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19th February 2015, 12:23 PM  #1 
diyAudio Member
Join Date: Jun 2012
Location: Sweden

SPL of ESL
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! Last edited by phazer99; 19th February 2015 at 01:58 PM. 
19th February 2015, 05:14 PM  #2 
diyAudio Member
Join Date: Nov 2004
Location: close to Basel

Hi,
the value for the air breakdown is too high. I remember 2kV/mm as theoretical maximum. Practical values are 15001700V/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 7080Hz). 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.
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19th February 2015, 08:42 PM  #3  
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Join Date: Jun 2012
Location: Sweden

Thanks Calvin.
Quote:
Quote:
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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. 

23rd February 2015, 10:12 PM  #4  
diyAudio Member

Quote:
High mechanical tension is not the best choice while considering factors like longterm stability and subjective sound qualities in midbass 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 midbass region. 300 Hz seems to fall in this range. Again, no proofs but a subjective opinion. Regards, Lukas. Last edited by Bazukaz; 23rd February 2015 at 10:15 PM. 

24th February 2015, 07:10 AM  #5 
diyAudio Member
Join Date: Nov 2004
Location: close to Basel

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 longterm stability is no more of an issue as with a lowtensioned 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 hightension 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
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24th February 2015, 08:07 AM  #6  
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Join Date: Jun 2012
Location: Sweden

Quote:
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). Last edited by phazer99; 24th February 2015 at 08:09 AM. 

24th February 2015, 02:50 PM  #7 
diyAudio Member
Join Date: Nov 2004
Location: close to Basel

Hi,
no, I just meant to express my preference for hybrid ESLs over fullrangers. jauu Calvin
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24th February 2015, 05:00 PM  #8  
diyAudio Member

Quote:
I have seen you builing dipole towers of bass drivers to acompany curved panels. Just curious, what max. output is achievable with those towers at low frequency(for example, 40 or 50Hz) and realistic listening distance(like 23m) ..? Regards, Lukas. 

24th February 2015, 05:09 PM  #9 
diyAudio Member
Join Date: Jul 2010
Location: Lower Hutt, NZ

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 fullrange ESL (higher cutoff 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 highfrequency hybrid ESLs can reach above 105110 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 linesource 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 linesource ESLs means they typically sound about 5 dB louder at listening distance (3m) than pointsource ESLs. hope this helps Rod
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24th February 2015, 06:45 PM  #10 
diyAudio Member

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. 
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