SPL of ESL

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I'm trying to wrap my head around how to calculate the max SPL of an ESL panel. I've used the equations from The Design of Electrostatic Loudspeakers, which btw is a great resource for a beginner. Basically I want to check that my calculations are correct.

Let's take a single Acoustat Spectra 11 panel as example (the dimensions are approximate):

Effective area, A: 1.07 x 0.16 m2 = 0.1712 m2
D/S spacing, d: 0.0022 m
Breakdown voltage of air, Vb: 4000 V/mm
Optimal polarizing voltage, Vp: d * Vb * 1000 / 2 = 0.0022 * 4000 * 1000 / 2 = 4400 V
Peak signal voltage, Vs: 2 * Vp = 8800 V
Capacitance between stators, C = e0 * A / (2 * d) = 8.85 * 10^-12 * 0.1712 / (2 * 0.0022) = 3.443 * 10^-10 F
Signal series resistance, R: 660000 Ohm
Peak signal current, I: Vs / R = 8800 / 660000 = 0.0133 A
RMS current of sine wave, Irms = I / sqrt(2) = 0.0133 / sqrt(2) = 0.0094 A
Min freq of current drive, Fc = 1 / (2 * pi * R * C) = 1 / (2 * pi * 660000 * 3.443 * 10^-10) = 700.3 Hz
Distance to speaker, r = 2 m
Walker's equation for sound pressure above Fc, Prms = Vp * Irms / (d * r * 2 * pi * c) = 4400 * 0.0094 / (0.0022 * 2 * 2 * pi * 343) = 4.36 Pa
SPL = 20 * log10(Prms / P0) = 20 * log10(4.36 / 0.00002) = 106.8 dB

So with optimal polarizing and signal voltage the max SPL for a sine wave of frequency greater than 700 Hz is about 107 dB at 2 m. Is this a reasonable value?

Isn't the Fc value of 700 Hz a bit high considering that below this frequency the SPL drops 6 dB / octave?

For a frequency f below Fc the max SPL can be calculated as (assuming SPL drops 6 dB / octave):

spl(f) = SPL - log2(Fc / f) * 6 = 106.8 - log2(700.3 / f) * 6 dB

This value seem to be almost constant for a specific frequency regardless of the values of Fc or R. Is this a correct observation?

If you use DEQ to flatten the frequency response, what is the advantage of choosing a larger value for R (signal series resistance) and thus lower Fc? For low R values you basically get a voltage driven ESL, right?

The capacitance between the stators is overestimated as they are perforated. Is there a better formula for calculating the capacitance if you know the perforation area percentage?

Also, why isn't the insulation or the coating of the stator included in the voltage breakdown calculation? Is it too thin to make a difference?

Is the maximum diaphragm excursion half of the D/S spacing (before breakdown/arcing)?

Is there some free software that does all these calculations for you (and plot some nice frequency response graphs)?

Thankful for any comments!
 
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Hi,

the value for the air breakdown is too high.
I remember 2kV/mm as theoretical maximum.
Practical values are 1500-1700V/mm.

A d/s value of 2.2mm is quite alot for a 16cm wide panel.
Seems like an attempt to achieve a low lower bandwidth limit with a thin panel.
This indicates low mechanical tension (low Fs) by mainly thermal tensioning of the diaphragm (probabely no more than 70-80Hz).
This on the other hand results in considerable mechanical offset as soon as bias is applied.
Depending on the bias level one may end up with just 1mm d/s or even less.
This costs on maximum SPL, which will depend mostly on the choice of the crossover point.

I regard current steering unneccessary if the panel is segmented electrically.
The loss through acoustic phase cancellation will be alot higher than current steering could possibly counter.
There's a considerable voltage loss across the series resistor, that requires a increase in transformation factor.
All together a sure recipe for a highly restricted dynamic range and lame sound.
I'd use highest mechanical tension combined with high bias, electrical segmenting, voltage steering and a elevated xover frequency (>300Hz).

jauu
Calvin

ps: the formula for the capacitance is quite exact up to high values of open area.
 
Thanks Calvin.
the value for the air breakdown is too high.
I remember 2kV/mm as theoretical maximum.
Practical values are 1500-1700V/mm.
According the document I linked to it varies between 4.5 and 3 kV/mm for an air gap greater than 1 mm (Wikipedia states it's about 3 kV/mm). I suppose that is for dry air, it will decrease with increased humidity.
Depending on the bias level one may end up with just 1mm d/s or even less.
The bias level is about 5 kV on the Spectra, which is not far from the 4.4 kV I calculated.
I regard current steering unneccessary if the panel is segmented electrically.
Actually the Spectra panel is electrically segmented into two vertical halves where the high/mid half doesn't have any signal series resistor (the transformer resistance is about 1 kOhm), and the mid half have two 330 kOhm resistors.

