If the gap distance d is reduced, we can reduce both E _P and e ˜ in while keeping the sound pressure constant. Hence the sensitivity is increased. Although the capacitance is increased proportionately, the input current does not change because of the reduced voltage. Hence the reactive power is reduced, while the efficiency is increased. With a smaller gap, some very low bass is sacrificed, but this is usually in a region where there is little output anyway because of rear-wave cancellation.

A good place to start the design process is with the highest frequency of interest because we can use this to define the membrane thickness h independently of its radius a. From this we can work out the maximum tension T that it can withstand. Maximizing the tension enables a smaller gap d to be employed without the membrane touching the electrodes at the maximum input voltage at low frequencies, where the displacement is highest. We have shown (see Eqs. 15.84 and 15.86) that making the tension large enough to avoid contact automatically meets the condition for stability under the electrostatic force of attraction.

Knowing the tension, we can then define the fundamental resonance frequency f ₀ for any given radius a or choose a fundamental resonance frequency to find the required radius. Having found the radius, we can define the gap d, the polarizing voltage E _P , the specific resistance of the dust screen R _s (for the desired Q factor), and the sound pressure level.

Example 15.1. We now use the equations of Section 15.17, to derive the tension, resistance, gap width, and polarizing voltage required for the 280   mm diameter membrane for which we derived the on-axis pressure and directivity pattern in the previous sections while assuming an ideal membrane with no mass or tension. If we use a stator with a thickness of t   =   1   mm perforated with holes of radius a _h   =   1.5   mm and pitch b _h   =   5   mm in a triangular array, the porosity is given by

Then if we use a membrane with a thickness of h   =   12   μm and density of ρ _D   =   1400   kg/m³, the upper cut-off frequency is given by

Assuming the tensile strength of polyester is σ _max   =   60   MPa and the stress is 60% of the maximum, we obtain the tension from

Then, for a membrane radius of 140   mm, the resonance frequency is

We let Q   =   1 so that we can obtain the optimum specific acoustic resistance of the dust screen

Referring to Table 4.1 on p. 126, this value may be made up from using a 75   rayls dust screen on the front and a 47   rayls screen on the back, for example. The gap between the membrane and each stator is given by

These values are used in Eq. (15.93) to calculate the on-axis response shown in Fig. 15.25 for an electrostatic loudspeaker without a delay line. Also shown is the on-axis response of the same loudspeaker calculated from rigorous equations [10] from Section 14.10 based on an analytical model. This reveals an excellent correlation between the lumped-element model and the analytical one, except that the latter shows a 2.7   dB dip at 330   Hz, which is smoothed out by the simpler lumped-element model. Because the simple model cannot reproduce any harmonics above the fundamental resonance, this correlation is dependent on the membrane being well damped through the use of a dust screen. The simple model appears to overestimate the peak displacement by around 10% at the lowest frequencies, whereas the rigorous model keeps it just within the gap width of 1.6   mm.

Note that the plot of Walker's voltage equation shown in Fig. 15.25 uses the following version of Eq. (15.20) which is modified to include the resistance R _S of the dust screen

Next, we combine the on-axis response of a real membrane, based on the lumped-element model, from Eq. (15.93) with Eq. (15.52) for the far-field pressure of an ideal electrostatic loudspeaker with a delay line to yield that of a real membrane with a delay line

Assuming 1 2 e ˜ in = E P = 3200 V , d   =   1.6   mm, a   =   160   mm, and R _S   =   121   rayls, this gives a pressure of 4.31   Pa rms or 106.7   dB SPL with an input voltage of 2   ×   3200   V rms.

Although the high-frequency response appears to roll-off prematurely, in practice this can be compensated for by omitting some of the stray neutralizing capacitors.

Problem 15.1. A simple electrostatic loudspeaker with no delay or any kind of directivity control whatsoever has a diameter of 10   cm (i.e., radius of 5   cm). Assuming a maximum field strength of 2000 V/mm, calculate the maximum rms sound pressure in Pa and dB SPL re 20   μPa at 1   m on-axis at 1   kHz. Assume that 1 2 e ˜ in = E P .

Hint: From Eq. (15.20), p rms ( r , 0 ) = 2 π f ε 0 a 2 ( E P / d ) 2 / ( 2 2 r c ) and use Eq. (1.18) to calculate the sound pressure level in dB SPL.

Problem 15.2. Suppose that the loudspeaker has a gap width of d   =   0.224   mm. Calculate the polarization voltage E _P , minimum tension T needed for stability (from Eq. 15.86), resonance frequency f ₀, and specific resistance R _S needed for unity Q. Assuming the maximum membrane stress before breaking is 55   MPa, calculate the maximum allowable tension according to Eq. (15.113).