Just when I thought I understood vented speaker impedance!

[witwald]

There are three unique frequencies in the impedance curve where the phase response passes through zero degrees indicating a likely resonance. This happens at the impedance minimum where cone movement is minimum, and again at the two impedance peaks where cone motion is maximum due to the back emf like you clearly outline.

Aren't the impedance peaks close to, but not quite at the point where the phase response passes through zero?

 

[Rick Sykora]

Kimmo's circuit has Reb in series with Lceb. Benson's circuit has it in parallel. They'll have somewhat different effects, won't they? So, they are similar yet distinctly different.

Yes, thanks for pointing this out.

Deleted as was more work to reword. Will revisit later.

 

[2020]

I know absolutely nothing. If I had to engineer speakers

"**** it! Seal the ***** and ship it with a big ass power supply!"

 

[Rick Sykora]

I like your analysis Tommy. I kept coming back to what causes what. There are three unique frequencies in the impedance curve where the phase response passes through zero degrees indicating a likely resonance. This happens at the impedance minimum where cone movement is minimum, and again at the two impedance peaks where cone motion is maximum due to the back emf like you clearly outline.

I think where Rick was going with this is ... what factors in the system are dominant at those particular frequencies and therefore responsible for the local maximums in cone velocity? I absolutely agree with you that it's a large continuous system and it is maybe unfair to try and isolate certain components without regard to how they interact with the other components (stiffness, mass, resistance, inductance, enclosure, port, etc).

This exploration feels similar to analyzing the impedance chart of a woofer playing in free air and being able to clearly see that the resonant frequency is easily predicted by a combination of the two dominant mechanisms at that frequency: suspension stiffness and mass. There are certainly other components in the system like inductance, voice coil resistance, magnetics, eddy currents, heat transfer, and so on .... but those other components are insignificant at the mechanical resonant frequency that they can be minimized or simplified in order to better understand what is happening *right there*.

I think this is what Rick is trying to do, to explore what is happening *right there* at the lower and upper impedance peaks.
At least, this is what I'm trying to do haha.


edit: oh and welcome to the forum! this looks like your first post

Yes exactly, while I get back emf is the electrical reason for the peaks, it does not help further one’s understanding of what the measurements mean. While the Bassbox definitions could be improved, I think they are directionally useful.

 

[Justin Zazzi]

Aren't the impedance peaks close to, but not quite at the point where the phase response passes through zero?

Yep, good catch! The impedance peaks are just slightly higher frequency than the phase zero crossings. I believe this is because the peak velocity is at peak impedance, but the peak displacement is at zero phase. I'm mostly sure, it's been a while since that class!

Looking at my class notes, one of the definitions of resonance is when the mechanical system is at peak velocity (the peak emf and thus peak impedance). That matches nicely.

 

[witwald]

The impedance peaks are just slightly higher frequency than the phase zero crossings. I believe this is because the peak velocity is at peak impedance, but the peak displacement is at zero phase.

Analysis of dynamical systems is somewhat tricky. For a sinusoidal signal, the peak velocity occurs at the same frequency as the peak displacement, as velocity is the time derivative of the displacement signal. The two are displaced 90° in phase. For example, if the displacement is represented by a sine wave, x(t) = sin(ωt), then the velocity is represented by a cosine wave scaled by the circular frequency of the oscillation, v(t) = ω·cos(ωt).

The slight shift in the position of the 0° phase point is due to the damping in the system, which causes the impedance peak to be slightly higher in frequency. For the first peak, the phase is –8.5°, while for the second peak the phase is –9.2°.

From these results, it appears that Kimmo's model might be incorrect. It seems to change the phase relationship between the impedance peak and the 0° phase point. As it contradicts what I've seen Thiele, Small, and Benson use, I would treat it with caution before relying on it. Producing an impedance curve is one thing, but getting it to match the physical system is what we really need the model to do. I think that it's relatively easy to create a circuit that doesn't represent the physical system.

 

[witwald]

I'd like to offer up an example of the application of a model of the impedance of a vented box loudspeaker. In the plot shown below, the measured vented-box impedance of an actual woofer is represented by the dashed line, and the simulated impedance using initial estimates of the component values in the equivalent circuit is represented by the solid line.

