #51
It's really interesting that nobody couldn't invent a general formula which can be apllied to all!
- Thread Starter
- #52
How much comparable power do we need for different frequencies?
Does anyone know how much power we need for each frequency? I mean, I wanna know, how much relative power do I need for 20hz, 30hz, 50, 100 ... Lets say; at 20hz, for each additional 3 db, I have to double the power comparing 30hz for 30hz for each additional 3db, I have to double the power...
tmuikku said:
Yeah, SPL is ultimately about volume displacement, simplified it means cone area * excursion. To maintain same SPL octave lower in frequency, the volume displacement needs to quadruble. So, since any woofer or tweeter has static size, as it stays ~the same size no matter what frequency it tries to reproduce, it means higher frequencies need way less excursion than lows to maintain flat SPL. This means highs can get away with way less power to move smaller cone less, than at same SPL at some lower frequency bigger cone needs to move way more.
Two octaves down volume displacement is 16x already, so either the cone area must have gone up or the excursion, which would need more power in general. Sensitivity is often the voltage sensitivity, how much SPL is measured at 1m at standard 2.83V or 1V or what ever input. For highs this is very low, compared to lows. For example, if you have 90db at 100Hz, and count 7 octaves up it's 12800Hz which needs 4^7 = 16384 times less volume displacement to reach 90db. Typical 15" driver has about 850cm^2 cone area, while a 1" tweeter dome is roughly 5cm^2, this is difference of 170. 16384 / 170 is roughly 100, so the woofer still needs to have about 100x excursion of the tweeter even though it's much larger, to product same SPL at 100Hz as the tweeter does at 12800Hz.
Consequently 12800Hz wavelength is roughly 2,7cm, so almost equals diameter of the tweeter. Wavelength of 100Hz is 3.4m, so if you wanted a woofer to be have as little excursion at 100Hz as tweeter has at 12800Hz, the woofer would need to be roughly 3.2m in diameter! 15" is way less, like 10x, hense much more excursion required.
Well, these numbers seem so outrageous I might have error, so please welcome anyone to double check But it's along the lines, lows are very tough to reproduce due to very very long wavelenght.
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Maybe someone make a formula with changable parameters such as frequency, cone diameter, xmax etc. ?
Unfortunately, my mathematical skills are not high enough.
Just for your reference, I edited and added two YouTube links and two diagrams in my above post #45.
For the details, please visit my post here#63 on remote thread;
- A nice smooth-jazz album for bass (low Fq) and higher Fq tonality check and tuning
For the details, please visit my post here#63 on remote thread;
- A nice smooth-jazz album for bass (low Fq) and higher Fq tonality check and tuning
My thoughts:
Although the "power" may be less for high for high frequencies, in a single amplifier configuration (no crossover before the amp) the high frequencies may represent the extreme swings of the signal, as the highs ride the lows, the amplifier must be able to swing full voltage and current and correctly modulate that high voltage and high current as the "low power" high frequencies ride the "high power" lows.
Example:
The high frequency swing pushes the limits of the voltage range, riding on the low frequency swing.
Full range and 10k high pass of the signal:
Essentially agree with you, and your point would be one of the important rationales (and pros) for DSP-based multichannel multi-SP-driver multi-amplifier audio setup where we do not need much powerful amplifier(s), but we do need small-power excellent S/N low distortion amplifiers, to directly dedicatedly drive tweeters and super-tweeters.
I have shared here on my project thread;
- Even Greg Timbers uses "reasonable and low-budget" Pioneer Elite A-20 for compression drivers (super tweeters) in his extraordinary expensive multichannel stereo system with JBL Everest DD67000 which he himself designed and developed: #435
And, therefore, I have been repeatedly emphasizing that it is critical/important selecting amplifiers in policy/stance of "right-person-in-right-place"; I wrote here as follows:
(Almost) all of the home-use Hi-Fi amplifiers, I mean integrated amps and power amps, are designed for full range operation, i.e. to cover ca. 20 Hz - 30 kHz. This means that we should be very much careful in evaluating and selecting each amplifier to directly and dedicatedly drive each of the SP drivers, in my case woofers (WO), Be-(midrange)-squawkers (Be-SQ), Be-tweeters (Be-TW) and horn super tweeters (ST). These BE-SQ, Be-TW and ST are highly efficient in response to amp's power input.
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Do you have a source for this statement? Seems excessive.
Below is taken from this Linkwitz Lab page.
In the highlighted formula to estimate SPL from a speaker driver:
x is peak-to-peak displacement amplitude in m, SPL is proportional to 20 log10(x)
f is frequency in Hz, SPL is proportional to 40 log10(f)
d is piston diameter in m, SPL is proportional to 40 log10(d)
r is listening distance in m, SPL is proportional to -20 log10(r)
SPL change is proportional to 40 log10(f). If we have frequencies of f and f/2, we get 12 dB reduction:
At f, SPL = ... + 40 log10(f) + ...
At f/2, SPL = ... + 40 log10(f/2) + ... = ... + 40 log10(f) - 40 log10(2) + ... = ... + 40 log10(f) - 12 + ...
Since SPL is only proportional to 20 log10(x), to compensate for the 12 dB reduction, x has to increase to 4x.
- Thread Starter
- #59
So using an active speaker with a dedicated separate amps for tweeters and woofers works much better then a passive speaker + power amp(feeding tweeters and woofers simultaneously) combo. ? (Amps have same power)
Linkwitz showed an example of how he used the nomographs (you can do the same with the formulas he provided too) to size the drivers of a speaker system. I'd suggest that you read through it first.
He didn't show how to calculate (from fundamental physics) how much voltage (power is a mostly irrelevant metric) to drive the drivers to produce his SPL targets. However, you can easily do that by reading the sensitivity spec or frequency response graph off the speaker driver datasheets. Remember Linkwitz's calculations are for radiating to full space. Speaker drivers are measured when mounted to an infinite baffle, so they are radiating to half space and for LF, that means 6 dB higher SPL than Linkwitz's numbers.
The electrical power efficiency (i.e. the SPL-to-power relationship at a given frequency) can be obtained by calculating the current required using the impedance plot, and multiplying it with the signal voltage. That will give you the apparent power (in VA). If you also have the phase plot, you can multiply the apparent power and the cosine of the phase angle to get the real power. But please note that this does not take into the account of any cross-over filters (and as the Linkwitz speaker designs are active multi-ways, the speaker drivers are directly connected to the amplifiers).
If you are interested in the electrical power required from basic physics, I can start a new thread to show the mathematics on how this can be done from the Thiele-Small parameters.
