Speaker acceleration, displacement and SPL

[radulescu_paul_mircea]

So, some time ago I said that the driver excursion is a secondary effect of the acceleration of the diaphragm but it is the acceleration that implies the sound pressure level.

I read this in a blog post of Geoff Martin from Bang and Olufsen when he said that one graph he made isn't real because in it the SPL was following the driver excursion but in fact it should follow the drivers acceleration.

I am not very good with math and for now I only have basic knowledge in this but this is why I would really like to discuss it with you guys

[radulescu_paul_mircea]

So what is important for me is what is happening when there isn't sufficient displacement capability for achieving the pressure levels some of us are getting in our measurements but we have enough motor force and amplifier power to generate the acceleration.

One problem I had was when I measured a Martin Audio MLX sub with a B&K sound level meter put on 17 ms fast averaging at 4 meters and I got 137 dB peak from a single sub. That would imply the drivers were moving around 74 mm p-p at the test tone of 42 hz which I think isn't possible. So the I thought it might be an explanation in the fact that the amp and the driver motor could make the diaphragm to accelerate enough to create that pressure but not for the entire quarter of the cycle but only at the beginning of the stroke. But that seems to me that the driver is not generating a sine wave anymore that would mean mechanical clipping the signal .IDK...

[Ricci]

You can simulate velocity and accelerative forces and how they relate to driver excursion and other behaviors using HR, Akabak or ABEC3.

Fun fact...I've seen a lot of people over the years saying that higher xmax subs sound slower and short xmax pro audio subs sound faster. With enough power a super excursion sub like a XXX will move its cone at a greater speed with higher acceleration.

 

Do you have a way to look at the captured waveform itself in comparison to the original drive signal? What was the drive signal? You may to some extent be right that the peaks are compressed or clipped some. This does happen but significant clipping off of the waveform peaks results in very nasty sounds and extremely high distortion levels. It's not something that would go unnoticed to the ear of most. Compression or flattening of the peaks such as being heavily into a limiter is less overtly nasty.

 

Here are some examples. Some are better than others. Some of these are extremely distorted. Note that these are considered a passing result or very close in a few cases as far as the original CEA-2010 distortion limits are concerned. Some of these have relatively low harmonic distortion but extremely modified waveform shapes. Some of these are quite close to the original wave shape and others are far from it. I put this together for a post a long time ago but don't think I ever posted it. I can't remember which subs these were anymore but hopefully I have it written down somewhere. I do know that you might be surprised because it was a cross section of alignments and types of subs.

 

 

 

[andy497]

The 31.5 hz sweeps where max output continues for a cycle or more after input stops just boggles my mind.  Crazy.  And it's interesting that this type of "distortion" slips right by CEA-2010 testing.  I guess you'd see that as ringing in a waterfall graph?

[Kyle]

So, some time ago I said that the driver excursion is a secondary effect of the acceleration of the diaphragm but it is the acceleration that implies the sound pressure level.

I read this in a blog post of Geoff Martin from Bang and Olufsen when he said that one graph he made isn't real because in it the SPL was following the driver excursion but in fact it should follow the drivers acceleration.

I am not very good with math and for now I only have basic knowledge in this but this is why I would really like to discuss it with you guys

 

Driver excursion can tell you SPL if you know the frequency too which gives you the time component (t) which is needed to solve the SPL equation. (that and also the  size of the cone).

[SME]

So, some time ago I said that the driver excursion is a secondary effect of the acceleration of the diaphragm but it is the acceleration that implies the sound pressure level.

I read this in a blog post of Geoff Martin from Bang and Olufsen when he said that one graph he made isn't real because in it the SPL was following the driver excursion but in fact it should follow the drivers acceleration.

I am not very good with math and for now I only have basic knowledge in this but this is why I would really like to discuss it with you guys

 

You are correct that SPL relates more directly to acceleration than displacement, at least as long as you are measuring anechoic or ground plane.  When you go indoors, you get room gain which allows higher SPL with less acceleration and displacement.

 

The mathematical relationship between acceleration and displacement is very direct.  It's easiest to describe with calculus: the cone velocity is the first derivative of the displacement with respect to time (dX/dt).  The acceleration is the second derivative of the displacement (d^2X/dt).

 

For sine wave signals, velocity v = dX/dt = j*om*X.  The value om = 2*pi*f, where f is the frequency.   The j here is the imaginary number sqrt(-1) and this essentially tells you that the velocity is 90 degrees phase shifted, which should make intuitive sense if you think about it.

 

For sine wave signals, acceleration a = d^2X/dt^2 = -om^2*X.  There is no j here, but there is a minus sign, indicating that the acceleration is 180 degrees out of phase with the displacement.  This also basically tells you for a given sub, you will always need 4X the displacement every time you half the frequency to achieve the same acceleration, which you need to maintain the same SPL in an anechoic or ground-plane environment.

