ESL Theory

This section is devoted to a look at some of the “basic” theory of electrostatics, and how it might be applicable to a speaker. As you might expect, there is a little more to it than just the “pith ball attracts pith ball” stuff that you get in school. It will involve, for a start, some simple mathematics, which I hope will be easy to follow.Some diagrams and graphs will help too.

To be 10000% pedantic, we have to cover two things - electrostatics and loudspeakers - sounds simple enough, hey?

Electric Charge

As far as anyone has been able to determine, in this universe at any rate, elementary electric charge is carried by sub-atomic particles we call electrons. You knew that, right? Just wanted to be sure.

Electric Field

The electric field surrounding even an elementary charge can be conceptually viewed as the volume of space in which an electric force acts. Maxwell and others conceptually modelled this as lines of force, like this:

Field Strength

The electric field is weaker the further we get from the centre point of the field. This is diagrammatically represented above by drawing the lines further apart as we "leave" the centre, indicating a less dense (weaker) field. There is, of course a mathematical representation for this variation in field strength, as shown below:

where F(s) is the field strength in Newtons per Coulomb, Q is the value of the charge in Coulombs, R is the distance from the charge "centre" and K is the permititivity of the dielectric medium that the field is propagated in.

Field Forces

Those forces that are exerted by the field on other charged objects in the field (other fields), is expressed by: F

F = F(s) x Q

The laws governing forces between two charged objects, for instance a diaphragm and a stator, were developed by the French Physicist Charles Coulomb. This is written in equation form as:

Single-Ended Speakers

Weird name? Unlike their single-ended amplifier namesakes, there is little good that can be said for single-ended electrostatic loudspeakers. However, a discussion of the field would be incomplete without them being given some mention. So...

A "single-ended" electrostatic speaker has only one stator and, of course, a diaphragm that moves to produce sound. Conceptually, it looks like this:

Without going into the deepest detail, it is clear that single-ended electrostatics have some problems. For a start, since there's only the one stator, then all of the restoring force on the diaphragm has to be supplied elastically from the elastic potential energy stored in the diaphragm, and the diaphragm is connected mechanically to the frame of the speaker. Also, just to prevent itself being "sucked in" to a stator, the diaphragm will have to be stretched pretty tightly. Any external mechanical force can only be applied to a single ended speaker in the plane of the diaphragm. Let's look at a simple calculation to see just how big these "obstacles" to good performance really are:

 

 

Using the equation for forces shown above, and assuming a field strength of 10 Newtons/Coulomb and an electrical charge of 1 Coulomb then we get:

F = (10 x Q) / d2

Using a gap of 0.75mm (7.5 x 10-4 m) which is about the distance between the diaphragm and the stator in a Quad Treble Panel (assuming no other stator is there) then we get:

F = (10 x Q) / (2.54 x 10-4)2

Now, the Charge Q is equal to the capacity between the diaphragm and the stator multiplied by the difference in voltage. That is: Q = C x V. Substituting in this equation, where C is replaced by the expression given above, and the speaker diaphragm is only 15cm square with a 0.75mm gap and only 100 Volts applied, we get:

    Q = [(8.85 x 10-12 x 0.15 x 0.15)/ 7.5 x 10-4] x 100

which works out to:

    2.655 x 10-8 Coulombs

Now, our electric force equation above (assuming our previous field strength of 10 Newtons/Coulomb) tells us:

    F = (10 x 2.655 x 10-8) / (7.5 x 10-4) = 0.531 Newtons

This looks like it'll be easy to build - eh? However, we need a mechanical spring rate in the diaphragm at least equal to this, or the diaphragm will just collapse into the stator and stay there. To get the spring rate, we just have to apply Hooke's Law, like this:

    k = F / d and get, k = 0.531N / 7.5 10-4 = 708 N/m for a 0.75mm air gap.

Remember that our speaker framing, or whatever, has to hold the diaphragm in place at the edges, but, the external forces can only act in the plane of the diaphragm, and the restoring force has to be at right angles to the diaphragm. So, the angle?