After performing a nonlinear curve fit, using a modified Marquardt nonlinear optimization code, we obtain the results shown below. The match between the measured vented-box impedance of the woofer (dashed line) and the simulated impedance using the optimized estimates of equivalent circuit component values (solid line) is very good. The magnitude and phase response are both very well represented by the circuit model. The behavior around the three 0° phase-response zero crossing points has been correctly captured by this particular model, which is comforting.

The equivalent circuit that was used here was simply the one from Small's journal paper:

The measured vented-box impedance data was kindly provided by Jim Verrenkamp of Nova Sound Pty Ltd, Melbourne, Australia.

 

[Head_Unit]

This inductor is there to represent the voice-coil inductance, but was deliberately neglected in Small's model because it has negligible effect at low frequencies

Not really was my experience. Of course the voice coil has more like a semi-inductance, but some woofers I measured back in the day would roll off significantly above even 100 Hz. (Begging the question: are those "low frequencies"?). In LEAP, the simple "inductance" parameter would yield simulations not rolling off much until higher frequencies, but if I input Chris Strahm's complex parameters then the simulations would look pretty close to what I would measure at the near-field surface of the woofer. R.I.P. Chris, you literally transformed my work experience (Doug Rife also).

I'm pretty sure Small himself (and Thiele and you go! @witwald also Benson) would say that those models while enormously useful are limited and not completely representing actual speakers. I actually met Dick Small once on a business trip and he was a very nice guy, even after I couldn't resist teasing him a bit about him "making my amplifier twice as powerful"* ha ha.


*Ad for KEF loudspeakers, when they were using enormous conjugate networks to make the speakers' impedance more resistive. The ironic part is, as The Power Cube showed later, with many amplifiers this advertising slogan was rather correct.

 

[witwald]

Of course the voice coil has more like a semi-inductance, but some woofers I measured back in the day would roll off significantly above even 100 Hz.

What do you mean when you say "roll off significantly above even 100 Hz"? Can I presume that you are referring to the sound pressure response?

Although I agree that some woofers have a large voice-coil inductance that significantly affects their output at frequencies above 100 Hz, the effect on impedance has a smaller bearing on the type of impedance modeling that we have been discussing here. Some of those woofers, back in their day, might have been pro sound woofers from Electrovoice and JBL, where high sensitivity was an important design goal, including those destined for guitar amps.

In any case, it's easy enough to include the inductance in the circuit model.

(Begging the question: are those "low frequencies"?).

I suppose that they are, to some degree, but once 150 Hz is exceeded the "low" descriptor might be less applicable.

In LEAP, the simple "inductance" parameter would yield simulations not rolling off much until higher frequencies, but if I input Chris Strahm's complex parameters then the simulations would look pretty close to what I would measure at the near-field surface of the woofer. R.I.P. Chris, you literally transformed my work experience (Doug Rife also).

I must admit I am a bit confused by that. The "inductance" just serves to filter higher frequencies by virtue of its increase in impedance. Of course, if the drivers are 4-ohm ones, then the effect of the inductance will be much greater. Strahm's models of inductance using complex parameters were accurate representations of the nonlinearity of the "inductance" effects, and a number of similar approaches have been identified. They all serve to try to represent the magnitude and phase of the impedance rise due to the "inductance", as correct knowledge of the complex impedance is critical for accurate crossover network modeling and design.

I'm pretty sure Small himself (and Thiele and you go! @witwald also Benson) would say that those models while enormously useful are limited and not completely representing actual speakers.

I think that [the enormous usefulness of these equivalent circuit models] tends to go without saying. The equivalent circuit models are not perfect by any means, and try to capture the main essence of what is occurring. I think that they do this very well. Keep in mind that, when designing a loudspeaker driver, the aim is to create a device that behaves linearly. The simple models are likely to work better and better the closer one gets to that ideal.

[...] Edit to make clearer what was meant.

 

[Tommythecat]

Yes exactly, while I get back emf is the electrical reason for the peaks, it does not help further one’s understanding of what the measurements mean. While the Bassbox definitions could be improved, I think they are directionally useful.

I can understand the desire to understand which parameters have the 'strongest' effect on certain regions of the impedance response, but the Bassbox definition is categorically incorrect because it labels those features as independent resonances. That is the main point I wanted to get across and a myth that should be dispelled. And yes, the 'definitions' listed probably point in the right direction as someone attempted to ascribe causes to the '3 resonances' by exploring which parameters have the largest impact on those features.

