XMAX.HTM --- Part of Manual for Driver Parameter Calculator --- by Claus Futtrup.
Created 18. January 2004, last revised 19. January 2004. Ported to XHTML 1.0 on 2. October 2004. Last modified 25. October 2004.

This document was originally a section of DATAINFO.HTM but it became a bit large for the document and consequently was moved into this separate document.

### The old definition of Xmax

In the traditional fashion, Xmax is calculated based on geometric properties in the motor system: Hg and Hc. Hg is height of the magnet gap. Hc is the height of the voice coil. Half the difference is Xmax (this is how it is defined in DPC):

```        Xmax = ABS(Hc - Hg) / 2
```

Xmax is a parameter often used, especially in the advertisement of subwoofer drivers, where such a parameter would be most relevant. The parameter is supported in DPC, though I do not consider it a paramter of much value. If I had no other choice than to rely on this parameter, then OK, but usually price and quality are related. The world is overflowing with cheap drivers, supposedly subwoofer units, who have a high volume displacement capability (Vd = Xmax * Sd). There is more to the world than Xmax because traditional electrodynamic drivers are inherently nonlinear. For example you could try to measure distortion levels instead.

If no specification is given for Xmax, then John Murphy uses the following assumption on Xmax:

```        Xmax = Nominal size / 50 (both in meters)
```

Since Ashley specifies a rule for nominal size (NomDia = Dd/2 * 2.54 = Dd * 1.27), this can be combined with John Murphys rule (this definition is not utilized in DPC):

```        Xmax [mm] = Dd [m] / 25.4
```

### Early attempts to improve the Xmax definition

Realizing the limitations of the definition of Xmax, early attempts have been made to improve this definition.

Mark R. Gander, Moving-Coil Loudspeaker Topology As An Indicator Of Linear Excursion Capability, starts by defining what we already know:

```               Hc - Hg
Xmax = -------
2

Hc = Voice Coil Height
Hg = Air Gap Height
```

Gander limits this to overhung geometries only (where Hc > Hg). He quotes Richard Small (Closed box), where he describes this definition explicitly, but Small also mentions Xmax in previous paper (Direct Radiator).

I can find Xmax mentioned as far back as A. N. Thiele, 1961.

There are no references of this definition mentioned anywhere, just that it is commonly specified by loudspeaker manufacturers. It seems like the origin of this definition fades into the very old times. On the other hand it proves that even for very old speakers, the focus on linearity (and non-linearity) was based on the magnet/motor system geometry.

Gander goes ones step further, and defines:

```               Hc - Hg
Xmax = ------- + 0.15 * Hc
2
```

Gander defines this equation for underhung (where Hc < Hg) and equal hung geometries (where Hc = Hg).

It is a good idea never to let Xmax = 0, because even a motor system with a very nonlinear behavior will have a peak and a certain amount of reduction in Bxl is allowed before the nonlinearity becomes obvious.

I dislike the fact that this definition allows first equation above to almost become zero, before a discontinuity happens when an equal hung geometry is present. I recommend that any system, where Xmax < 0.15 * Hc is changed to this value.

Douglas J. Button, Design Parameters and Trade-Offs in Large Diameter Transducers, concludes, based on work by Barlow, that:

```               Hc - Hg
Xmax = ------- + 0.25 * T
2

where T = Top Plate thickness (=Hg).
```

He states that this is a more precise method than Ganders method of adding 15 %. The method could be made valid for all geometries (overhung, underhung and equal hung).

Anyway, the whole basis for these definitions has moved, since a lot has happened in recent years regarding motor systems. Very large gap geometries combined with very small (or even larger) voice coils have appeared, and the basis for these initial definitions may have shifted considerably.

Other models have been discussed, like the AES2-1984 recommendations. These techniques have been shown to be very ambiguous, by Wolfgang Klippel, and this is probably also the reason why they have not been generally applied. Klippels suggestion is to look at various kinds of distortion to set a limit, and though this method is probably closer to reality, it can be concluded that, generally, none of the "improved" models have appeared (consistently) in datasheets from loudspeaker driver manufacturers.

The Driver Parameter Calculator does not utilize any of these definitions, since the old definition is still the one most commonly used.

### References to early attempts

References:

Mark R. Gander, Moving-Coil Loudspeaker Topology As An Indicator Of Linear Excursion Capability, 64th AES Convention New York City, preprint no. 1554, November 1979.

Richard H. Small, Direct-Radiator Loudspeaker System Analysis, JAES vol. 20, June 1972. Reprinted with permission from IEEE Transactions on Audio and Electroacoustics, vol. AU-19, pp. 269-281 (December 1971).

Richard H. Small, Closed-Box Loudspeaker Systems, Part I: Analysis, JAES vol. 20, No. 10, pp. 798-808, December 1972.

