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Contrasseur

Xmax Investigation

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Ricci    652

I'll try to answer a few questions and post some additional data when I get a chance. Knee deep in a bathroom remodel at the moment. I do have the captured waveforms from the burst tests. I'll see if I can post a couple.

 

Xmax is an ideal parameter for the most part usually only based on gap geometries or on a free air (ideal) distortion measurement. When you place a driver in a high Q sealed alignment the harmonic distortion grows greatly especially below the system resonance. This is why a driver like the xxx exhibits much lower output in the standard cab at 10Hz than predicted while meeting the THD thresholds. If the driver were IB the THD would drop and the passing score would increase. A driver like the LMS would see very little increase most likely.the XXX still moves more air than the LMS ultimately it just has a lot of distortion comparatively in the same airspace.

 

I think the posts by Nathan are spot on for the most part and account for a lot of the discrepancy. There are some drivers we know for a fact cannot be driven past a certain point yet the rough xmax calculation based on the burst data implies it has been. For example the RF simply cannot be driven past somewhere around 45mm. Perhaps a bit more but no way can it travel 60mm.

 

Excuse any errors In on my phone.

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Steve81    7

Hi Contrasseur,

 

I've got a formula and some thoughts (already mentioned) that might help you square some things away... The formula I use is from Acoustics by Leo Beranek: P(rms) = Square Root Of 2 * Pi * Frequency Squared * Rho * Sd * Excursion / Distance

 

P(rms) is recorded in Pascals

Rho is the density of air, which varies by things like temperature, humidity, elevation, and so on.

Sd is the radiating area of the driver in square meters

Excursion is also noted in meters, as is distance

 

Here are a couple examples of backdooring excursion through that via excel. With the LMS, that seems to track Nathan's comment re: 39mm of excursion, and Josh's <65mm comment for the T3S2-19. Rho is altered between the two to account for temperature differences on the day of measurement, and assumes a ~600' elevation, and 50% humidity (and yes, Pi goes to more than just 2 digits).

post-3442-0-32916900-1464718910_thumb.png

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SME    211

I've been wanting to weigh in here for a while but have been busy.

 

First, I think a half-space to full-space transition is very unlikely.

 

Second, THD vs. excursion is not constant at all constant with frequency.  THD during sine sweeps is a consequence of non-linearity in various system parameters.  AIUI, the Klippel system measures linearity of the driver suspension stiffness (K), the magnetic flux (B ), and the inductance (Le) with respect to cone displacement.  Because the relative influence of these parameters on the overall system behavior varies with respect to frequency, the distortion that results due to non-linearity of these parameters will also vary.

 

With respect to the trends observed here, I believe that what is happening is that a lot of drivers have better BL linearity than K linearity.  I've mentioned in a few other places already, system stiffness/compliance (note that these quantities are merely inverses of one another) has a very strong impact on ULF efficiency.  The same is true of BL, but unlike BL, stiffness has less importance as higher frequencies.  An area where things get interesting is the resonance frequency.  The resonance frequency is determined by stiffness and mass.  However, while the woofer is near resonance, the effects of the stiffness and mass essentially cancel.  (Edit: This isn't really the best description of what is happening at resonance.  See below for the interesting results with respect to stiffness/compliance.)

 

I've been doing some mathematical analysis to try to better understand the extent to which non-linearity of system parameters influences distortion.  I started with just system stiffness and assumed it took a form like:

 

    k = k1*(1 + k3*X^2)

 

Edit: Typo corrections here.  "k1*(1 * k1/k3*X^2)" has been corrected to "k1*(1 + k3*X^2)"

 

Edit: This shape is plotted below for various k3 values:

 

post-1549-0-81815900-1464936889_thumb.png

 

We define X as the fraction of peak displacement.  In other words, X=+/-1 when displacement is at the peak.  By doing so, we can let k1 be the stiffness without distortion and k3 be the fraction by which the stiffness increases at the peak of excursion.  The X^2 is a parabola, so this models a suspension that gets stiffer by an equal amount with inward and outward displacement.  This form for k is very useful because it is a simple model that will produce odd-order harmonic distortion similar that that exhibited by a real driver.  Note that this model ignores mechanical damping, which is often very small anyway.

