The Heresy of the T/S Parameter Bl

[smowry]

Before this thread get locked (just in case), I'll give an example of time domain simulation.

In this thread I showed the frequency domain derivation of the loudspeaker transfer functions.

Mathematical Model of Dynamic Loudspeaker Driver and Physics of Thiele/Small Parameters

There are discussions in another thread regarding the merits of maximizing the amplifier “damping factor” by reducing the amplifier output impedance (which is usually assumed to be purely resistive) to essentially zero or even negative. The theory is that this will maximize the electrical...

www.audiosciencereview.com

Here, the loudspeaker input to output transfer function is computed using a time domain simulation. This is basically a numerical analog of the typical measurement process. An input is given to the model, and the sound pressure output is computed at many time steps. The transfer function (i.e. frequency response) is the FFT of the output divided by the FFT of the input. It arrived at the same results as the frequency domain method. I kept Bl constant in this simulation but it doesn't have to be.

The big difference is that in time domain simulations, the factors Bl, Kms, Sd, Le, etc., can be made dependent on the diaphragm displacement and velocity, time, temperature, etc. Frequency domain analyses require the system to be LTI (linear time invariant), and time domain analyses don't. If you want to deal with nonlinearities, you'll need to simulate in the time domain, and since F = ma is (one of) the fundamental physically law in loudspeaker physics, you have to calculate force. Software like Loudsoft FINEMotor Pro let you simulate your motor design and provide you with the Bl(x) and Le(x) functions. Or you can perform your own FEA.
View attachment 506654
[Edit] Added the plots of the time domain simulation results (input, displacement, velocity, acceleration, current). Updated the ZIP file contents.
View attachment 506692

Figure from the previous frequency domain study.

@olieb complained that I used too much Mathematica, so this time I use FOSS (free open source software) I thought of using Python first, since it is familiar to a lot of people. But I decided to go with Julia since it runs compiled code and should therefore be faster. The printout of the Julia "notebook" is HTML format in the ZIP file.

Thanks for sharing this. It seems like you, Lars and I are the only one's making quantitative posts.

I have a few questions.

1. Is the following an example of the inputs to the transducer simulation?

2. One will need to use FEA to obtain the Kms(x) and Bl(x) simulation. Is this correct?

3. How does your simulation handle Le(x,i)? You show Le as a constant @ 1 kHz.

4. Rms and Sd are also shown as constant. How does your simulation handle these quantities? The Purifi surround has addressed what I call Sd modulation; however, other manufacturer's have not.

5. Suppose someone says they want a 160mm coaxial transducer with a 30mm dome tweeter. What is the simulation and design process?

 

[Snickers-is]

Thanks for sharing this. It seems like you, Lars and I are the only one's making quantitative posts.

Did you miss the fact that NTK just showed why your claim is incorrect?

 

[NTK]

Thanks for sharing this. It seems like you, Lars and I are the only one's making quantitative posts.

I have a few questions.

1. Is the following an example of the inputs to the transducer simulation?
View attachment 506708

2. One will need to use FEA to obtain the Kms(x) and Bl(x) simulation. Is this correct?

3. How does your simulation handle Le(x,i)? You show Le as a constant @ 1 kHz.

4. Rms and Sd are also shown as constant. How does your simulation handle these quantities? The Purifi surround has addressed what I call Sd modulation; however, other manufacturer's have not.

5. Suppose someone says they want a 160mm coaxial transducer with a 30mm dome tweeter. What is the simulation and design process?

1. Yes. Those were the numbers used by the motor! function for the simulation

2. Yes, or from measurements. Of course FEA let you simulate before building any hardware.

3. If we know what Le is a function of, e.g. x. Then just define Le as: Le(x) = some function of x

4. For varying Sd, you probably may need to go all out and simulate the cone and surround flexing, like what Tectonic does modeling their BMRs, and simulate the interactions between the cone/surround and the air (COMSOL jargon "acoustic-structure interaction"). Extremely computationally intensive.
https://www.hifi-selbstbau.de/image...N-4B/Tectonic_BMRWhitePaper_Rev2.0_2019-1.pdf

5. Again. It can be extremely computationally intensive if you want to consider the interactions between all different parts (e.g. modeling Doppler distortions). You may need supercomputers for those or spend money at Amazon AWS/Google Cloud/Microsoft Azure.