A. N. Thiele, Loudspeakers in Vented Boxes: Part I. Reprinted in the JAES, and again in the Anthology on Loudspeakers, vol 1. The original is from the Proceedings of the IRE Australia, vol. 22, pp. 487-508 (August 1961).

Douglas J. Button, Design Parameters and Trade-Offs in Large Diameter Transducers, 91st AES Convention New York, preprint no. 3192, October 1991.

Wolfgang Klippel, Assessment f Voice-Coil Peak Displacement Xmax, JAES vol. 51, no. 5, May 2003, pp. 307-323.

AES2-1984, AES Recommended Practice Specification of Loudspeaker Components Used in Professional Audio and Sound Reinforcement, Audio Engineering Society (revised 1997).

### Nonlinear approaches to Xmax

Regarding excursion (ie. low frequency related), especially Bxl nonlinearities and Cms nonlinearities are present. These static nonlinearities are actually quite well understood. At the moment I am aware of 2 different commercial systems available for measuring these parameters. One is the relatively old DUMAX measurement system, by DLC Design, another is the Klippel Distortion Analyzer, by Wolfgang Klippel.

One of the later articles on this is:

W. Klippel, "Adaptive Nonlinear Control of Loudspeaker Systems," J. Audio Eng. Soc. vol 46, pp 939-954 (1998).

The founder of DLC Design, David Clark, has written an article, "Presicion Measurement of Loudspeaker Parameters," J. Audio Eng. Soc. vol 45, pp 129-141 (March 1997), which describes how the DUMAX measurement system has worked, more or less unchanged since 1991.

The nonlinearities of drivers has been treated by several people, among who I would like to mention the work by Morten Knudsen (Aalborg University) and the work by Erling Sandermann Olsen (Denmarks Technical University, Lyngby). Both from Denmark.

Morten Knudsen, Loudspeaker modeling and parameter estimation, AES paper at the 100th AES Convention, May 11-14 1996, Copenhagen. The preprint number is not given on my copy.

Erling Sandermann Olsen (DTU) & Knud Bank Christensen (B & O), Nonlinear Modelling of Low Frequency Loudspeakers - A More Complete Model, AES preprint 4205 presented at the 100th Convention in Copenhagen, May 11-14 1996.

Each nonlinear description is a bit different from each other. Naturally there are many ways to describe a given nonlinearity, and the complexity of the topic makes it difficult for people to agree on a single model. Things are not as simple as the Thiele/Small parameter model.

In MEASURE.HTM I have treated the topic of Bxl and Cms nonlinearities when trying to reduce the problem of these nonlinearities by keeping a reasonably constant cone excursion at lower frequencies.

The following information will go deeper into the discussion on Bxl and Cms nonlinearites versus the inadequate Xmax specification.

Both parameters can have different kinds of nonlinearities:

1. Symmetric nonlinearities, which create 3. harmonic distortion 2. Nonsymmetric nonlinearites, which create further 2. harmonic distortion

Situation 1 occurs when the voice coil is properly placed in a nonlinear but symmetric magnetic field (inside and around the gap), and the spider (Cms) is made symmetric.

There is rarely (if ever) a driver, which is perfectly linear at any excursion level. There is (almost) always nonlinearities as soon as the driver cone (or dome) is moving. Magnetic power decreases and spring resistance increases, creating a kind of compressed (nonlinear) sound signal. The question is to what degree.

Situation 2 occurs when the voice coil is not placed exactly symmetric in the gap and/or the spider (Cms) is not exactly symmetric and/or the magnetic spread of the field in the air gap is nonsymmetric. This is normally the case - if nothing else, then at least because of production tolerances.

Nonsymmetric nonlinearities can be studied by applying a sinewave signal to the driver at a fairly large power (enough for high excursion levels) and utilizing a stroboscope. Sweep up and down in frequency to observe DC-offsetting. The driver will simply be moving its operating center / rest position either in or out of the air gap as if a DC-current had been applied to the signal. Even order (primarily second) harmonic distortion creates this behaviour.

A clever driver designer will try to play the different nonlinearities against each other, to get a lower distortion level, but an optimum will only occur for a given power level, which must be chosen to fit the expected operating level of the driver. Of course the best choice is to have both Bxl and Cms as symmetric as possible, and to have the voice coil centered as precise as possible. This is one of the differences between a high-quality (normally expensive) driver and a low-quality (normally inexpensive) driver, which cannot be observed in the T/S parameters.

The ear tolerates quite high levels of nonlinearities from Bxl and Cms before increased distortion is observed by the listener, otherwise the distortion is noticed less directly / more in a indirect manner.

DUMAX, being the old system on the market, has created consensus that Bxl may decrease to 71% times its max level (the parameter is called Xmag), whereas Cms may decrease to 25% of its max level (the parameter is called Xsus), ie. suspension stiffness may increase 4 times, before significant distortion is observed. In case of a nonsymmetric system an average between inwards and outwards excursion is calculated.