 

Chugging through the math (and hopefully not goofing anything up), I got the following results for the ratio of third harmonic relative to the fundamental.  It is helpful to define a useful quantity, om0 (the "om" is traditionally symbolized by a lowercase Greek omega), which is related to the resonance frequency of the undistorted system (fb).  We also define om in terms of the frequency of interest (f).  To be clear, om0 is a property of system while om is a variable that changes with frequency:

 

    om0 = 2*pi*fb = sqrt(k1/m)

    om = 2*pi*f

 

    |H3| = 9 * k3 * om0^2 * sqrt{ 1 + 9*om^2*(Le/R)^2 } / sqrt{ [om0^2 - 9*om^2]^2 + 9*om^2*[om0/Qtc + Le/R*(om0^2 - 9*om^2)]^2 }

 

(Edit: This formula and those that follow have been completely re-written into "simpler" forms.)

 

Ugh! Lots of math.  Well, the math getting there was much worse.  One way to simplify is to assume zero inductance, which is probably not a bad approximation at low frequencies and may be a good approximation at fairly high frequencies for some drivers with very low Le/R:

 

    |H3| = 9 * k3 *om0^2 / sqrt{ [om0^2 - 9*om^2]^2 + 9*om^2*(om0/Qtc)^2 }    (assuming no inductance)

 

That's a lot better.  We can simplify things further by considering some special cases.  First, consider very low frequency, so low that 3*om is much less than om0 (we're well below 1/3th of the resonance frequency):

 

    |H3| = 9 * k3 / sqrt{ 1 + 9*[om/(om0*Qtc)]^2 }    (assuming no inductance and frequency well below 1/3 of the box resonance)

    |H3| = 9 * k3    (assuming no inductance, frequency well below 1/3 of the box resonance, and frequency well below 1/3 of the box resonance times Qtc)

 

These assumptions are pretty restrictive considering that typical box resonances are in the 25-50 Hz range, but they give insight into 3rd harmonic distortion for the lowest of low frequencies.  (Low single digits in most cases).  In this situation, the percent change in stiffness at the peak is related directly to the 3rd harmonic distortion of the displacement.  However, the third harmonic distortion in the output is 9 times greater than the third harmonic distortion of the displacement.

 

Let's look at high frequencies now, where om is much greater than om0.  We'll consider the results both with and without inductance.  Where we consider inductance, we'll go ahead and assume the frequency is high enough that the inductance rise is fully established, or mathematically that 3*om is much greater than R/Le.

 

Let's assume that it's much larger than R/Le, R/(BL)^2, that we're well above the resonance frequency, and that 2*pi*f^2*m is much larger than (BL)^2/Le:

 

    |H3| = 9 * k3 * om0^2  * Le/R / sqrt{ 9*om^2 + [om0/Qtc - 9*Le/R*om^2]^2 }    (frequency well above resonance and fully developed inductance)

    |H3| = k3 * om0^2 / om^2    (assume frequency well above resonance, fully developed inductance, and frequency well above 1/3 of resonance divided by Qtc)

    |H3| = k3 * om0^2 /  sqrt{ om^2 * [om^2 + (om0/(3*Qtc)]^2] }    (assume no inductance and frequency well above resonance)

    |H3| = k3 * om0^2 /  om^2    (assume no inductance, frequency well above resonance, and frequency well above 1/3 of the box resonance times Qtc)

 

As frequency goes up, the influence of k3 diminishes to nothing.

 

What about at the system resonance?  At the resonance, om = om0.  Let's see what happens:

 

    |H3| = 9 * k3 * sqrt{ 1 + 9*om0^2*(Le/R)^2 } / sqrt{ 64 + 9*[1/Qtc - 8*Le/R*(om0)]^2 }    (at resonance)

    |H3| = k3 / sqrt{ 64/81 + 1/(3*Qtc)^2 }    (at resonance and assuming zero inductance)

 

When ignoring inductance for simplicity, we obtain a fairly simple expression for the third harmonic distortion.  To get an idea of how this compares to the third harmonic distortion in the low frequency limit, let's look at |H3| for some values of Qtc:

 

    |H3| = 0.624*k3    (at resonance with zero inductance and Qtc=0.25)

    |H3| = 0.900*k3    (at resonance with zero inductance and Qtc=0.5)

    |H3| = 0.994*k3    (at resonance with zero inductance and Qtc=0.707)

    |H3| = 1.05*k3    (at resonance with zero inductance and Qtc=1.0)

 

We can see that the third harmonic distortion due to stiffness non-linearity will be much smaller at resonance than for lower frequencies, and lower Qtc alignments have less distortion than higher Qtc alignments.