And I wouldn't say the Spectra sounds dull, but the sound is definitely improved with DEQ.

The Spectra panel was just used as an example as I know most about it, but I'm more interested if I got the math correct so I can use it to compare different ESLs.
 
Hi,

I'd use highest mechanical tension combined with high bias, electrical segmenting, voltage steering and a elevated xover frequency (>300Hz).

jauu
Calvin

area.

Hi,

High mechanical tension is not the best choice while considering factors like long-term stability and subjective sound qualities in mid-bass range. I suspect tension has influence in resonance modes of membrane, but no proofs. Subjective differences between highly stretched vs thermally annealed membrane were significant, with the latter being much more neutral sounding in mid-bass region. 300 Hz seems to fall in this range. Again, no proofs but a subjective opinion.

Regards,
Lukas.
 
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Hi,

thanks for hinting to that detail.
Sometimes certain details just slip through.
I should have noted that the term highest mechanical tension does not exclude thermal treatment, at least not for flat panels.
Thermal treatment always reduces mechanical tension, but the tension of a mechanically stretched and thermally treated membrane will be higher than using thermal tensioning alone.
With the proper membrane material long-term stability is no more of an issue as with a low-tensioned membrane.
The amount of tension directly affects the amplitude response, but mainly at the Fs and the neighbouring frequency range.
If one stays off of the Fs by more than an octave the differences in amplitude response are negligible.
Not negligible are the considerably higher efficiency and larger dynamic range possible with the high-tension panel.
The Q is: "Do I let the frequency range where a ESL is a inferior transducer dominate the outcome over the complete bandwidth, or do I restrict the bandwidth and preserve all advantages that this transducer can offer?
Its a decision the designer has to make in advance.

jauu
Calvin
 
The Q is: "Do I let the frequency range where a ESL is a inferior transducer dominate the outcome over the complete bandwidth, or do I restrict the bandwidth and preserve all advantages that this transducer can offer?
Its a decision the designer has to make in advance.
So, if I understand you correctly, you're basically saying that the Spectra panels are better suited for low frequencies and above that a panel with less D/S would be preferable. I'm actually considering a three way system consisting of (per side) one ER Audio minipanel, one or two Spectra panels and one Dayton Audio UM15-22 in a dipole U- or H-baffle. XO freqs would be about 100 and 350 Hz. What do you think about such a setup?

Btw, the minipanel is segmented so it would sort of be a four way system (or even five or six depending on how I connect the two Spectra panels).
 
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Hi

Baxandall has a nice simple formula for the maximum SPL at the listener position based on the dielectric breakdown of air...

Pmax=(F/A)max.A.f/2/c/r

where:
(F/A)max is the maximum force per unit area that can be applied by the electrostatic motor before breakdown occurs, this a constant for air = 50 N/m^2

A is area of panel in m^2 (bigger is better)

f is the low cutoff frequency in Hz- determined either by XO cutoff in hybrid or resonant frequency in full-range ESL (higher cut-off is best for high SPL).

c is speed of sound = 330 m/s

r is distance between panel and listener in m

to convert pressure to db => 20.log(Pmax/0.00002)

NOTE: This formula applies to ESLs behaving like a point source, like the QUAD ESL's and small square units. Only large ESLs or high-frequency hybrid ESLs can reach above 105-110 dB. Note too that if listener position fixed (3m say), the only design parameters that have a direct link to max SPL are area and cutoff frequency.


The formula is slightly different for line-source ESLs like the acoustats and tall segmented designs (ideally floor to ceiling, but > 1.5m usually enough)...

Pmax=(F/A)max. W/2.sqrt(f/c/r)

where W is width of panel in m. In this case only width and cutoff frequency directly affect max SPL. The weaker sqrt(.) dependence on f and r for line-source ESLs means they typically sound about 5 dB louder at listening distance (3m) than point-source ESLs.

hope this helps

Rod
 
Hi,

Yes I know about those formulae, and have seen PDF file floating somewhere. Also bolserst has made a spreadsheet to calculate max. SPL.
My question to Calvin about bass towers was to approximately evaluate how big an ESL should be for adequate low frequency power.Still, I am not convinced that an ESL can not produce good bass, as my relatively modest panels with 0.34 m2 area is enough for casual listening, and if fact it's bass output sounds really good; just there is almost nothing below about 45 Hz. Of course there are a lot of challenges building a full range, but IMO there are chances that the result is quite rewarding.