    A = tan-1 (L / d) = tan-1(75 / 0.6) = tan-1(125) = 89.54 degrees

We need a spring rate, as previously noted of 708 N/m, so, to produce the restoring force in the diaphragm necessary to limit the diaphragm excursion to 0.6mm, at this angle:

 

    k = tan(A) x 708 = tan(89.54) x 708 = 8.9 x 104 N / m

This is, then, the spring force required to limit the diaphragm movement to limit diaphragm movement to 80% of the air gap (0.75mm), about 0.6mm. The actual force needed to achieve this spring rate is shown opposite:

    F = k x distance = (8.9 x 104) x (7.5 x 10-2) Newtons = 6, 675 Newtons

OK, so, how are you going to build this, rather small, 15cm x 15cm speaker if it is single-ended? You need a frame, or something that can withstand a lateral stretch of about 6.7 tonnes! Secondly, there's a good chance that the material in the diaphragm will tear at the corners. Plus, there's other problems as well...

Like, frequency doubling and distortion. A speaker like this can generate 4% to 6% distortion in the mid and high frequencies. This might be alright in a moving coil bass driver, but it is totally unacceptable in a mid-range unit. It is possible to reduce distortion in a push-pull electrostatic to less than 0.4% at all frequencies, an with careful design, even a little less.

 

Symmetrical Electrostatic Speakers

Early electrostatic speaker designers changed to the symmetrical (push-pull) format at a very early stage (see Patents Section) In this design of speaker, we have a restoring force being applied to each side of the diaphragm by "opposing" electrical fields. This means that the diaphragm only (in theory) needs to support its own weight. This is true over small distances, but the diaphragm still needs intermediate support (ribs) to prevent it being sucked into a stator. The mechanical situation is, however, much improved, and diaphragm tensions need be only a fraction of those needed in even a small single-ended speaker. Plus, and most importantly, the frequency doubling effect and other nasties can be completely eliminated, to produce a speaker with little more distortion than some electronics!

Here are a few useful equations for use when doing calculations on symmetrical electrostatic speakers :

The capacity of a symmetrical speaker is given by the expression, where C1 and C2 are the relevant capacitances between the central diaphragm and each stator:

Force exerted electrically on a symmetrical speaker's diaphragm is then given by:

(which is clearly zero if your speaker was made accurately and C1 = C2, and keep d1 = d2 and the voltages are kept equal on each side of the diaphragm.)

 

The equation opposite relates fundamental resonance f to the mass M and tension T of the diaphragm, for a speaker with a square diaphragm:

The equation opposite relates fundamental resonance f to the mass M and tension T of the diaphragm, for a speaker with a circular diaphragm (Area = p r 2):

The equation opposite relates fundamental resonance f to the mass M and tension T of the diaphragm, for a speaker with a rectangular diaphragm ( A = a x b ):

This equation allows you to calculate the actual voltage between plate and diaphragm, given the insulation resistance between the plate and the diaphragm, R1; and the resistance in series with the speaker capacity, R2.

One last, simple, but important relationship between series resistance, R; Half Period of Charging, T ; and the speaker capacitance, C, is:

T = RC

Constant Charge Operation

Additionally (and complementary) to the idea of symmetrical operation is the idea of constant charge operation as described by Hunt in ‘Electroacoustics’ pages 187~188 and summarised most elegantly in the following diagram:

This circuit has “novel advantages” and among these can be counted: 1) Vanishingly low harmonic distortion; 2) Evenly distributed driving force on the diaphragm; 3) Completely linear force acting everywhere between the two stators and 4) The high voltage diaphragm is shielded from contact with the surroundings by the stators. The original idea for this circuit was suggested to Hunt by Carlo V. Bociarrelli of Philco Corporation who (according to Hunt) “qualitatively argued” the advantages of constant charge operation. Hunt’s analysis expanded greatly on this idea to show other “virtues” could be ascribed to this arrangement.

The circuit itself, though not its unusual virtues, has been proposed in various patents and other publications dating back at least to H. Riegger’s disclosure in German Patent No. 398,195 (filed March 10, 1920) issued July 2, 1924. [Hunt]

A fundamental error made by most electrostatic speaker designers is to take this diagram literally. The schematic shows the “protective” resistance Ro and many take this to be simply a resistor in series with the resistance of the coating on the diaphragm. The coating being something with relatively low resistance like graphite. Yes, you can build a speaker like this, but it will not perform as well as one in which the entire resistance Ro is distributed on the diaphragm. In other words, you need a high resistance (~1012 ohms/sq in the original Quad ESL) on the diaphragm itself. This ensures that there can be no arcing from the diaphragm to the stator as the diaphragm approaches the stator (E field decreases).  It does not prevent stator-to-stator arcing however, as many have discovered!