 

Let's look at one more case.  Suppose we use a frequency that's 1/3rd of the system resonance, om = 1/3*om0.  This is a pretty low frequency, but it happens to fall right in the area that is of interest in many ULF applications.  It is physically relevant because the harmonic that's produced is at the system resonance:

 

    |H3| = 9 * k3 * Qtc * sqrt{ 1 + om0^2*(Le/R)^2 }    (at 1/3 of resonance)

    |H3| = 9 * k3 * Qtc    (at 1/3 of resonance and ignoring inductance)

    |H3| = 2.25*k3    (at 1/3 resonance with zero inductance and Qtc=0.25)

    |H3| = 4.5*k3    (at 1/3resonance with zero inductance and Qtc=0.5)

    |H3| = 6.36*k3    (at 1/3resonance with zero inductance and Qtc=0.707)

    |H3| = 9*k3    (at 1/3resonance with zero inductance and Qtc=1.0)

 

This is a very interesting result.  It again suggests that Qtc has an influence on distortion.  And in fact, if Qtc is greater than 1, the distortion at 1/3 of the resonance will actually be *higher* than it is for the lowest frequencies and a 3rd harmonic distortion peak will appear in this region.  In a sense, the resonance of the system helps amplify the 3rd harmonic.  A lower Qtc means more back-EMF and more damping for this unwanted harmonic.

 

There's a lot more insight that could be gained with more analysis like this.  I wouldn't be surprised if Klippel has outlined some of this in his papers.  Looking at B non-linearity is a bit harder because it affect both the mechanical and electrical balances.  It can also vary with the current as well displacement, which has been mentioned here already.  So called flux modulation distortion occurs due to the interaction of the permanent magnetic field of the driver with the temporary magnetic field induced by the current in the coil.  This could also account for a big increase in distortion relative to displacement in lower frequencies, but I still think compliance stiffness non-linearity is the most common reason for this trend.  Why?  Because flux modulation distortion should also affect frequencies well above the resonance unless inductance is extremely high.

 

OK.  Hopefully I won't have to revise any of these formulae, but I might after I review my work at a later point.

 

Edit: Correction: the original formulae I made were for 3rd harmonic of displacement vs. fundamental of displacement.  However, pressure relative to displacement scales with 1/f^2 (12 dB/octave).  So since we really want to know the 3rd harmonic vs. fundamental for pressure, everything gets multiplied by 3^2 = 9.  Because all the results are scaled equally, the trends are still the same.  On the other hand, we see that for very low frequencies, distortion is extremely sensitive to non-linearity in the stiffness.  Stiffness distortion of 10% 3rd harmonic will result in 90% 3rd harmonic distortion of the sound.

 

Edit: I caught and corrected an algebra error I made in the formulas.  The overall interpretation doesn't change much.  The big change is that the Qtc must be greater than 1.0 rather than 0.33 for there to be a 3rd harmonic distortion bump at 1/3rd of the resonance frequency.

 

Edit: I rewrote the formulas in terms of fewer variables to make them more useful.  I also rewrote and improved the interpretation of the results.  Hopefully in time, I'll be able to do some plots to make it easier to *see* the trends I wrote about above.

 

Edit: Here is a plot that characterizes the third harmonic distortion behavior in this model for k for different Qtc alignments.  These assume no inductance for simplicity.  The x-axis is what might be called reduced frequency.  It's the ratio of the frequency to the box resonance frequency.  The y-axis is the factor that you multiply k3 by to determine the level of the distortion harmonic relative to the fundamental.  What kind of high frequency roll-off do you think that is?  It's 12 dB/octave.

 

post-1549-0-08646200-1464936986_thumb.png

 

There is a subtle caveat to consider when looking at this plot.  The "k" value in the model is for the overall stiffness of the system rather than the driver alone.  It is not a model of distortion in driver Kms.  What this means is that if you wanted to compare distortion for different Qtc alignments for a particular driver, you can't really use this plot directly.  The k in this model is the sum of Kms and the air spring.  To use a driver in a higher Qtc alignment, you necessarily shrink the box and increase the stiffness of the air spring.  That actually makes the system stiffness more linear by overwhelming the deviation contributed by the Kms non-linearity.

 

As such, it's not entirely clear that lower Qts = lower distortion as far as Kms non-linearity is concerned.  With a small adjustment to the model, I may be able to resolve this without issue.  Stay tuned.

 

Edit: As a related caveat that I missed that may be more obvious is that changing the Qtc changes the fb as well.  If you take a driver and build sealed systems with different Qtc values, the fbs change also.  This has the effect of shifting the absolute frequency scale relative to the x-axis for each curve, thus visually shifting the curves relative to one another.  Depending on the situation, it's possible that the curves could overlap in places.