Regards,
Lukas.
 
Hi Lukas

My ESLs are 2.2 m high by 400 mm wide, resonant frequency just below 40 Hz as best I can tell. I have lost about 10 dB in sensitivity due to poor engineering decisions when I was building them (can only run at 2.6 kV instead of 5 kV, and I could gain an extra 4-5 dB through better choice of intersegment resistance and lower capacitance transformer).

But still they go louder than I need and the bass is fantastic, absolutely seamless. Also, having two speakers with line-source behaviour fills a room with almost uniform SPL, and averages out the nasty bass resonances I had with conventional speakers. Imaging is fantastic.

ESLs have a reputation for fantastic female vocals and acoustic instruments like guitars, but on my ESLs you can add great male vocals and low frequency instruments like double bass, cello, and drums.

So yes, I disagree with Calvin too.

Much of the poor reputation of ESLs ESLs in respect of bass performance comes from ESLs that behave as a line source for midrange-treble frequencies, but point source at low frequencies - because they are not tall enough. The QUAD ESL-57 is a classic example. They measure OK at 1m but bass falls off faster than everything else as you move away.

The ideal line source ESL has enough height to benefit from the reflections from the floor and ceiling to make it look like an infinite line source. At 1.5 m tall, they will loose a little bass below 100 Hz. Obviously depends on ceiling height, but in my house I think 2m would possibly be indistinguishable from perfect. I suggest curtain-rail/window height is good compromise. WAF deteriorates rapidly as speakers go beyond that :p .

regards
Rod
 
Hi,

Well, I see you are another full-range positive person around :). Actually I have started another full-range project, about a year ago. The area is about 0.8 m2 per speaker, so I guess it should do well in low frequency range. The wire gluing is finished, and the last spacer is being bonded via epoxy on the table.
I have decided to design it, according to my experience, somewhat different compared to floating "rules of thumb". For example, somewhat higher D/S spacing compared to optimum for max. SPL. This a) somewhat relaxes requirements for high precision required, and b) IMO helps from dust particles from being trapped between membrane and structure, which results in lesser buzzing noises. Also the ratio of D/S spacing to unsupported width is optimized for membrane stability and low tension. Not sure when I'm going to finish it.. But curious about the result :)
And I completely agree with you that line source ESLs are superior to more square-shaped, like quad 57.
Also I have found a subjective feeling that tilted line sources are inferior compared to higher, which extend above the ears.

Regards,
Lukas.
 
Hi,

the peak SPL of the bass tower is more than sufficient to easily cope with the Panel if xovered at ~50Hz to the subwoofer.
If used wo sub and equed down to 35Hz the bass limits first.
Anyway a dipole does not just suffer from acoustic phase cancellation, but from a certain freq-range on it actually gains from acoustic phase adding.
Depending on the dimensions around 60-80Hz will occur the even-point, from where on it offers more SPL than a direct radiator.

Me thinks we don't disagree about ESL bass. :)
The seamless integration and typical dipole's sonic character have their charme and can sound very good indeed.
But I just wanted more than very good ... I wanted exceptionally good. ;)
I was never happy with the FRs tendency to soft one-note bass and low dynamics.
I like my bass hard, clean, precise and loud, like founded on granite.
One won't get that level of precision with the typical high-Q (>>1) of ESLs, regardless under how many damping mats You hide the panel.
One probabely needs to have listened to such a system and have experienced how much more precise, dynamic and linear the dipole/hybrid can play with similar seamless integrated lower mids to bass.

jauu
Calvin
 
My question to Calvin about bass towers was to approximately evaluate how big an ESL should be for adequate low frequency power.Still, I am not convinced that an ESL can not produce good bass, as my relatively modest panels with 0.34 m2 area is enough for casual listening, and if fact it's bass output sounds really good; just there is almost nothing below about 45 Hz. Of course there are a lot of challenges building a full range, but IMO there are chances that the result is quite rewarding.
I've built dipole U-baffle woofer towers inspired by Calvin's design, but I only use 4 drivers per tower. The front baffle is aesthetically pleasing at 0,75 x 0,2 m and the woofers integrate very well with the Spectra panels (steep XO at 250 Hz). According to Linkwitz max SPL calculator I should get about 110 dB at 100 Hz and 1 m, and the towers are usable down to about 40 Hz which is enough for most music. I use closed subs to cover the lowest 20 Hz (they don't integrate that well though).

Using the formulas golfnut referenced to it would require an ESL panel area of almost 1 m^2 to get the same SPL, which would be quite huge in my small room and I'm trying to keep the size small :) But since I have some extra Spectra panels lying around it would be interesting to try them for the mid-bass region, say about 100-400 Hz. One panel would give about 96 dB at 100 Hz and 1 m, which is probably enough in my small room. Below that I would use 15 inch dipole woofers.

