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30th September 2008, 08:17 PM  #1 
diyAudio Member
Join Date: Aug 2008

Derived Physics Model of a Planer ESL
Hi
Would any one be interested in a peer reviewed model of a planer ESL? The model first determines the applied electrostatic pressure applied to a coated diaphragm. I then solve the forced, damped wave equation with clamped edges. I was able to determine the mechanical response to a harmonic load, both in the transient and steady state. This is currently solved for. Later… Soon I will finish it by considering the pressure waves it can generate. I will determine the radiation or directivity functions (plots) everywhere on and off axis. I will also determine SPL vs. frequency. I will compare a planer source with a cylindrical one as well. Happy Building Bryan 
5th October 2008, 01:17 AM  #2 
diyAudio Member
Join Date: Apr 2004
Location: Maine, USA

Bryan,
Perhaps you can clarify what it is that you're offering. I'd be interested in having a way to model ESLs, but I'm not sure whether you've written a standalone computer program, some Matlab code, worked out some analytic solutions... Can you supply a bit more detail? Few 
5th October 2008, 02:55 AM  #3 
diyAudio Member
Join Date: Sep 2006

There are many variables that can't be measured or estimated. Very hard thing to simulate.
What are you hoping to learn? Ideal panel shape? 
5th October 2008, 03:10 AM  #4 
diyAudio Member
Join Date: Aug 2008

Hi Few,
This is only analytic at the moment, I have been using mathematica a little to show the motion of the membrane. The model is in three parts: Part 1 Electrostatic Model (done) Objective: Determine the force density (pressure) that is applied to the film. Method: I model the stators as flat plates; the film is halfway between the plates. Since most coat one side of the film, the film has what is called a free charge distribution on one side and an induced charge distribution on the other. There are three regions of interest to the left and right of the film and the region inside the film. The electric potential satisfies the Laplace equation of electrostatics in all three regions. Once the boundary conditions are known we can solve the potential in all three regions. For example the potential is constant on both plates, is V on one and grounded (0) on the other. The other boundary conditions come from requiring that the potential be continuous across the regions, etc.. With the potential know the electric field is easy to find, I then compute the pressure exerted on the thin film. This derived pressure has the form P(t)=Po +P1sin(wt), it is harmonic, the constants Po and P1 depend on the bias potential the signal potential, distance between plates, thickness of plates, permittivity of the air and the Mylar. Part 2 Mechanical Model (done) Objective: Determine the displacement amplitude of the membrane Method: With the forcing known from part 1 we can solve the damped inhomogeneous (forced) wave equation. This will give us the displacement amplitude of the membrane, z(x,y,t). I find both the steady state solution and the transient solution. The full solution is just transient plus steady. The transient solution depends on the initial conditions; the steady state solution is the long time behavior of the membrane. For example say we wanted to understand how fast the speaker could respond to changing frequencies, this information is in the transient part. In speaker magazines you may have seen that they test speakers by passing a voltage signal that is very sharply peaked at time t=0, called a Dirac impulse function, then they determine how long it takes the membrane to settle, again this information is in the transient part. I use the steady state solution and find the velocity of the film, I perform a spatial average, so I make assumption later in the acoustic part that the ESL is essential a rectangular piston. Part 3 Acoustic Model, Radiation from a Piston Source I am working this out now, should not be much longer, we need to estimate the directivity of such a source. In other words the radiation pattern of the source, how narrow is the major lobe, are there many side lobes, how does the pattern change with changing frequencies, etc? Also, determine SPL vs freq. We need to determine the pressure generated on and off axis. Since I assume that most panels have their longest dimension about L=1 meter. If i recall correctly, if kL<<1 we are in near zone, and if kL>>1 in the far zone. k=2pi/wavelength. The lowest freq. my ESL will produce is about 400Hz or (c=freq*wavelength) wavelength=0.86 m. this gives the magnitude of the wave vector, k = 7.3, as we increase the freq, decrease the wave length we see k will only increase. I treat each elemental area of the membrane as a simple source, then I integrate (sum) over the source to find the pressure at any point at any time. These integrals are well known, the Fraunhofer and Fresnel integrals , depending on which zone we are. The last part is to invert the problem we know what we want in terms of output acoustically, how does this optimal output depend on the input we control? I worked hard on this model in order to help build a better ESL and learn much about them. I hope this model will help others build better ones as well. Happy Building, Bryan 
5th October 2008, 03:17 AM  #5 
diyAudio Member
Join Date: Aug 2008