 

The reason the curves are presented this way is because it makes the plot more generally useful.  If an absolute frequency scale were to be used, you'd also have to fix the resonance frequency or some other parameters to do the plot, making it less generally useful.

Edited by SME
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TTS56A    3

A question, the spl calculated here ---> http://www.baudline.com/erik/bass/xmaxer.htmlis refered at 1 meter free space or half space?

 

I asked it because by looking at the formula it seems to be calculated in full-space, but i'm not sure. However i think all point sources which would be a subwoofer in the lower range can be considered always in half-place (+6 db). Even if the sub is flying at 20 meters. Different the case when the mic is not close to the ground, so the comb filtering will be achivied, but the overall gain is still +6 db.

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Contrasseur    17

I asked it because by looking at the formula it seems to be calculated in full-space, but i'm not sure. However i think all point sources which would be a subwoofer in the lower range can be considered always in half-place (+6 db). Even if the sub is flying at 20 meters. Different the case when the mic is not close to the ground, so the comb filtering will be achivied, but the overall gain is still +6 db.

It's for full space. Unless the ground (grass and dirt in this case) is a perfect reflector at the frequencies in question, these subs will not behave as if radiating in half-space. I hypothesize that much of the sound at 10Hz travels through the ground. We know that sometimes low frequencies can travel through earth (see: worm charming). I seem to be alone in this, but I think that the lowest frequencies travel through the ground and the higher frequencies begin to reflect.

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Contrasseur    17

I've been wanting to weigh in here for a while but have been busy.

 

First, I think a half-space to full-space transition is very unlikely.

 

Second, THD ....

Edit: Correction: the original formulae I made were for 3rd harmonic of displacement vs. fundamental of displacement.  However, pressure relative to displacement scales with 1/f^2 (12 dB/octave).  So since we really want to know the 3rd harmonic vs. fundamental for pressure, everything gets multiplied by 3^2 = 9.  Because all the results are scaled equally, the trends are still the same.  On the other hand, we see that for very low frequencies, distortion is extremely sensitive to non-linearity in the stiffness.  Stiffness distortion of 10% 3rd harmonic will result in 90% 3rd harmonic distortion of the sound.

These formulas look really great! When I get a chance I'll graph them and see what difference they make by frequency.

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TTS56A    3

Basically if sub is well decoupled from the ground, it can't directly trasmit vibration on the earth. Why do you think that very low frequencies can't be reflected?

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SME    211

It's for full space. Unless the ground (grass and dirt in this case) is a perfect reflector at the frequencies in question, these subs will not behave as if radiating in half-space. I hypothesize that much of the sound at 10Hz travels through the ground. We know that sometimes low frequencies can travel through earth (see: worm charming). I seem to be alone in this, but I think that the lowest frequencies travel through the ground and the higher frequencies begin to reflect.

 

Very low frequencies transmit very well through both the air and the ground.  However, this does not mean that low frequencies readily transmit *between* the air and the ground.  The degree to which energy is transmitted between materials is governed by the impedance difference.  Earth generally has much higher impedance than air, so most energy in the air stays in the air and most energy in the ground stays in the ground.

 

Furthermore, if a transition from full-space to half-space were occurring for lower frequencies, you would expect to see a 6 dB drop in the sub response in the vicinity of the transition.  So if the transition occurred at around 20 Hz where a typical sealed system is rolling off with ~12 dB/octave slope, the response slope in the region around 20 Hz would actually get even steeper before levelling out both above and below this transition area.  But if you look at the response measurements, the roll-off is very smooth.  It's only the max "clean" output that's reduced below 20 Hz.

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Contrasseur    17

Furthermore, if a transition from full-space to half-space were occurring for lower frequencies, you would expect to see a 6 dB drop in the sub response in the vicinity of the transition.  So if the transition occurred at around 20 Hz where a typical sealed system is rolling off with ~12 dB/octave slope, the response slope in the region around 20 Hz would actually get even steeper before levelling out both above and below this transition area.  But if you look at the response measurements, the roll-off is very smooth.  It's only the max "clean" output that's reduced below 20 Hz.

Ugh, yep, data doesn't support it. Must be one or the other. The question is, are they radiating in full space or half space?

 

Evidence for full space

-Highly linear LMS Ultra 5400 distorts at calculated ~38mm throughout the displacement limited region.

-LMS Ultra's excursion would only be 19 mm if it were half space.

 

Evidence for half space

-Most drivers produce clean output at impossibly high levels in the displacement limited region. Such SPLs can only be produced with excursions we know are beyond Xmech or by half-space radiation

-They're literally sitting on the ground during testing?