Speakers in General

There are some things that all speakers of whatever persuasion (electrostatic or electromagnetic) have in common. The discussion in this section revolves around those things that apply to both. So, let's start with the "real basics" and work our way up to a better understanding of speakers in general.

Acoustical Properties of Air

To understand that sound is "caused" by the pressurization and de-pressurisation of the air is one thing, but to quantify this impression, we need to know certain physical facts, concerning air and sound. Many of these require a datum for measurement, and that is usually sea level. Some of these are summarized in the table below:

Velocity of Sound:

343 m/s or 1130 ft/s

Air Density:

1.18 kg/m3

Pressure:

105 Newtons/m2

Acoustic Impedance:

405 Rayls

To discuss sound waves of course we need to talk about frequency, or the number of times per second (as it happens) that the pressure near our ears changes. We interpret this as a longitudinal wave, as a convenient model. The change from maximum pressure (top of a wave) through minimum pressure (bottom of a wave) and back to maximum again is known as the Period of the wave. The frequency is just the number of times that this is happening per second. There is a simple relationship between the Period (T) and Frequency (f) of a wave that we will find useful, as follows:

T = 1 / f

Another useful property of wave motion that you probably already know about is Wavelength. The wavelength is equal to the distance between two corresponding equal points on the wave form. For instance, the distance between two maxima, or between two minima. The wavelength is simply described, as follows:

l = c / f

where c is the velocity of sound, f is the frequency in Hertz and l is the wavelength.

We have expressions to model Acoustic Resistance not unlike those that you may have seen for electrical resistance. You just have to be clear on what each equivalent physical quantity is. Not at all unlike R = V / I for an electrical circuit, we can write:

R = P / u

where R is the acoustic resistance of the medium, P is the RMS pressure in dynes/cm2 and u is the particle velocity in cm/s

That is where the completely simple concepts end. In order to perform useful calculations regarding speakers, we need to understand something called the ka factor. It appears in a lot of loudspeaker calculations. In one area, for example, it relates the size of the speaker to its acoustical output. The actual value of k is the wave number, and is given by:

k = w / c = (2 x p x f)/c = 6.28 x f/ l

If you use the metric system and the value of c is in m/s then the value of k is:
1.831 x 10-2 x f. To calculate the acoustical resistance of the air to a particular speaker, you have to take some properties of the speaker into account, thus:

Rr = pc x A x R 1 x (2ka)

where Rr is the radiation resistance, pc is the resistance of the media, A is the driving area of cone or diaphragm in m2 , R1 is a Bessel function f is the frequency in Hertz, a is the cone/diaphragm radius and k = 6.28/l

The Bessel function bit sounds a little fearsome, and it could be for non-mathematicians, so you can estimate it with R1 = x2 / 8 where x = 2ka.

Radiated Power and Frequency

This really just refers to the way in which a speaker will have different outputs depending on frequency for the same electrical input. Below the resonant frequency of the diaphragm, the response of the speaker is controlled by the compliance of the diaphragm or suspension. The output in this region falls rapidly at about 12dB/octave. The air mass generally controls the speaker performance above resonance until we reach the limit of radiation resistance. The reactance of the air mass however eventually falls to a value that is less than the radiation resistance and the speaker puts out less energy and starts to become directional.

Directivity

That ka thingy also affects directivity, by which we mean the way in which a speaker "sprays" sound at you at different frequencies. A polar plot of this is shown below that indicates how the radiation (spraying) angle changes as a function of ka and how acoustic output decreases with this change.

This plot only shows the radiation pattern for specific values of ka, but the loss for any value can be calculated by:

 

P / Pref = (2 x [J1(x)] / (x))

where P is the acoustic Power at some angle to the speaker's axis, Pref is the acoustic power at the same distance on axis, J1(x) is a Bessel function and
(x) = ka sinq

Hopefully this has been of some use in introducing you to the concepts surrounding loudspeakers, including electrostatics. The mathematics is a little more involved than is possible to represent easily on the web, but if you're really interested, a few hours with a good book, pencil and paper will clear things up in all likelihood.