 

So what is important for me is what is happening when there isn't sufficient displacement capability for achieving the pressure levels some of us are getting in our measurements but we have enough motor force and amplifier power to generate the acceleration.

 

This is kind of a complicated question.

 

First, the formulas I gave are for continuous sine wave signals.  A burst signal is not a continuous sine wave.  It contains additional higher frequency content that is necessary to make the signal start and stop playing.  So it's entirely possible to read higher peak SPL values during the start or stop part of the signal, depending on what's going on.  If you want to analyze output with CEA burst type signals, you really need to look at the level of the fundamental as opposed to the overall SPL.

 

Second, you have to define "sufficient displacement" more carefully.  Displacement may be hard limited or soft limited.  If hard limited, then things are bashing into each other and you definitely won't be able to ever exceed that limit without breaking something.  If you are soft limited, on the other hand, then by definition there is no precise point at which you are limited.  This could happen because some turns in the coil are leaving the magnetic gap, in which case, motor force will be reduced at the extremes of cone travel.  This could also happen because the suspension stiffness is higher at the limits of its travel, in which case, restoring force will be stronger than it would be (if linear) at the extremes of cone travel.  How this impacts actual measured distortion is a bit complicated.

 

Ultimately, the cone moves back and forth because of the oscillating forces acting upon it.  If either of those forces is distorted, than the output will be distorted.  However, the relative importance of these forces to the motion of the cone depends a lot on the frequency being tested relative to the acoustical parameters of the system.  For example, the restoring force of the suspension has a stronger impact on lower frequencies than higher frequencies.  The motor force is most relevant at low and high frequencies but not so much at resonance.  And then there is the simple fact that the relationship between displacement and acceleration amplifies different frequencies (e.g., the fundamental vs. distortion harmonics) differently.

 

The details get complicated, and I have made several posts around this forum on this subject already.  At some point, I may do a detailed write-up of the qualitative mechanical behavior of a driver in a sealed box system, but it'll probably be a while before I get around to it.

[radulescu_paul_mircea]

Very good answer SME, thank you for the time. I am very bad with math for now. I am studying for a masters degree in Acoustics and Vibrations and Noise Control. It's my first year now so even though I get most of the things, the math is killing me.

For now I am trying to deduce the theory in my mind using visualization. This lead me to the graphs attached bellow but for now, I still can't find what is wrong in my representation...or in fact how to put the pressure graph. Because if it is a function of acceleration it cannot be because it would be also a function of displacement and at least for now, I know it shouldn't be.

Ricci, I don't have access to the subwoofer anymore and the sound pressure measurements we took were with a hand held (very expensive) device in free field on a beach so I don't have any data except my memory and experience..for the measurements I took I used sine waves, CEA bursts and music. The results Vere very similar on the peak hold, where I got those huge values keeping the distortions from getting to strong. We tried to use them as they would be used in real life.

Still, in the future I will have more and more access to study things and for start I will try to purchase a better soundcard and mic and to calibrate them to the meters from the university.

[radulescu_paul_mircea]

Here is an animation that tells me I possibly got it all wrong..it could be that in fact the pressure increases in tandem with the piston speed and the particle velocity is the same. But the acceleration is in tandem with the displacement except is phase shifted 180°

http://resource.isvr.soton.ac.uk/spcg/tutorial/tutorial/Tutorial_files/Web-basics-nature.htm

[SME]

Very good answer SME, thank you for the time. I am very bad with math for now. I am studying for a masters degree in Acoustics and Vibrations and Noise Control. It's my first year now so even though I get most of the things, the math is killing me.

For now I am trying to deduce the theory in my mind using visualization. This lead me to the graphs attached bellow but for now, I still can't find what is wrong in my representation...or in fact how to put the pressure graph. Because if it is a function of acceleration it cannot be because it would be also a function of displacement and at least for now, I know it shouldn't be.

 

Pressure, displacement, velocity, and acceleration are all inter-related quantities that vary with time.  If the parameters of the sub system and environment are known, it is possible in principle to describe any one as a function of another.

 

In your picture, the pressure and acceleration curves should look essentially the same, but the displacement curve should be inverted.  If you were to draw a cone velocity curve, it should be shifted so that it crosses zero at the same time that the other curves hit their peaks and valleys, and it should hit its peaks and valleys when the other curves cross zero.  Cone velocity is always zero at the extremes of motion.  Think about it.  In order for the cone to change from moving in to moving out or vice verse, it *must* hit a it must go through zero velocity in between.  Those velocity zero points are at the extremes of displacement.