Problem with time domain nonlinear simulations is that simulation of one condition may not tell you a lot for a different operating condition. They are more difficult to interpret and you'll need to run a lot of them.

 

[MAB]

Isn't Bl very useful for a range of applications?
Like effective capacitance of a driver:
Ceff = Mms/(Bl)^2
An equivalent circuit can be made, many critical parameters can be calculated, explicitly or with FEA.
Not saying Beta isn't useful.

 

[smowry]

My claim is that if Bl if considered alone it is misleading and potentially meaningless, for example if referenced from a Data or Specification Sheet; I posted numerous examples. Whereas if Beta were considered alone from a Data Sheet, it is not misleading. I also pointed out that Beta = B^2(Volume of Conductor)(conductivity). This is intuitive to transducer engineers and eliminates the myth that the more turns, N, the better. No one has disproved my claim. For every claim that I made, I provided examples and/or math proof. Beta helps to better understand 2D DC FEA simulation where the coil is just a box.

I did not consider time domain analysis. Obviously, NTk's work has merit.

I did point out that transducer manufacturers typically ignore Beta in their specifications.

 

[smowry]

Gentlemen:

I have a few simple questions. If transducer manufacturer's included Beta in their published specifications and data sheets as Peerless does, would that in any way be detrimental to them, their product, or the industry? My claim is that doing so would help avoid confusion and provide additional meaningful information. I consider the ignoring of Beta to be negligence and potentially ignorance at best. If one rakes parameters in order of significance then Beta would be relatively high.

Why is Beta not included in transducer manufacturer's specifications? Several folks including Lars have indicated that they use Beta and that it is useful, but that's obvious. My objective of starting this thread was to get the message out, that Beta needs to be added to data and specification sheets and not to insult or belittle anyone. Some folks claimed that they had never heard of Beta. Well they cannot claim that now. Yet, I was insulted and belittled. As I indicated, I will not make a claim without documentation.

Two of my primary examples were to replace (Bl)^2/Re with β in the Qes and η0 equations. Qes and η0 do not go as Bl, they go as β. Finally, I claimed that ignoring Beta is consistent with the loudspeaker industry's state of development that is significantly behind the other associated A/V industries, including TV's and video monitors, computers and electronics, and digital audio. Below is another reason that the industry lags behind.

A blast from the past.

Here's another example of why the Loudspeaker Industry is so developmentally delayed. Some folks might remember this. https://pearl-hifi.com/06_Lit_Archive/14_Books_Tech_Papers/Mowry_Steve/The%20_Truth_About_Be_Domes.pdf...

www.audiosciencereview.com

 

[smowry]

Now with regards to my claim of how Dr. Bose insisted that his R&D engineers and scientists use Beta rather than Bl, I am not the only witness to this.

If you question my claim then you can contact a well respected colleague, Mr. Eric Johanson. Please do not belittle or insult him. He has listed his contact information on LinkedIn. Note he is also retired. https://www.linkedin.com/in/eric1johanson1acoustics/overlay/contact-info/

 

[smowry]

I neglected to post this above. Bl(x) = B(x)2πNrc N/A

Note that this implies the more turns the better, when in fact for the same coil section the higher the N, number of voice coil turns, the lower β.

My goal is not to replace Bl globally on specification and data sheets or anywhere else with β. My goal is to include β where applicable such that folks will not be confused or misled. All my examples were with respect to this goal.

Now continue to heckle and belittle me. You will only compromise the status of the ASR forum. In fact the heckling does not reflect on me, rather it reflects on the hecklers. I came to the ASR to share, especially with young and upcoming engineers and technicians. I am retired in Phuket and the weather has turned spectacular. My daughter is visiting me from Dubai. She brought rock lobsters home for diner last night. Rose and her sister Mary are flight attendants for Emirates. Interesting fact, to Thai nationals rock lobster cost 1,800 THB / kg, while for tourists rock lobster cost 5,200 THB / kg! As for me, I just send my Thai wife Jane.

