ESL Efficiency

Hello all,
I often read that ESL efficiency is inversely proportional to the SQUARE of the distance between stators. How come? the ESL is effectively a plate capacitor (?), where the field strength equals V/d (Voltage difference between stators divided by the distance between them), and not V/d^2.
Looking forward to your (well reasoned) replies.
 
The force on the diaphragm is proportional to the product of the field strength and the charge on the diaphragm. If signal voltage and bias voltage are held constant and the spacing is doubled, the field strength is cut in half and the charge on the diaphragm is cut in half(because diaphragm-to-stator capacitance is halved with bias voltage held constant; Q=CV). So, the force on the diaphragm is reduced by a factor of 4.

The formula for calculating far-field pressure from ESL dimensions and voltages is:
P = ( Vsig * Vpol * e0 * A * f ) / ( d^2 * c * r )

Where:
P = sound pressure (N/m^2)
Vsig = signal voltage applied to stators
Vpol = bias voltage applied to diaphragm
e0= electric constant
A = area of ESL (m^2)
f = frequency (hz)
d = stator to diaphragm spacing (m)
c = speed of sound (m/s)
r = listening distance (m)

Complete analysis and derivation of above equation(and many more) can be found in the the following reference:
P. J. Baxandall, “Electrostatic Loudspeakers,” Chapter 3 in
Loudspeaker and Headphone Handbook, J. Borwick, Ed.
(Reed Educational and Professional Pub., Woburn, MA,
1998), pp. 106–195.
 
Hi

I'm sure you do not mean efficiency, but sensitivity, which is what Bolserst describes.

Efficiency is strictly the ratio: acoustic power out divided by electrical power in.

For conventional speakers this is normally about 1 %, so they are very inefficient.
Horns can be as good as 10% or more. Simple ESLs have an efficiency that is very close to 100%.

The most useful figure is the mid-band sensitivity, e.g. 85 dB for 2.83 V in (1W into 8 ohms). For wide range ESLs, it is very difficult to get the sensitivity above 90 dB. 80 dB to 85 dB is more common. With hybrid ESLs it much easier to get the sensitivity up to match high sensitivity conventional speakers.
 
...Simple ESLs have an efficiency that is very close to 100%.
..... For wide range ESLs, it is very difficult to get the sensitivity above 90 dB
That's interesting thanks. High efficiency with relatively low sensitivity appears contradictory? Perhaps conventional sensitivity measurements don't transfer over to ESL's? Or something else?


Calvin did some in room measurements of his ESL's and got dB levels at the listening position that seemed to equate to Sn~103dB@1W/1m equivalence for an 8ohm conventional driver; assuming data extracted from his post "50W@8Ohms. The last power value translating to ~110dB@4m and beginning of clipping (since the measurement was fullrange the maximum value of SPL above 200Hz is even higher than that!)" can be plugged into Peak SPL Calculator
 
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Thanks for your reply bolserst!
Now let's make it interesting:
Experiment a: one electron (charge e) is placed at the center of the diaphragm, which is disconnected from the HV supply, and an audio voltage (V) is applied to the stators, which are spaced d length units apart. The electric field intensity between the stators is E=V/d, and the magnitude of the electric force (F) exerted upon the electron (ie upon the diaphragm) is F=eE=eV/d. Thus, the electric force on the diaphragm is halved for each doubling of the distance between stators.
Experiment b: Like experiment a, except that now N (large number) electrons are evenly distributed across the whole area of the diaphragm. What changes in the previous analysis, except for a multiplication by N, that is, F=NeE=NeV/d ?
If nothing changes (and if the previous analysis is correct in the first place) wouldn't it be wiser to charge the diaphragm and then disconnect it from the HV supply, thus trapping a constant charge on it? This way we could outsmart physics, and have the force reduced by a factor of 2 for each doubling of d, instead of by a factor of 4 !

Thanks (in advance) for your reply!
 