 

When I get home I'll model the enclosures and see whether measured sensitivities match up to whole space or half space radiation.

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SME    211

Ugh, yep, data doesn't support it. Must be one or the other. The question is, are they radiating in full space or half space?

 

Evidence for full space

-Highly linear LMS Ultra 5400 distorts at calculated ~38mm throughout the displacement limited region.

-LMS Ultra's excursion would only be 19 mm if it were half space.

 

Evidence for half space

-Most drivers produce clean output at impossibly high levels in the displacement limited region. Such SPLs can only be produced with excursions we know are beyond Xmech or by half-space radiation

-They're literally sitting on the ground during testing?

 

When I get home I'll model the enclosures and see whether measured sensitivities match up to whole space or half space radiation.

 

It's half-space.  Is the LMS-U really only getting to 19 mm half-space before failing CEA?  I'm thinking 38 mm may be closer to the truth.  You are aware that most piston excursion / SPL calculators compute half-space results, right?  That's how it's done because of full-range speakers where the radiation is closer to half-space above the major baffle step transition.  Also, I believe most sims give half-space results as well.  These are more generally useful.

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Ricci    652

It is half space with the SPL reported being referenced to 2m distance from the DUT. This approximates a 1m full space measurement. The actual measurements are all halfspace though. All cabs are on the ground as is the mic.

 

If you are trying to use simulations to match the data measured for various systems you will find virtually none of them match up exactly. Some require a large amount of fudging in the simulator via the inductance, BL and other parameters to make the curve match. TSP's and a basic simulator leave a lot to be desired as far as capturing the complex behavior of a system. As this discussion points out you cannot assume a pure reproduction of the waveform from the sub and use that assumption to calculate the driver excursion. The waveform being reproduced is anything but.

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Kvalsvoll    75

Just to clarify the half-space radiation question;

 

Half space is the correct radiation model, there is NO low frequency absorption in the ground.

 

So SME and Ricci are right, and that makes 3 of us.

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Kvalsvoll    75

Great thread, lots of good information and knowledge have been posted.

 

The thing about xmax is that this number is only one of many parameters of significance for what is actually coming out of the speaker.

 

When you find there is a mismatch between xmax numbers and calculated vs measured max spl output, that is to be expected, because the calculation model used does not include enough information, the model is not accurate enough.

 

First, if the driver suddenly stopped moving at xmax, there would still be 3dB spl for free due to the crest factor of the produced square wave compared to a pure sine wave.

 

But it does not stop at xmax, actually if the motor force drops at xmax the output will increase.

 

The force acting on the diaphragm is Bl x i , where Bl will start to drop significantly around xmax.

But i - current - depends on the electrical impedance, and when Bl drops the q increases and the impedance peak drops to a lower level, causing more current to be drawn from the amplifier.

Around resonance there will be more output than expected, and below resonance the output may drop because the current increases.

The mechanical properties (xmech) are very important for maximum usable displacement, this usable displacement can be more or less than the xmax number suggests.

 

A driver in a ported box behaves quite different, and for horns, like the ones I design, the xmax parameter alone does not say much about the performance of a driver.

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Contrasseur    17

Ok so I modeled some of these systems and used Josh's measured sensitivities. As Ricci mentioned before, everything's fucked.

 

These are small-signal parameters though. In the bottom of the bandwidth at low voltages, behavior should be dominated by Thiele-Small parameters and therefore quite predictable. Why are they so useless?

 

post-4069-0-82698400-1464903450_thumb.png

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Contrasseur    17

First, if the driver suddenly stopped moving at xmax, there would still be 3dB spl for free due to the crest factor of the produced square wave compared to a pure sine wave.

 

No it won't. These are CEA 2010 Max Burst SPL measurements. They only measure the portion of the total SPL produced by the fundamental. 

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Kvalsvoll    75

No it won't. These are CEA 2010 Max Burst SPL measurements. They only measure the portion of the total SPL produced by the fundamental. 

 

If you filter out everything except the fundamental from a square wave you end up with a sine wave with peak amplitude 3dB higher than the orignal waveform.

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Contrasseur    17

Hi Contrasseur,

 

I've got a formula and some thoughts (already mentioned) that might help you square some things away... The formula I use is from Acoustics by Leo Beranek: P(rms) = Square Root Of 2 * Pi * Frequency Squared * Rho * Sd * Excursion / Distance

 

P(rms) is recorded in Pascals

Rho is the density of air, which varies by things like temperature, humidity, elevation, and so on.