One way to get some extra dB in the low range would be to use U-baffles for the ESL panels. Is that a big no-no, or acceptable for the low range? I'm just trying to figure out my options here before building something. :)

P.S. The "full range" planars I've heard, some Quad's and Magnepan 1.7 didn't impress me much in terms of bass SPL.
 
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Hi,

According to Linkwitz max SPL calculator I should get about 110 dB at 100 Hz and 1 m, and the towers are usable down to about 40 Hz which is enough for most music. I use closed subs to cover the lowest 20 Hz (they don't integrate that well though).

Using the formulas golfnut referenced to it would require an ESL panel area of almost 1 m^2 to get the same SPL

According to bolserst max. SPL calculator indeed you would need about 1m2 of membrane area to get 110dB SPL at 1 meter. The spreadsheet, as I understand, assumes that all membrane moves as whole, however in practice the motion close to the edges is reduced. Still, a tall radiating surface, as already stated, shows slower rate of SPL drop with increasing distance compared to smaller drivers. So at higher listening distances, like 3 meters, it's likely an ESL with 1m2 area would favour purely in terms of maximum output. Of course, if it's within excursion limits.

One way to get some extra dB in the low range would be to use U-baffles for the ESL panels. Is that a big no-no, or acceptable for the low range? I'm just trying to figure out my options here before building something. :)

I have tried to use squared shaped baffles(like "U' but with sharp edges). It sounded really poor, IMO. Because the membrane of ESL is mostly transparent to the sound, the U baffle will create cavity resonances to a larger or lesser degree, depending on it's shape. I recall some people using pieces of PVC pipe to reduce this effect. Still, advantages compared to just a simple flat baffle in terms of floor area are not that much significant IMO. I have seen pictures of flat baffle made from thick plexiglass or something similar, completely transparent. A nice solution in optical terms.

Regards,
Lukas.
 
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I have tried to use squared shaped baffles(like "U' but with sharp edges). It sounded really poor, IMO. Because the membrane of ESL is mostly transparent to the sound, the U baffle will create cavity resonances to a larger or lesser degree, depending on it's shape. I recall some people using pieces of PVC pipe to reduce this effect. Still, advantages compared to just a simple flat baffle in terms of floor area are not that much significant IMO. I have seen pictures of flat baffle made from thick plexiglass or something similar, completely transparent. A nice solution in optical terms.
Yes, you have to keep baffle dimensions small enough to avoid resonances (and comb filter effects) in the operating range. That's why I'm considering using the Spectra panel with flat or U-baffle only between 100-400 Hz and a mini panel without baffle for the top range. I want to keep things visually pleasing, i.e. slim and not too high (that's why the U-baffle is preferable).
 
Yes, you have to keep baffle dimensions small enough to avoid resonances (and comb filter effects) in the operating range. That's why I'm considering using the Spectra panel with flat or U-baffle only between 100-400 Hz and a mini panel without baffle for the top range. I want to keep things visually pleasing, i.e. slim and not too high (that's why the U-baffle is preferable).

Hi,

About a year and a half ago I have tried to integrate a small panel with considerably larger one for bass. The bass unit was just an experiment, two 1m*30cm wide panels stacked on top of each other, and the treble unit was 1.2m*12 cm. Treble panel was placed either on side(as close as possible) or top of just a single larger unit. I experimented in frequency ranges like 400-800 Hz. The integration was not superb, least to say(active XO using DSP with different slopes, two step-ups). My conclusion was, that best radiation pattern for an ESL is a long line source, and of course symmetrical. Just an opinion though, maybe you would come up to a different result.

Regards,
Lukas.
 
About a year and a half ago I have tried to integrate a small panel with considerably larger one for bass. The bass unit was just an experiment, two 1m*30cm wide panels stacked on top of each other, and the treble unit was 1.2m*12 cm. Treble panel was placed either on side(as close as possible) or top of just a single larger unit. I experimented in frequency ranges like 400-800 Hz. The integration was not superb, least to say(active XO using DSP with different slopes, two step-ups).
Yes, maybe it's tricky to integrate the panels. My panels would be slightly smaller though, bass panel about 1.07x0.23 m and treble panel about 0.5x0.16 m, which should make things a bit easier. The other option I have is to build a two or three way system using the mini panel together with one 15" woofer and optionally one or two 7" woofers (all in dipole configuration). This should bring the sound source centers closer together for more seamless integration (but with less ESL goodness).
 
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