Analysis of dynamical systems is somewhat tricky. For a sinusoidal signal, the peak velocity occurs at the same frequency as the peak displacement, as velocity is the time derivative of the displacement signal. The two are displaced 90° in phase. For example, if the displacement is represented by a sine wave, x(t) = sin(ωt), then the velocity is represented by a cosine wave scaled by the circular frequency of the oscillation, v(t) = ω·sin(ωt).

The peak velocity does not always occur at the peak displacement - it's in your analysis above! The velocity is scaled by the angular velocity (omega = 2*pi*frequency). At higher frequencies you can have lower displacement, but the larger angular velocity causes the cone velocity to be larger.

Think of a vented box: peak displacement occurs below the system resonance and as you approach zero frequency (zero angular velocity). The velocity would then approach zero, while the displacement approaches its maximum.

 

[Head_Unit]

I think that that tends to go without saying [that Thiele-Small-Benson are limited models].

I beg to differ a bit-that is true for those who already know. For newbies, limitations maybe not understood at all, particularly how the parameters "change" as you play louder and louder. It took me a long time to really understand that "eh this stuff is a good first approximation but don't take it TOO seriously."

As for "I must admit I am a bit confused by that. The "inductance" just serves to filter higher frequencies by virtue of its increase in impedance. Of course, if the drivers are 4-ohm ones" well come to think of it I was working in autosound, so 4 ohm and even 2 ohm, often huge coils with many windings, very heavy cones sometimes. Not like back-in-the-day pro woofers, where for various JBLs the "ideal" Q=0.7 sealed box would have such a small calculated volume such that you could just silicon the driver face down onto a piece of wood (which also took me some time to realize that at full crank in a concert the operating temperature probably drove the impedance into a different alignment suitable for the big vented boxes they were used in).

 

[witwald]

I beg to differ a bit-that is true for those who already know. For newbies, limitations maybe not understood at all, particularly how the parameters "change" as you play louder and louder. It took me a long time to really understand that "eh this stuff is a good first approximation but don't take it TOO seriously."

I'm not entirely sure what you mean when you say "don't take it too seriously". Sure, the parameters may change at higher driver excursion levels, but that's because the driver is behaving nonlinearly due to electromechanical issues, and that's not a limitation of the model but of the driver itself. The Rvc of the driver can increase quite a bit due to heating at high power levels, but the model can handle that as you can run the simulations at different temperature levels. Of course, once a woofer is outside its linear excursion zone, then the models may not provide as accurate results. But how inaccurate are they really? Is there any data available that we can see?

(which also took me some time to realize that at full crank in a concert the operating temperature probably drove the impedance into a different alignment suitable for the big vented boxes they were used in).

I'm curious to know if there has been any modeling of that type of behavior to show that it is indeed what might have been happening.

 

[witwald]

Not like back-in-the-day pro woofers, where for various JBLs the "ideal" Q=0.7 sealed box would have such a small calculated volume ...

I guess that the small sealed-box volume resulted from those drivers having Qts values of around 0.20, rather than 0.50.

 

[fpitas]

I know absolutely nothing. If I had to engineer speakers

"**** it! Seal the ***** and ship it with a big ass power supply!"

If you think this is bad, you should try to design a complete 3-way speaker.

 

[witwald]

The peak velocity does not always occur at the peak displacement - it's in your analysis above! The velocity is scaled by the angular velocity (omega = 2*pi*frequency). At higher frequencies you can have lower displacement, but the larger angular velocity causes the cone velocity to be larger.

The example I was providing was more specific to the driver resonance frequencies as manifested in the impedance peaks that they generate. It's certainly true of the second impedance peak of a vented enclosure.

Think of a vented box: peak displacement occurs below the system resonance and as you approach zero frequency (zero angular velocity).

Agreed and understood. We can easily see what happens to a woofer when its terminals are attached to a 9-volt battery (DC signal at 0Hz).

It's interesting to note that the very ω-factoring effect that you describe actually causes a peak in the woofer's cone velocity. This peak occurs with the first impedance peak.

The velocity would then approach zero, while the displacement approaches its maximum.