Sd is the radiating area of the driver in square meters

Excursion is also noted in meters, as is distance

 

Here are a couple examples of backdooring excursion through that via excel. With the LMS, that seems to track Nathan's comment re: 39mm of excursion, and Josh's <65mm comment for the T3S2-19. Rho is altered between the two to account for temperature differences on the day of measurement, and assumes a ~600' elevation, and 50% humidity (and yes, Pi goes to more than just 2 digits).

I've updated all of my calculations to include this formula for comparison. I consider this one to be more accurate because it accounts for temperature differences as well. Otherwise, they show very good agreement. The discrepancy is less than 5%, less than .5dB.

post-4069-0-39569900-1464906798_thumb.png

 

However, we're still seeing these SPLs which were MEASURED TO BE CLEAN at excursions we know are unrealistic.

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Contrasseur    17

It is half space with the SPL reported being referenced to 2m distance from the DUT. This approximates a 1m full space measurement. The actual measurements are all halfspace though. All cabs are on the ground as is the mic.

 

If you are trying to use simulations to match the data measured for various systems you will find virtually none of them match up exactly. Some require a large amount of fudging in the simulator via the inductance, BL and other parameters to make the curve match. TSP's and a basic simulator leave a lot to be desired as far as capturing the complex behavior of a system. As this discussion points out you cannot assume a pure reproduction of the waveform from the sub and use that assumption to calculate the driver excursion. The waveform being reproduced is anything but.

The waveform is actually quite close when we're only considering CEA 2010 passing results. Didn't you read post 22 and post 16?

 

Cone excursion is not only related to the generated SPLs, IT'S REQUIRED to produce ANY NOISE AT ALL. If it were possible to create more output from the same displacement, wouldn't someone have leveraged this? We all know "there's no replacement for displacement" because there is NO WAY AROUND it, right? Obviously ported boxes and horns can use acoustical impedance and resonance to leverage more spl per cone excursion, but sealed boxes have no such acoustic effects at frequencies below the baffle step like these.

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Funk Audio    38

No it won't. These are CEA 2010 Max Burst SPL measurements. They only measure the portion of the total SPL produced by the fundamental. 

I think what your missing is that the cone does not have to move as far to produce the fundamental at a given output if it contains the right proportions of distortion components, which is enough under the passing "clean" CEA2010 limits to reduce actual cone travel significantly as per my example posted before. What your calculating from the output is the "effective" displacement the driver has at that frequency with passing distortion levels, and in the end isn't that what really matters to us, if the "effective" displacement is higher than xmax or even xmech does that really matter, that's why this testing is done and so valuable as it seems your starting to see "sims" are highly inaccurate in many cases.

 

Also don't forget what I said first most motor designs will have more linear xmax than their "static" rating near resonance, so as long as the mechanical parts can take it you will get much more travel around resonance. Drivers with high mechanical clearances past xmax will show a more pronounced difference, drivers with basically none like the LMS will show less difference.

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SME    211

I have edited my post on distortion from stiffness non-linearity, fixing errors and re-writing the algebra and interpretations.  I hope to maybe do some illustrative plots soon, but for now, I want to present the results on my analysis of magnetic flux (B ) non-linearity.  This analysis assumes B only varies with displacement and ignores variation due to current (i.e. flux modulation distortion).

 

The model I used was:

 

    B = B1*(1 - B3*X^2)

 

This shape is plotted below for various B3 values:

 

post-1549-0-56858600-1464934786_thumb.png

 

Whereas system stiffness usually increases at the extremes of excursion, magnetic flux density usually drops.  The parameter B3 defines by what fraction B is decreased at peak displacement, where we define X = +/-1.  The results follow:

 

    om0 = 2*pi*fb = sqrt(k/m)

    om = 2*pi*f

 

    |H3| = 9 * B3 * sqrt{ (om0^2 - om^2)^2 + om^2*[om0/Qtc + 3*Le/R*(om0^2 - om^2)]^2 } / sqrt{ (om0^2 - 9*om^2)^2 + om^2*[3*om0/Qtc + 3*Le/R*(om0^2 - 9*om^2)]^2 }

 

We can look at the same set of limiting cases that we looked at for H3 due to stiffness in my previous post.  First up, if we can assume zero inductance, the above simplifies to:

 

    |H3| = 9 * B3 * sqrt{ (om0^2 - om^2)^2 + om^2*om0^2/Qtc^2 } / sqrt{ (om0^2 - 9*om^2)^2 + 9*om^2*om0^2/Qtc^2 }