ak_47
The only parameter that I don’t know very well is the mechanical damping coefficient the membrane feels, but this can be measured. Give ESL Dirac impulse input, measure amplitude; fit the amplitude to a decaying exponential. You need special equipment to measure the amplitude. Yes, there are many parameters; most can be known very well, at least orders of magnitude estimates. A better question is how does the performance suffer when the parameters drift over time? We can understand this by using the current model and make small perturbations on all variables. bryan 
5th October 2008, 03:28 AM  #6 
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Join Date: Jun 2004
Location: Edmonton area, Alberta

I would be very interested in reading such a paper.

5th October 2008, 06:45 AM  #7 
diyAudio Member
Join Date: Nov 2004
Location: close to Basel

Hi,
it would be most interesting to have a working model of an ESL, but so far part3 seems to be the only one which could offer some practical usefulness. I think that it can´t be as easily determined as you think bryan. In part 1 the assumption is that the membrane stays centred between the stator plates which it doesn´t in praxi. While by the principle of superposition this doesn´t affect the alternating signal forces on the membrane it does decrease the SPL limits seriously by introducing a mechanical Offset. It will be very difficult to determine the amount of offset, which depends on the mechanical tension and the dimensions and material parameters of the membrane. So besides determining the parameters it will be needed to take mechanical parameters into account for the electrical model and electrical parameters into the mechanical model. This work has been done by Peter Baxandall. So what news do you expect to discover? You imply some assumptions like pistonic movement but do they hold up? Will the results based on ´wrong assumptions´ still be close enough to the real world behaviour? It will be probabely close enough to reality with regard to distribution character. So point No.3 seems the most promising with regard to usefulness to me. jauu Calvin 
5th October 2008, 09:27 PM  #8 
diyAudio Member
Join Date: Aug 2008

Calvin,
Part 3 is what we want, but to get there you needed to look at part 1 and part 2. I agree looking at the integrals that give the pressure offaxis in the near zone is anything but trivial. I have to see how others handle this. "In part 1 the assumption is that the membrane stays centered between the stator plates which it doesn´t in praxis. While by the principle of superposition this doesn´t affect the alternating signal forces on the membrane it does decrease the SPL limits seriously by introducing a mechanical Offset." When I determine the pressure the electrostatic force applies to the membrane I evaluate at the mean position of the membrane. This is not a terrible assumption. The amplitude of vibrations is small enough to permit this. If you wanted to avoid this assumption you will end up with an almost impossible problem to solve. Essential it amounts to solving the wave equation and Laplace’s equation together, they are coupled. The forcing to the wave equation becomes some kind of KleinGordon type. How does superposition decrease SPL? What do I hope to learn? 1. Basic output levels, how much force to apply to hear anything a. we already know this from exp. 2. The radiation pattern a. how does it change with changing freq b. what conditions have to be met so that it has a desirable pattern. I’m looking for a pattern that has a wide major lobe, but not to wide, I don’t want the sound to have a directionless quality to it. But not to sharp as to avoid the head in the vice problem 3. Solve the Inverse problem a. We will be able to input the desired patterns, levels, etc and derive optimal controlled parameters. I then hope that these values are close to what we normally use. I worked on this model because whenever I would talk with an "audiophile" they would always use qualitative adjectives to described the sound. What I wanted to learn was for everything that they were saying there has to be some quantitative measure. It turns out that much of what they described is the pattern the source is producing. 
6th October 2008, 10:03 AM  #9 
diyAudio Member
Join Date: Nov 2004
Location: close to Basel

Hi,
superposition does not decrease ´normal´ SPL but the maximum SPL because of decreased membranestator distance, hence lower excursion limit. jauu Calvin 
6th October 2008, 04:57 PM  #10 
diyAudio Member
Join Date: Aug 2008

Calvin,
Superposition is a method to solve the wave eqaution. It in no way restricts the stator membrane distance. Say we have a string that is of length L and clamped at the edges, you would find that the eigenfuctions are sin(n*pi*x/L). Where n=1,2,3..., we use a sum or a superpostion of these building block solutions to satisfy the inhomogenous part. Bryan 
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