Absolutely correct.

 

[Head_Unit]

I guess that the small sealed-box volume resulted from those drivers having Qts values of around 0.20, rather than 0.50.

Yeah, those big 15" and 18" on the face of it would be excellent midbass driver candidates!

 

[Rick Sykora]

I like your analysis Tommy. I kept coming back to what causes what. There are three unique frequencies in the impedance curve where the phase response passes through zero degrees indicating a likely resonance. This happens at the impedance minimum where cone movement is minimum, and again at the two impedance peaks where cone motion is maximum due to the back emf like you clearly outline.

I think where Rick was going with this is ... what factors in the system are dominant at those particular frequencies and therefore responsible for the local maximums in cone velocity? I absolutely agree with you that it's a large continuous system and it is maybe unfair to try and isolate certain components without regard to how they interact with the other components (stiffness, mass, resistance, inductance, enclosure, port, etc).

This exploration feels similar to analyzing the impedance chart of a woofer playing in free air and being able to clearly see that the resonant frequency is easily predicted by a combination of the two dominant mechanisms at that frequency: suspension stiffness and mass. There are certainly other components in the system like inductance, voice coil resistance, magnetics, eddy currents, heat transfer, and so on .... but those other components are insignificant at the mechanical resonant frequency that they can be minimized or simplified in order to better understand what is happening *right there*.

I think this is what Rick is trying to do, to explore what is happening *right there* at the lower and upper impedance peaks.
At least, this is what I'm trying to do haha.


edit: oh and welcome to the forum! this looks like your first post

So circling back to my OP since there appears to be enough consensus that the Bassbox defs need to be better, here is a strawman to hopefully do so:

1. first peak is high impedance primarily due to port mass
2. fB remains fB without change
3. second peak is high impedance primarily due to driver mass

Also suggest some more meaningful abbreviations would help like fPD and fDD instead of fL and fH. PD is for Port Dominated and DD for Driver Dominated.

 

[witwald]

1. first peak is high impedance primarily due to port mass

The first (and second) impedance peak is due to the motional impedance of the driver voice coil vibrating in the magnetic gap. The first impedance peak isn't primarily due to the mass of air in the port. If you consider the equivalent circuit model, at lower frequencies (where the effect of Lceb is reduced), it is apparent that Cmep tends towards being in parallel with Cmes. Hence, the mass of the air in the port, as represented by Cmep, is added to the effective moving mass of the driver Cmes.

 

[Rick Sykora]

The first (and second) impedance peak is due to the motional impedance of the driver voice coil vibrating in the magnetic gap. The first impedance peak isn't primarily due to the mass of air in the port. If you consider the equivalent circuit model, at lower frequencies (where the effect of Lceb is reduced), it is apparent that Cmep tends towards being in parallel with Cmes. Hence, the mass of the air in the port, as represented by Cmep, is added to the effective moving mass of the driver Cmes.

If so, how would you change the values of the components in the model to illustrate (just as Justin did earlier)?

 

[witwald]

If so, how would you change the values of the components in the model to illustrate (just as Justin did earlier)?

That's relatively easy to demonstrate.

The first impedance peak is at about 18.5 Hz, which is ω = 2*pi*18.5 = 116 radians/sec. At that frequency, the impedance of the inductor Lceb is Zlceb = j*Lceb*ω = j*56E-3*116 = j6.5 ohms. The impedance of the capacitor Cmep is Zcmep = 1/(j*116*180E-6) = -j47.9 ohms. Hence, it is apparent that in that arm of the circuit the impedance is dominated by the capacitor, and is j6.5-j47.9 = -j41.4. This is equivalent to a capacitor whose value is Cequiv = 1/(116*41.4) = 208 μF. This can then be added to Cmes because it is in parallel with it.

As a good first approximation, this can be simulated by simply short-circuiting Lceb in the equivalent circuit. In using this approach, we are relying on Zlceb being much smaller than Zcmep, which is true. This results in a single impedance peak at around 19.7 Hz, which is quite close to the 18.5 Hz peak in the original model. If we short-circuit Lceb (i.e., set it to zero) and change the value of Cmep to 208 μF, we get the resonance frequency of the impedance peak to drop to 18.5 Hz, just as it was in the original model.