 

Now let's look at the limit for very low frequency, where f is much less than 1/3 of the box resonance:

 

    |H3| = 9 * B3 * sqrt{ om0^2 + om^2/Qtc^2 } / sqrt{ om0^2 + 9*om^2/Qtc^2 }    (assuming no inductance and frequency well below 1/3 of the box resonance)

    |H3| = 9 * B3    (assuming no inductance and frequency well below 1/3 of the box resonance and well below 1/3 of the box resonance times Qtc)

 

Here we see a result analogous to what we saw for stiffness non-linearity.  In the low frequency limit, both magnetic flux and stiffness linearity are equally important for third harmonic distortion.

 

Now let's look at the high frequency limits:

 

    |H3| = 3 * B3 * sqrt{ om^2 + [om0/Qtc - 3*Le/R*om^2]^2 } / sqrt{ 9*om^2 + [om0/Qtc - 9*Le/R*om^2]^2 }    (assume frequency well above resonance)

    |H3| = B3    (assume frequency well above resonance, fully developed inductance, and frequency well above 1/3 of resonance divided by Qtc)

    |H3| = 3 * B3 * sqrt{ om^2 + om0^2/Qtc^2 } / sqrt{ 9*om^2 + om0^2/Qtc^2 }    (assume frequency well above resonance and no inductance)

    |H3| = B3    (assume frequency well above resonance, no inductance, and assume frequency well above 1/3 of resonance divided by Qtc)

 

This indicates that 3rd harmonic distortion in the high frequency limit *is* sensitive to magnetic flux linearity, but the sensitivity is not nearly as much as for the very low frequencies.

 

Now let's look at what happens at resonance.  The result here is very interesting:

 

    |H3| = 9 * B3 / sqrt{ 8*Qtc^2 + [3 - 24*Qtc*Le/R*om0]^2 }    (assume frequency at resonance)

    |H3| = 9 * B3 / sqrt{ 8*Qtc^2 + 9 }    (assume frequency at resonance and no inductance)

 

We can calculate third harmonic distortion for different Qtcs at resonance:

 

    |H3| = 2.50*B3    (assume frequency at resonance, no inductance, and Qtc=0.25)

    |H3| = 1.80*B3    (assume frequency at resonance, no inductance, and Qtc=0.5)

    |H3| = 1.41*B3    (assume frequency at resonance, no inductance, and Qtc=0.707)

    |H3| = 1.05*B3    (assume frequency at resonance, no inductance, and Qtc=1.0)

 

Here again, we can see that lower Qtc reduces third harmonic distortion at resonance, this time for magnetic flux non-linearity.  A particularly interesting result is that for most typical Qtc alignments, the third harmonic distortion due to B non-linearity at resonance is *lower* at resonance than in the high frequency limit.  This combined with the result for stiffness non-linearity predicts that for a reasonably low Qtc, distortion will be lower at resonance than for surrounding frequencies.

 

Edit: I got the above formulas wrong the first time.  The inverse is actually true.  At resonance, higher Qtc alignments are better for 3rd harmonic distortion due to B non-linearity.  This actually makes more intuitive sense.  With high Qtc resonance, the motor force is very low.  The driver is basically moving without any help from the motor at all.  Also, my previous conclusion that this model predicts lower distortion at the resonance than the surrounding frequencies is only accurate for Qtc > 1 or so.  To the extent that a distortion minimum at resonance is observed in the real world, this model does not account for that.  A B(i) non-linear model may better account for these situations.

 

Lastly, let's look at distortion at 1/3 the resonance frequency:

 

    |H3| = B3 * sqrt{ 64*Qtc^2 + ([3 + 8*om0*Le/R*Qtc]^2 }    (at 1/3 of resonance frequency)

    |H3| = B3 * sqrt{ 64*Qtc^2 + 9 }    (at 1/3 of resonance frequency and no inductance)

    |H3| = 3.60 * B3    (at 1/3 of resonance frequency, no inductance, and Qtc = 0.25)

    |H3| = 5.0 * B3    (at 1/3 of resonance frequency, no inductance, and Qtc = 0.5)

    |H3| = 6.40 * B3    (at 1/3 of resonance frequency, no inductance, and Qtc = 0.707)

    |H3| = 8.54 * B3    (at 1/3 of resonance frequency, no inductance, and Qtc = 1.0)

 

This is a remarkable result.  It indicates that third harmonic distortion due to magnetic flux non-linearity is always amplified by the resonance, but here as well as before, lower Qtc gives superior results.

