Chapter Fifteen

Electrostatic loudspeakers

Abstract

After a short introduction to the construction and general principles of an electrostatic loudspeaker, there follows an analysis of various methods for broadening the directivity pattern as well as the effects of using a delay line feeding discrete rings rather than a theoretical continuous delay. Design formulas are derived for designing a delay line to imitate an oscillating sphere including a method to neutralize stray capacitance. Next, a lumped-element model based on an analogous circuit is developed, which is valid when there is sufficient resistance (usually in the form of a dust screen) to suppress the membrane modes. From this, useful design formulas are derived.

Keywords

Alternating signal voltage; Capacitance; Cut-off frequency; Figure-of-eight directivity pattern; Fundamental resonance frequency; Sound pressure level; Electrostatic loudspeaker; Stator; Membrane; Directivity control; Oscillating sphere; Delay line; Stray capacitance; Walker's equation; Constant impedance; Stability; Static current; Motional current; Radiation impedance; Dust screen

Part XXXXII: Fundamentals of electrostatic loudspeakers

15.1. Introduction to electrostatic loudspeakers

Although the number of electrostatic loudspeaker manufacturers in the world can be counted on two hands, this type of loudspeaker has been regarded as a benchmark for sound quality for more than 60   years because of its low distortion, low coloration, and outstanding transient response. The main drawback is that it is difficult to produce high sound pressure levels, especially at lower frequencies, unless the diaphragm is very large. This can then lead to beaming at high frequencies, although we shall examine a method to counter this.
Instead of passing a current through a coil within a magnetic field to drive a rigid cone or dome radiator, an electrostatic loudspeaker simply consists of a perfectly flexible charged membrane held under the tension between two perforated electrodes or “stators.” The signal across the stators creates an electric field that pulls the charged membrane back and forth to produce sound waves, which exit through the perforations in the stators. It is the same force that makes your hair stand on end when you rub a balloon on your clothes and then hold it close to your head. The membrane is thinner than a human hair, which makes this the closest thing to moving air particles directly without any mechanical structure.
Compared to a dynamic loudspeaker, this simple transduction method turns the whole design paradigm on its head. Whereas the radiation load of a dynamic loudspeaker only accounts for a small portion of the total moving mass, including the voice coil and diaphragm, the mass of the gossamer-thin membrane of an electrostatic loudspeaker is tiny compared to that of the air which it is moving. Therefore, the stiffness of the air in an enclosure would raise the fundamental resonance frequency so high that all the bass would be lost. Indeed, the fundamental resonance frequency would occur when the enclosure depth is roughly one quarter of the wavelength. To prevent this, electrostatic loudspeakers mostly operate in free space. Hence, a large diaphragm is needed to prevent the antiphase rear waves from canceling those from the front at the lower frequencies.
Because the membrane is so light and there is no enclosure, an electrostatic loudspeaker is acoustically transparent over its working frequency range, which starts at the fundamental resonance, as determined by the tension and radiation mass, and ends where the inertia of the membrane starts to have an effect.
Another major difference from dynamic loudspeakers is that, to avoid problems with vibration modes or cone “breakup,” the diaphragm of a dynamic loudspeaker must be made as rigid as possible, which means that there is a lower limit to how thin it can be made. To reduce the moving mass and thus improve efficiency and transient response, the thickness can only be reduced further by employing expensive exotic materials. Even then, the inevitable “breakup” cannot be eradicated altogether—only moved up to frequencies above those to be reproduced by the driver. Higher frequencies must be reproduced by another smaller driver with the added complexity of a crossover. Another approach is to use highly damped materials, but these reduce the high-frequency radiation efficiency due to a shrinking effective area and, in the case of domes, narrow the directivity pattern because the dome effectively becomes a ring source.
By contrast, the membrane of an electrostatic loudspeaker may be perfectly flexible with the restoring force provided purely through tensioning. Although the driving force is uniform over the whole membrane, this does not make it immune to vibration modes, unless the membrane were somehow freely suspended (such as the resilient disk in Chapter 13). As we saw in the Chapter 14, the clamped perimeter does result in modes, but these can be effectively damped out by the acoustic flow resistance of a fine mesh cover material, which also serves to exclude dust and moisture. In any case, at higher frequencies, where the radiation load becomes more resistive than massive, the modes are effectively damped by the radiation resistance.
Various methods have been used to prevent high-frequency beaming. One of the most common is to use separate “wide” and “narrow” electrostatic drive units for the lower and higher frequency ranges, respectively, with a crossover network to separate the frequency ranges feeding each one of them. Another method is to curve the membrane, but this adds distortion to an otherwise linear transduction mechanism and curtails the lower frequencies due to increased stiffness. If you curve a piece of paper, you can support small objects on it, which is why such designs typically use shallow curves that only spread the sound out over relatively narrow angles. In this chapter, we shall focus on the use of delay lines to control the directivity pattern.
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