 

[smowry]

Isn't Bl very useful for a range of applications?
Like effective capacitance of a driver:
Ceff = Mms/(Bl)^2
An equivalent circuit can be made, many critical parameters can be calculated, explicitly or with FEA.
Not saying Beta isn't useful.

Thank you for posting this. It represents exactly what I have been preaching. I do not wish to insult or belittle you. I want to help make things clear and avoid confusion.

My search gave a different result. I believe it was Lars that posted the following concerning the ratio of mass to beta.

I claim that the ratio of Mms/ (Bl)^2 is misleading and effectively meaningless.

Note that if you see (Bl)^2, chances are that it is misstated and actually means beta!

 

[NTK]

Ran a few simulations with a Bl dependent on the diaphragm displacement x. Below is from Purifi's datasheet.

I picked off the data at x = -8, -6, ..., 6, 8 mm. Computed a fitting function using a 4th order polynomial (see below, but note that the Purifi datasheet x is in mm. I need to keep everything in SI units so my x is in m).

Here is the spectrogram of a sweep to 1000 Hz with 2.83 V input.

Here is the spectrogram with 5x 2.83 V. See the extra harmonics at the lower frequencies. Because of the high diaphragm excursion at low frequencies, Bl deviated significantly from a constant, and its nonlinearity created copious amounts of harmonic distortions. The distortions went down and then away at the higher frequencies because diaphragm excursion drops to the square of frequency for the same SPL, and it keeps itself in its linear range for low excursions (just like the above case with 2.83 V).

The Julia source file and printout in HTML are in the attached ZIP archive.

 

[Snickers-is]

When you add Re into the equation, you effectively apply voltage. While useful in some cases, it is not given that this is what you are looking for.

Many equations do contain the calculation of beta, but that is not the same as a proof of Bl being misleading. It is the same as with capacitors, or battery cells. Farads or Amp*hours are not misleading, but if you want to know the actual capacity of energy storage, you have to add voltage.

Why is it that topics like these has to be discussed with an accusation towards someone or something? It appears like an unnecessary distraction that completely overshadows the interesting parts of a topic.

As I said before, it you want to know the actual power dependent force of a driver, Re is not sufficient, meaning beta is not the "true revealing quality parameter" you seem to claim. You need to add in the complex impedance model for this to be accurate. I did mention this before, and @NTK showed this rather clearly.

This is where the quality of a speaker motor can be determined. We see situations where the combination of Bl(x) deviation, Bl(i) deviation, polar Bl deviation together with modulated impedance account for in excess of 60% loss in Bl within the drivers stated X-max on expensive high end drivers. Beta would only account for a part of this. It would appear a bit like if the amplifier was required with a power senseing feedback that partially compensated for the modulation in impedance.

Yes, you could argue that beta would be pretty accurate to determine the sensitivity in the pass band of a driver, but seriously, would you even consider the pass band to be the range that separated the good from the bad designs?

If we can agree that the reactive range below the pass band is where we have higher excursion, higher power (at least in bass reflex designs) and but far the highest inductive coupling, therefore this is the range where we see the biggest differences between different designs, then we should really be able to see why beta has very limited validity as an individual parameter too.

It is not unique for beta, all the T/S parameters has to be considered as a set. But to me, beta does not stand out as something far more useful than the other T/S parameters.

And Bl is not a suspicious scam, it is just the Lorenz force determined inside the complex layer of the electrical impedance.

 

[MAB]

Thank you for posting this. It represents exactly what I have been preaching. I do not wish to insult or belittle you. I want to help make things clear and avoid confusion.

My search gave a different result. I believe it was Lars that posted the following concerning the ratio of mass to beta.

View attachment 506785

I claim that the ratio of Mms/ (Bl)^2 is misleading and effectively meaningless.

Note that if you see (Bl)^2, chances are that it is misstated and actually means beta!

OK, thanks. Yes I see now.

The equation I quoted is from Joseph D'Appolito's book: Testing Loudspeakers, Chapter 2, equation 2.10.
The equation as published is clearly wrong, now that I think about it.

 

[Snickers-is]

OK, thanks. Yes I see now.