The Klippel Distortion Analyzer, being more critical / discriminating works with Bxl decrease down to 0.77-0.78 and Cms decrease to a similar level. I do not think that this more discriminating nature has any hold in reality. Such limits should be based on listening tests, and the people behind DUMAX have performed such listening tests.

When talking about the linear range of a driver, the engineer is always trying to refer to a nonlinear range, which is sufficiently inaudible to be tolerable. Different listeners will have different opinions. For example JBL uses 10% THD as a reference for acceptable distortion level (for describing the limits of linear excursion of a driver).

Since the voice coil can be placed nonsymmetric related to Bxl and/or Cms, the DUMAX system works with the Xmag and Xsus parameters in both forward and reverse cone motion (ie. out of and in to the magnet system). This is a simple way of trying to describe the excursion behaviour at fairly high levels, but obviously the behaviour at lower and higher levels are of interest too.

For example a driver with an Xmax figure a bit higher than the competitors is not worth much, if shortly after the Xmax excursion level, the driver bottoms (the excursion is physically limited, eg. by the magnet system) - this will create a loud noise, perhaps in excess of 140 dB. The sound is highly unpleasant, and such a bad behaviour would require that the listener is very careful at keeping enough headroom for dynamic sounds in the music without getting close to bottoming.

This is why DPC supports the Xlim parameter (sometimes named Xmech), which is the physically limited excursion level - the damage level. A good woofer driver design has Xlim = 2 * Xmax (or more, depending on the linearity of Bxl and Cms - more linear behaviour requires more headroom). DLC recommends that Xsus = 1.4*Xmag or more, as a minimum if the woofer is supposed to play a good bass.

The ideal solution would be graphs showing Bxl and Cms as a function of cone excursion, or alternatively 2. and 3. harmonic distortion versus cone excursion (since this is really what's important). Both DUMAX from DLC and the Klippel Distortion Analyzer can provide you with such information.

As far as I understand DUMAX performs an quasi static measurement. The diaphragm is moved pneumatically to an certain x position and the small signal parameters are estimated for this working point. If you do this for a couple of working points you can estimate the nonlinear curves. Note that the estimated small signal parameters will give you the slope of the nonlinear curves, i.e. to get the real nonlinear parameters you have to integrate the curves.

In contrast to this the Klippel equipment measures the nonlinear driver parameters dynamically. The driver is exited by "realistic" (large amplitude) audio or noise signals and the nonlinear curves are determined by an system identification. This will give you an realistic model of the driver, secondary effects like creep in the suspension and temperature effects may be included.

Please notice, by the way, that Xmax is a single value. The voice coil may be placed at a nonsymmetric position (to minimize the total harmonic distortion). This would result in a situation where Xmax inwards is a different measure than Xmax outwards. Normally an average of these two values is used for Xmax.

It should be obvious by now that Xmax really is not worth much in the real world and that it is quite easy to manipulate a driver to have a decent Xmax-value, but lousy distortion qualities at intermediate and high power levels.

### Measuring

Measuring some kind of an Xmax value is not a simple task, if only impedance measurement techniques are available. It would be much easier to evaluate Harmonic Distortion like eg. JBL suggests.

Xmax may be measured / detected by measuring T/S parameters at increasing power levels, until nonlinearity in Bxl (or Le) is found. This technique is only recommendable when utilizing the compare resistance method suggested in MEASURE.HTM, where excursion is held reasonably constant.

Measuring (or calculating) nonlinear parameters, Xmax or other parameters, is partially a valuable tool for the clever driver designer, partially also a good way for a loudspeaker system designer to evaluate the quality of the driver(s) he is considering for his system. Values like Xlim, Xmax, or even Xsus and Xmag etc. for that matter, are methods to simplify a qualitative measure of a given driver, for easy comparison with other drivers.

### Some points

Generally, woofers must be able to "run" into the nonlinear range when they are getting pushed, whereas for midranges (or tweeter) where the excursion level is under control (by the crossover), the stiffness nonlinearity is normally not an issue.

For midranges, DLC recommends that Xsus is smaller than Xmag because this prevents the driver from generating excessive intermodulation distortion. For (sub-) woofers the motion out of the gap produces a fair amount of intermodulation distortion, but this is less of an issue than the total harmonic distortion when using a driver in this frequency range. Besides, subwoofers are used in a limited frequency range, so the intermodulation distortion will be limited to this frequency range. Full range drivers are driven above and below resonance, and here the compromize according to DLC is to keep the Xsus and Xmag about equal.

The whole case around Xmax and related parameters is only an issue with drivers, whose cone travel is not limited by eg. a crossover. This is the case for woofers and subwoofers.

I cannot state my warnings too much. The market is overflowed with cheap supposedly long stroke subwoofers, with a high Xmax figure. Be aware that there are other things to be considered, also other than the T/S parameters. If you choose a subwoofer (or woofer) driver entirely on data like eg. the Xmax figure, then you may end up with a dissappointing system.