 

Some important things to keep in mind:  Pressure depends on cone acceleration, cone surface area, and measuring conditions.  The relationship between pressure and cone acceleration does not depend on frequency.  However, frequency is a key quantity with regard to the relationships between acceleration, velocity, and displacement.  If you invert frequency, you get a quantity with units of time.  This time may be called the period, and describes how long it takes for a cycle to complete.  Acceleration is a quantity that describes how quickly velocity is changing with respect to time.  So you see that for lower frequencies that have longer time periods, the cone velocity will be higher for the same acceleration and therefore the same pressure.  Velocity itself is a quantity that describes how quickly the displacement is changing with respect to time.  And as before, the lower the frequency, the longer the period and the further the cone moves for the same peak and RMS velocity.

 

Low frequencies need drivers with lots of displacement because they need to be able to sustain high acceleration for a lot longer in each back-and-forth motion than high frequencies do.

 

It's very cool to hear about your masters study.  Best of luck with that!

[SME]

Here is an animation that tells me I possibly got it all wrong..it could be that in fact the pressure increases in tandem with the piston speed and the particle velocity is the same. But the acceleration is in tandem with the displacement except is phase shifted 180°

http://resource.isvr.soton.ac.uk/spcg/tutorial/tutorial/Tutorial_files/Web-basics-nature.htm

 

The cone velocity and particle velocity of the air at the cone boundary will be phase shifted 90 degrees relative to the acceleration, displacement, and pressure.  This means it will cross zero at the extrema of the others, and vice verse.  If the speaker is in an I.B. / ground plane or anechoic environment then at some distance, depending on the frequency, the phase of the pressure and particle velocity will realign and the wave will essentially become a plane wave.  (The situation indoors is a lot uglier.)

[LTD02]

...

Fun fact...I've seen a lot of people over the years saying that higher xmax subs sound slower and short xmax pro audio subs sound faster. With enough power a super excursion sub like a XXX will move its cone at a greater speed with higher acceleration.

...

 

 

 

 

as you know...

higher xmax subs tend to have a reduced top end frequency response in part because of inductance effects and in part because of increased moving mass.

lower xmax subs tend to have an increasing top end frequency response.

 

so without equalization, pro audio drivers will tend to sound faster at some average output level because...they are. 

[Ricci]

as you know...

higher xmax subs tend to have a reduced top end frequency response in part because of inductance effects and in part because of increased moving mass.

lower xmax subs tend to have an increasing top end frequency response.

 

so without equalization, pro audio drivers will tend to sound faster at some average output level because...they are. 

 

Right that's why they sound "faster" but that does not mean they are, which was my point. You seem to be talking about frequency response or the speed at which the cone is changing direction. I am talking about the maximum velocity that the moving assembly can reach which are two different things. I'm only pointing out that it is funny that technically, the cone of a high inductance, long throw, heavy mass, low bl driver can reach just as high or even higher maximum velocity than a high BL, light mms, low inductance pro driver. It doesn't have anything to do with explaining subjective impressions like "speed", "more punchy" and "faster", but does say something about the accuracy of those terms.

 

If you look at the potential velocity limits of the voice coil / moving assembly based on the measurements of various systems tested here, I think you'll see what I mean.

[SME]

Right that's why they sound "faster" but that does not mean they are, which was my point. You seem to be talking about frequency response or the speed at which the cone is changing direction. I am talking about the maximum velocity that the moving assembly can reach which are two different things. I'm only pointing out that it is funny that technically, the cone of a high inductance, long throw, heavy mass, low bl driver can reach just as high or even higher maximum velocity than a high BL, light mms, low inductance pro driver. It doesn't have anything to do with explaining subjective impressions like "speed", "more punchy" and "faster", but does say something about the accuracy of those terms.

 

If you look at the potential velocity limits of the voice coil / moving assembly based on the measurements of various systems tested here, I think you'll see what I mean.

 

If you completely ignore power handling and amp limits, then the "fastest" woofer will be the one with the *greatest* Xmax because for linear operation at any given frequency, cone velocity will be proportional to excursion.  But really, this discussion is getting silly.  At least I think we can agree that literal woofer speed isn't really a metric we should worry ourselves about. 

 

as you know...

higher xmax subs tend to have a reduced top end frequency response in part because of inductance effects and in part because of increased moving mass.

lower xmax subs tend to have an increasing top end frequency response.

 

so without equalization, pro audio drivers will tend to sound faster at some average output level because...they are. 

 

To the extent this correlation may be observed in the real world, it probably reflects design priorities rather than any physical limitations of a high excursion system.  A driver with high Xmax is usually designed for deep bass, and therefore, the designer may purposely choose to increase moving mass and give less attention to inductance in order to save money on a product that otherwise serves its intended purpose very well.

 

If you want a driver with high Xmax and low Mms and low inductance, such drivers are definitely out there if you look, but you should expect to pay more.

[Kyle]

An aside. Inductance causes 2 problems.

