An ESL consists of a thin conductive membrane and one or two drilled electrodes. They form a capacitor where high voltage drags the membrane towards the electrode which also shifts air and thus produces sound. The sound signal is amplified in voltage and a high bias voltage is added to attract the membrane more or less with respect to the music signal. Some kilo volts are sufficient for the bias.
The simpler design has a nonlinear characteristic which result in some harmonics. The electrostatic force is quadratic:
Let there be a plate capacitor with the area A the distance d and the permittivity ε (in air ε=ε0=8.854E-12F/m). The capacitance C results in:
By applying the voltage U a charge builds up:
The force can be calculated by the change in energy dE by membrane movement dd:
The force depends on the square of the voltage which is highly nonlinear. The parable can be linearized with a shift of the working point by the bias voltage:
With a bias voltage ten times of the music harmonics would be 1% only as it is squared. The force and thus loudness also depends on the bias voltage – the higher, the louder.
As with the displacement of the membrane the electrode distance d changes, another nonlinear behavior occurs but only during loud parts of music. It’s the same with the nonlinear force the membrane tension pushes itself back to its origin.
Example: A=5.5×11.5cm; d=3mm; Ubias=1000V; Umusic=100V;
Fmusic=3.11E-4N or as pressure p=F/A=0.0492Pa=4.92E-7bar and sound pressure level Lp=68dB which fits usual listening.
Let’s end all the theory by listening to the prototype
I decided to generate the bias voltage with the help of the audio signal, so no external power source is needed. A pumping effect appears with fluctuating bias voltage.
I used a car accident rescue foil from an expired pack. Two screws squeeze a cable onto a stack of residual foil at one side. The transformer is a small 230V to 15V 2x1W pcb type.
After some high voltage arcing through the membrane the fine conductive coating vaporized along a fine scratch or a fold. I replaced it with usual household aluminum foil which I bent a little to make it more flexible. The low frequency rendering is litle worse due to the higher stiffness causing resonances. The loudness and treble is not affected by the higher weight.
Spectra: The distortion is low as no harmonics (odd factors of basic frequency) are visible. When moving the microphone standing waves will x out certain frequencies causing these amplitude gaps even 1 meter away.
Symmetrical ESL with constant charge membrane
To get rid of nonlinearities, a symmetrical built ESL with a membrane between two perforated electrodes is essential. The membrane is charged to a static high voltage only. The surrounding electrodes form a capacitor with constant plate distance and carry the high voltage music signal. The membrane’s charge feels a force due to the electrostatic field which is linear to capacitor voltage. No current flows to the membrane which doesn’t require a good contact.
Calculation of membrane charge, electrostatic force and sound pressure level:
Bent plastic profile frame with perforated metal electrode attached and membrane contact molten into the frame.
Taped membrane around a cardboard, glued onto the frame with wire contacts from both sides.
With the higher electrode distance a higher voltage can be applied until corona discharge occurs. The symmetrical construction needs a center taped high voltage transformer. For this purpose four transformer connected in series will generate a voltage high enough to play music loud. Thanks to the symmetrical distortion free membrane movement. The applied DC Voltage doesn’t have to be much higher than the AC music signal on the electrodes to diminish nonlinear curves because there are none. The membranes polarity is unrestricted; maybe a negative polarity will cause corona winds to blow the near membrane away.
The 10M resistor maintains the quasi static charge of the membrane. The lamp is absolutely essential when using transformers. Once their core saturates the inductance drops thus a high current will overload the amplifier without this protection.