Both thought experiments a & b with diaphragm charge and stator voltages remaining the same will have the force reduced by a factor of 2 if spacing is doubled. In the case of experiment b, since Q=CV, the bias voltage(ie DC voltage that the diaphragm-to-stator capacitance is charged to) will double. So, no physics have been outsmarted. In the formula provided back in post #2, d would be replaced with 2*d, and Vpol with 2*Vpol resulting in P being reduced by a factor of 2 rather than 4 if diaphragm charge is held constant rather than diaphragm voltage. In practice, to get the same charge on the diaphragm of ESLs with large spacing higher bias voltages are used. To keep charge constant, diaphragm coatings are of high resistance and a high value resistor is placed between the bias supply and the diaphragm. Values are chosen to keep the charging time constant an order of magnitude or more higher than the period of the lowest audio frequency the ESL will be playing.

Another thing covered in the Baxandall ESL chapter is the fact that there is a limit to how much force can be generated per unit area before the voltage gradient in the gap between diaphragm and stator exceeds break down threshold for air and it starts to conduct. Example of corona display on stators when limit is exceeded posted here: First time ESL builder

There is also an optimum value of bias voltage and ratio of bias voltage to stator voltage to achieve the maximum possible force. More information posted here:
Measured my ESL attempt
 
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Thanks again bolserst, why should anyone bother to read books if they have you (kidding), and now to the next challenge:
I read that the QUAD 63's sensitivity is hardly affected by the thickness of the PCBs which constitute its stators, or in other (my) words, that flipping the PCBs "outside in" so that the copper side faces the diaphragm would hardly increase the speaker sensitivity. Assuming that it's true, what does it imply on the physical parameters? That the electric field intensity between the stators hardly increases despite their closer proximity (smaller d)? If that's the case, shouldn't there be a negligible field within the PCB's dielectric material (as the AVERAGE field intensity must always be E(av)=V/d) ?

Looking forward...
 
When the copper sides of PCB stators face outwards, the air-gap and the PCB dielectric form an AC voltage divider for the AC voltage applied to the copper sides of the stators. Basically you have two capacitances in series, the air gap capacitance Ca, and the PCB dielectric capacitance Cd. The voltage across Ca is what applies electrostatic force to the diaphragm. The voltage across Cd is a loss. In general the thickness of Ca is larger than Cd, and the dielectric constant of Cd is larger than Ca. In the case of the ESL63, this results in Cd being roughly 20 times larger than Ca. Doing the voltage divider math results in > 95% of the applied AC voltage appearing across Ca and driving the diaphragm. This is equivalent to < 0.5dB loss in SPL.

If the dielectric is thicker and the air gap is smaller, then it starts to make sense to put the conductive side facing the diaphragm. BTW, that is exactly what Quad did with the tweeter panel on the ESL-57 where the stators were actually thicker than the 0.5mm airgap.
 
Very succinct bolserst, as I've come to expect by now.
What you say implies that a PCB material of higher dielectric constant (Er) incurs smaller losses as its capacitance is higher and thus drops less voltage in the "AC voltage divider". Agree?
Another case:
I coated two perforated aluminum stators with an acrylic insulation spray (destined for motor wire insulation, PCB protection etc.).
The result was a drastically reduced output, to the extent of being not usable anymore.
What could be the physical explanation to that? After all, the insulating layer couldn't have a lower dielectric constant that that of air (E0), could it?

Thanks again!
 
...PCB material of higher dielectric constant (Er) incurs smaller losses as its capacitance is higher and thus drops less voltage in the "AC voltage divider". Agree?
Correct :up:

..aluminum stators with an acrylic insulation spray ...drastically reduced output

The likely cause is that the acrylic insulation spray is too good of an insulator. Any idea what the volume resistivity is? For stator insulation you want something pretty leaky like PVC or Nylon 66. FR4 PCB is not quite as leaky as those, but pretty close. Just like the air gap and dielectric capacitances(Ca & Cd) form a voltage divider for the AC signal, the air gap and dielectric resistances(Ra & Rd) form a voltage divider for the DC bias voltage. With PVC, nearly all the bias voltage appears across the gap. With really good insulators like Teflon, nearly all the bias voltage winds up across the insulator.

See Attachment #2 here: Measured my ESL attempt
Another post on the subject: How to construct a cube louver (Acoustat)

Thanks for posting those excellent links and all your help.
Glad to hear the posts have been helpful.
 
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Hi all,

... as always interesting contributions to read here ;-) ...