It fights the change in current so it directly reduces the amount of current. e.g. force (B*L*I) that is applied during a period. This becomes more problematic when changes happen at higher frequencies, e.g. higher frequencies

That's why the L/R time constant is valuable because you can see how much relative time the inductance takes to reach 0. This is approximately 5 times the L/R time constant http://www.google.com/search?q=l%2Fr+time+constant&ie=utf-8&oe=utf-8#q=inductance+l%2Fr+time+constant. For example, let use my old favorite LMS-5400 which has an L/R of 0.94ms or about 1/1000th of a second. @ 20Hz the current changes direction twice in a cycle so every 25ms or Pi/180. The time constant * 5 = ~5ms for the big LMS. So the 20Hz cycle is largely unaffected by inductance, But now lets try speeding that up 10 times. @ 200Hz the current changes direction ever 2.5ms. The LMS-5400 never fully discharges its inductor and this SPL is affect by the inductance value to some signification amount.


That is probably the biggest drawback but there is a more enigmatic effect with respect to distortion. Because the coil moves and because steel is a lossy inductor. The inductance is non-linear and distorted. This causes some more complex issues with respect to distortion that probably required FEA to understand better. copper or aluminum around the inside of the voice coil greatly helps reduce the magnetic energy of the inductor by virtue of stealing the current and shorting it out rather than having it stored as magnetic energy in the steel inside the motor, but this distortion is hard to quantify and isolate without computer modeling or kick ass measurement equipment like a klippel.

 

[Ricci]

If you completely ignore power handling and amp limits, then the "fastest" woofer will be the one with the *greatest* Xmax because for linear operation at any given frequency, cone velocity will be proportional to excursion.  But really, this discussion is getting silly.  At least I think we can agree that literal woofer speed isn't really a metric we should worry ourselves about. 

 

 

 

Exactly.

 

Wasn't trying to have or start a discussion. It was just a general comment on something that I find amusing when considered against commonly held beliefs.

[radulescu_paul_mircea]

copper or aluminum around the inside of the voice coil greatly helps reduce the magnetic energy of the inductor by virtue of stealing the current and shorting it out rather than having it stored as magnetic energy in the steel inside the motor, but this distort that is hard to quantify and isolate without computer modeling or kick ass measurement equipment like a klippel.

So that is why most of the time I observed that drivers with shorting rings have the magnet system warmer, as a trend. The ring is shorting the current that forms inside and turns it into heat. I knew that but I never thought about it...

Another reason to love the M-force system

[Kyle]

So that is why most of the time I observed that drivers with shorting rings have the magnet system warmer, as a trend. The ring is shorting the current that forms inside and turns it into heat. I knew that but I never thought about it...

Another reason to love the M-force system

 

I think you're exactly right, the energy is just turning into more heat instead of magnetic energy.

[LTD02]

Right that's why they sound "faster" but that does not mean they are, which was my point. You seem to be talking about frequency response or the speed at which the cone is changing direction. I am talking about the maximum velocity that the moving assembly can reach which are two different things. I'm only pointing out that it is funny that technically, the cone of a high inductance, long throw, heavy mass, low bl driver can reach just as high or even higher maximum velocity than a high BL, light mms, low inductance pro driver. It doesn't have anything to do with explaining subjective impressions like "speed", "more punchy" and "faster", but does say something about the accuracy of those terms.

 

If you look at the potential velocity limits of the voice coil / moving assembly based on the measurements of various systems tested here, I think you'll see what I mean.

 

i was highlighting the other half of the irony, where if the spl in the higher frequencies for a given average drive level is higher for the light driver, then the velocity is higher too, so the non-engineer who says that it "sounds faster" is actually right, at least in some sense. 

[radulescu_paul_mircea]

I managed to play with my Xoc1 sub and I used the Bruel & Kjaer 2270 hand held device. I don't have the manual for the device but according to my professor, the LCpeak measurement is measuring the peak pressure level even if it is just a single cycle at 20000 hz so 0,05 ms averaging. I will have to verify this claim as it seems a improbable (even though it costs 7400€ including calibration and taxes ).

The amp was a clone in bridge on a single sub free space with objects like walls and fences at 12-15 meters. dB SPLC sine at 46 hz averaged 119 dB at 4 meters. But the LC Peak was sometimes 128 dB at that distance... what was it measuring?

[radulescu_paul_mircea]

Update: the Lcpeak is the peak pressure level without any time constant. Also I have a set named Lce that is sound exposure level and it is showing almost the same values. I noticed that when I put the sub on a single channel the RMS sine SPL remains almost unchanged, but the Lcpeak goes lower with 3-5 dB

[radulescu_paul_mircea]

I have found a series of animations on another site to simply visualize the relations between displacement, acceleration and velocity so it is clear for me now that I don't know what is the reason I get those high LCpeak measurements )

http://www.acoustics.salford.ac.uk/feschools/waves/shm.php#velocity