The equation I quoted is from Joseph D'Appolito's book: Testing Loudspeakers, Chapter 2, equation 2.10.
The equation as published is clearly wrong, now that I think about it.

I would argue that it is not wrong, it is just a different expression. One is the capacitance, while the other one describes the stored energy.

It is like the difference between capacitor capacitance vs capacitor stored energy.

 

[MAB]

I would argue that it is not wrong, it is just a different expression. One is the capacitance, while the other one describes the stored energy.

It is like the difference between capacitor capacitance vs capacitor stored energy.

Yeah.
The heresy is me misreading D'Appolito's chapter 2.

 

[Audionaut]

not to refer to the term Bl or I would be ridiculed or shunned

Bl is just one neutral T/S parameter.
It does not claim by itself to describe “control” or “quality”; it only describes the force factor of the motor.

Beta is not an independent parameter, it is calculated from existing T/S parameters (typically Bl, Re, and Mms).
Without Bl, beta cannot be calculated at all, so Bl is still necessary as a fundamental input quantity.

Therefore, Bl is not “wrong” or “misleading” in a scientific sense.
The controversy is not about physics or measurement, but about interpretation:
treating Bl alone as a proxy for performance is an oversimplification.

In short:
Bl is necessary but not sufficient.
Beta is a derived metric.
The debate is mainly semantic and pedagogical, not scientific.

 

[smowry]

Yeah.
The heresy is me misreading D'Appolito's chapter 2.

@Lars Risbo below is also in defense of what you posted. I will use the same example to illustrate that in this case and although Bl is not used alone in this case, Bl is misleading and effectively meaningless. I will not use rhetoric, satire, nor metaphor. I will use simple math.

Ceff is the effective mechanical compliance.
So lets look at the 10 inch Purifi 4 Ω and 8 Ω woofers.

CJDA = Mms/Bl^2
CJDA = 0.098/164.84
CJDA = 0.000595 SI units

Ceff = Mms/β
Ceff = 0.098/42.01
Ceff = 0.0023 SI units

The guys at Purifi are cleaver. They use Cu and Cu Clad Aluminum wire for the 8 ohm and 4 ohm respectively. Can you tell me why? The wire material's impact on the parameters is not contained in Bl. It is contained only in Beta.

CJDA = Mms/Bl^2
CJDA = 0.118/246.49
CJDA = 0.000479 SI units

Ceff = Mms/β
Ceff = 0.118/38.51
Ceff = 0.003 SI units

4 Ω / 8 Ω
CJDA4 / CJDA8 = 595/479 = 1.24
1/1,24 = 0.80

4 Ω / 8 Ω
Ceff4 / Ceff8 = 0.0023 / 0.0030 = 0.77

I hope that Lars can explain the significance of the discrepancy. However, CJDA seems to be upside down.

 

[smowry]

Bl is just one neutral T/S parameter.
It does not claim by itself to describe “control” or “quality”; it only describes the force factor of the motor.

Beta is not an independent parameter, it is calculated from existing T/S parameters (typically Bl, Re, and Mms).
Without Bl, beta cannot be calculated at all, so Bl is still necessary as a fundamental input quantity.

Therefore, Bl is not “wrong” or “misleading” in a scientific sense.
The controversy is not about physics or measurement, but about interpretation:
treating Bl alone as a proxy for performance is an oversimplification.

In short:
Bl is necessary but not sufficient.
Beta is a derived metric.
The debate is mainly semantic and pedagogical, not scientific.

That's not even close to true.

Beta = (Bl)^2/Re N^2/W
Re = (rho)l/S ohms
Where S is the cross section of the wire m^2
l is the length of voice coil wire m
rho is the resistivity ohm m
Then Beta = (B^2)lS/rho
But lS = Volume m^3
Then Beta = (B^2)(Volume of conductor)/rho
Note I can acquire Beta directly from a 2D FEM, where the cross-section and wire material of the voice coil are known.

Bl = B2πNr
Where N is the number of turns of voice coil wire
r is the mean radius of the voice coil m

What is the number of turns in the figure above?

Have you ever used FEA? Have you ever designed a transducer?

The critical term within both parameters is B.
B is the magnetic flux density in the gap T. B is a solution provided by FEA that relates to magnets and ferromagnetic materials and geometry,

Beta can be calculated from Bl and Re; however, Bl can also be calculated from Beta and Re.