 

Edit: Corrections have been made here.  For high Qtc, the distortion can peak near the 1/3rd resonance, but for lower Qtc alignments, distortion here is down here relative to the low frequency limit.  Where as high Qtc would appear to benefit distortion performance near the resonance, low Qtc benefits at 1/3rd down.

 

Edit: Here is a plot that characterizes the third harmonic distortion behavior in this model for B for different Qtc alignments.  These assume no inductance for simplicity.  The x-axis is what might be called reduced frequency.  It's the ratio of the frequency to the box resonance frequency.  The y-axis is the factor that you multiply B3 by to determine the level of the distortion harmonic relative to the fundamental.

 

post-1549-0-67946000-1464936204_thumb.png

 

Edit: A similar caveat applies here as does for the plot for k non-linearity.  If you take a driver and build sealed systems with different Qtc values, the fbs change also.  This has the effect of shifting the absolute frequency scale relative to the x-axis for each curve, thus visually shifting the curves relative to one another.  Depending on the situation, it's possible that the curves could overlap in places.

 

The reason the curves are presented this way is because it makes the plot more generally useful.  If an absolute frequency scale were to be used, you'd also have to fix the resonance frequency or some other parameters to do the plot, making it less generally useful.

 

Please consider these results tentative and use at your own risk.  There was a lot of opportunity for error on my part here, so I may have to go back and revise.  At least now that I have these formulas in terms of fewer variables, I think I can maybe prepare a few example plots to help visualize these formulae.

 

Edit: Updates/corrections are on-going.

Edited by SME
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SME    211

For what it's worth (for a sealed system), if the motor force and stiffness are non-linear with respect to stroke (ignoring non-linearity with respect to current or temperature) and if the non-linear variation is smooth (i.e. no abrupt changes), then there is neither amplification nor compression of the fundamental tone when a sine signal is input.  This is the case with any woofer at and beyond Xmax.  This fails to be the case once excursion reaches Xmech because by definition, the variation is no longer smooth.

 

Something someone might want to try doing is computing compression for the CEA results for sealed systems.  This can be done with the data Ricci makes available, at least for the DIY builds for which the voltages are known.  I would l bet that the CEA results show very little compression?  Why?  Because they are burst measurements and they don't heat up the coil much.  Coil heating is the primary main cause of power compression.  That's not to say there aren't other mechanisms.  For example, I believe damping in the driver suspension may have have some non-linear effects on the response of the fundamental.

 

Ported systems on the other hand have port compression to contend with.

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Contrasseur    17

I think what your missing is that the cone does not have to move as far to produce the fundamental at a given output if it contains the right proportions of distortion components, which is enough under the passing "clean" CEA2010 limits to reduce actual cone travel significantly as per my example posted before. What your calculating from the output is the "effective" displacement the driver has at that frequency with passing distortion levels, and in the end isn't that what really matters to us, if the "effective" displacement is higher than xmax or even xmech does that really matter, that's why this testing is done and so valuable as it seems your starting to see "sims" are highly inaccurate in many cases.

 

Also don't forget what I said first most motor designs will have more linear xmax than their "static" rating near resonance, so as long as the mechanical parts can take it you will get much more travel around resonance. Drivers with high mechanical clearances past xmax will show a more pronounced difference, drivers with basically none like the LMS will show less difference.

Thanks, this is starting to make a lot more sense.

 

I get what's happening, just not exactly why. I think when I get the time to sit down with those equations that SME posted I'll have a lot more good data to work with.

 

That Bl*I variation Kvalsvoll makes a lot of sense too. 

 

Does anyone have Klippel data on any of these drivers? That might offer some clues too.

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Ricci    652

There is some compression in the burst tests as well.A large amount of it in some cases.

 

SME your math skills are impressive. I think you'll find that the data shows exactly what your formula appears to postulate.

 

Contrasseur I'm not disputing that displacement is required to produce the signal at the SPL recorded. That's simple. However the waveform produced by the system is not pure by any means and in some cases is grossly distorted despite receiving a passing grade. Harmonic distortion is not the only form of distortion to the waveform. The simple set of TSP specs commonly used for modeling is inadequate or incomplete to describe the behavior of many bass drivers even at small signals. Complex inductance effects must be included. Unfortunately virtually no MFG supply this info and very few simulators could use the info if it was available. This is another deep rabbit hole to go down.

I did get a few images put together of the captured waveform of some of the systems. Hopefully I'll get time to post them tomorrow. This is a good discussion I just haven't had time to participate much.

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