One comment/question: Why is it at all needed to isolate the wire stators in an ESL? Is it due to safety reasons, or acoustic reasons (e.g. lowering the self-resonance of the wires), or something else? I would reckon that the insulation would somewhat influence the sound quality due to its Dk and non-linear properties ... and maybe also due to adding thickness to the wires' diameter (as per bolserst & golfnut's article) ...

But again, maybe it is mainly due to safety reasons?

Cheers,

Jesper
 
Thanks again bolserst!
I tend to reply to you straight away, and only then delve into the links you supply where I usually find answers to many of my questions. Please forgive my impatience, as here I go again...

1) Is there any disadvantage in using "too good of an insulator" and then compensate for it with a much higher bias voltage?

2) FR4 PCB is relatively leaky, which is good DC wise (low bias voltage drop), but does it have a relatively high dielectric constant (Er) compared to other PCB materials, which would be good AC wise (low signal voltage drop) ?

Thanks again for the who knows what time, I'm losing count here...
 
Hi,

Dk and non-linear properties
Don´t forget that there is no(!) current flow through a capacitor, but just more or less charge on it´s electrodes.
Also, with low k (or low er) materials those non-linearities are negligable.
High-k materials on the other hand would allow for very thick and safe insulation without too much loss in efficiency.
The old Beveridge patents are worthwhile reading regarding this issue.

Raising the voltages means raising risks in case of fails or flashovers.
The ageing of materials will accelerate with rising kVs.
The too good insulators are typically also those with low er values (like PTFE or PE ... hence their low efficiency)

jauu
Calvin
 
…Why is it at all needed to isolate the wire stators in an ESL? Is it due to safety reasons, or acoustic reasons (e.g. lowering the self-resonance of the wires), or something else? I would reckon that the insulation would somewhat influence the sound quality due to its Dk and non-linear properties
The insulation in wire stators does provide safety for people touching the wires while music is playing and that is probably reason enough for most people to use it. But, it also avoids the possibility of diaphragm damage (or fire) from arcing if overdriven. Additional, it provides a soft limiting action when approaching the air ionization level. Dayton Wright mentions this in their literature as a bad thing, but if the alternative is destruction of the panel? Of course DW also enclosed their uninsulated stator panels in bags of SF6 gas to allow higher voltages and minimize the risk of arcing.

As far as the dielectric influencing sound quality, I agree that ideally stators would be uninsulated. Years ago I built a matching pair of wire panels; one insulated the other uninsulated. The uninsulated wire version used wire with OD of the PVC insulated version so the acoustic transparency was the same. There was no obvious change in sound quality or measureable difference in distortion between them. Even with prolonged listening and A/B testing I was not able to pick up on any differences that would make me want to switch to uninsulated wires. Of course I’m not saying that somebody with more critical listening skills might not pick up on something. For me though, it was verification that the added safety of PVC insulation did not come at the cost of an easily identifiable sonic impact.

There have been a few threads on this topic, here are a couple I had bookmarked:
Role of ESL stator insulation?
Insulated vs. bare metal stators
Stacked Acrylic ESL Insulator
 

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1) Is there any disadvantage in using "too good of an insulator" and then compensate for it with a much higher bias voltage?
Yes you could, but since the conductivity of air is humidity dependent you would need to adjust your bias based on the weather. Also, if the insulation is really good and humidity is low charge will collect on the outer surface of the insulation as you play music until it is nearly at the same potential as the diaphragm with resulting drop in SPL. With leaky insulation and higher humidity, this charge normally leaks off. This behavior is similar to what was discussed in the single-ended ESL thread: Odd Behavior in Single Ended ESL Test Setup

Also, as Calvin already mentioned, higher voltages are harder on materials and the consequences of failures will be worse.

2) FR4 PCB is relatively leaky, which is good DC wise (low bias voltage drop), but does it have a relatively high dielectric constant (Er) compared to other PCB materials, which would be good AC wise (low signal voltage drop) ?
There are many flavors of FR4. The ones I have measured had dielectric constants in the 3.5 to 4.5 range which is similar to PVC and Nylon 66.

The Beveridge stators Calvin mentioned were created from a mixture of epoxy, carbon (for adjusting conductivity), and barium titanate (for adjusting dielectric constant). If I remember correctly, he was targeting volume resistivity of around 10^10 ohm-cm and dielectric constant of about 50. At least one of the patents identified the makeup by weight of the mixture. If interested, US3668335 would be a good starting point.