Bl can be misleading. I have posted numerous examples.

Your proof is just rhetoric and false rhetoric at that. Where is your math?

Beta allows motor assemblies to be compared regardless of nominal impedance. Bl is again misleading is this regard. Many times a motor assembly will have a higher Bl but a lower Beta. I posted examples of this within this thread.

SPL 1 W @ 1 m goes as Beta and not Bl. Bl is again misleading in this regard. The higher the Bl, the higher the SPL is false. SPL cannot be determined without Beta. The higher the Beta is the higher the SPL.

Unfortunately your frivolous claim is just more Heresy, arrogant Heresy at that.

Dr. Small's classic papers were published in 1976, 50 years ago. Transducer Engineers did not have FEA back then. The development process was empirical, build samples and test. This was a stochastic process at best. The primary use of the T/S parameters was for enclosure design back then.

I could not design a transducer without Beta. Additionally, I could not design a transducer with just Bl. Without Beta, I cannot determine electrical damping, Qes or SPL. I cannot compare designs with different voice coils, yet only Peerless specifies Beta within their specifications. It's time to look beyond the 50 year old T/S parameters. We don't need to eliminate Bl, rather we need to add Beta to the list of specifications.

 

[Snickers-is]

That's not even close to true.

Beta = (Bl)^2/Re N^2/W
Re = (rho)l/S ohms
Where S is the cross section of the wire m^2
l is the length of voice coil wire m
rho is the resistivity ohm m
Then Beta = (B^2)lS/rho
But lS = Volume m^3
Then Beta = (B^2)(Volume of conductor)/rho
Note I can acquire Beta directly from a 2D FEM, where the cross-section and wire material of the voice coil are known.

You are missing the entire pint. You can not determine B without a thorough analysis of the entire field that surrounds the coil. That can be almost manageable with an under hung motor, but for an over hung motor it is quite unrealistic unless you run an FEA analysis. The output of the FEA analysis will typically be Bl, as the tool to determine B, because that is a more correct value than static un loaded field.

So while determining cross section of the conductor is not hard, but the most precise way we currently have of determining B is to go via measured Bl, because it keeps dynamic variation in impedance outside, while including all properties of the normal flux field intact.

So determining beta just by measuring or predicting B is very unreliable to the extent that it is absolutely useless.

@Audionaut 's post is not incorrect.

Unfortunately your frivolous claim is just more Heresy, arrogant Heresy at that.

Why do you have to behave like this? Do you have any reason to accuse him of anything? Is your knowledge not enough for you to be able to discuss this topic in a grown up manner?

 

[Audionaut]

Your own derivation actually confirms the point: beta is not an independent physical quantity.
It is algebraically derived from magnetic flux density B and from the geometry and material of the voice coil.
Those same quantities also define Bl.
The fact that beta can be obtained directly from FEM does not make Bl misleading or obsolet, it simply means that both parameters originate from the same underlying motor physics.
FEM can just as well be used to compute Bl from the same field solution.
Physically, the dominant term in both expressions remains B, the flux density in the gap.
Beta is therefore not a new degree of freedom, but a different representation of the same motor properties, with resistance and conductor volume folded in.
Sensitivity (SPL/1 W) is not determined by beta alone, but by the full set of parameters (Bl, Mms, Re, Sd, etc.).
Beta = convenient composite metric.
Bl = fundamental motor constant.
One does not invalidate the other.
So the issue is not scientific correctness, but interpretation and emphasis:
Bl is necessary but not sufficient.
Beta is a derived metric.
The disagreement is about representation, not physics.
Finally, repeatedly framing this as “heresy” is inappropriate in a scientific and engineering forum.
“Heresy” is a religious term, historically used by inquisitors to condemn dissent, not to resolve technical questions.
In science, disagreements are not settled by theological rhetoric.

 

[Audionaut]

Have you ever designed a transducer?

No, but this thread is slowly giving me an idea of why it wasn't wrong of me to always view BOSE speakers with a healthy dose of skepticism.
That was also my last post in this thread.
I really don't feel like being insulted.