# Basis for Hofmann's law

#### fers

I see people often speak of Hofmann's law when referring to a fundamental tradeoff between the size, the low frequency extension and the efficiency of an enclosure design. It is often mentionned that all enclosure types are subjected to the same tradeoff. After seeing references to this "iron law" being broken with some new driver technology by Brane Audio, I was puzzled as what they proposed (successfully or not, this is beyond the point of this thread) would only solve the problem of a driver having to work against a reducing compliance in smaller and smaller enclosures. This made me think that I misjudged what is referred to as "Hofmann's law", as I thought the theoretical basis for such a general law had to be concerned with the radiation impedance available at a certain size, irrespective of driver technology. The only "derivation" I could find for it by Henry Kloss is concerned only with the limitation tentatively addressed by the Brane Audio approach (while not a completely rigorous derivation, I believe the details can be rigorously filled in for that setting).

This leaves me unsatisfied with the way it is broadly applied to any enclosure types with, to my knowledge, little theoretical basis. I must admit I did not attempt to seriously derive anything of the sort as I assumed previous work might already have been done on the topic, hence this thread. I am looking for something as close as could be to a theoretical derivation of something akin to Hofmann's law, maybe beginning with the case of simple vented enclosure. It probably needs to be more subtle than the way it is usually applied: It is possible to make small, loud and efficient enclosure over a very restricted bandwidth, so bandwidth probably needs to enter the equation somehow. Is it really all about dealing with the decreased compliance of a smaller enclosure or does the radiation impedance associated to a certain size has anything to do with it?

If based on radiation impedance considerations, I would expect such a "law" to not necessarily be as hard as touted as it seems to mostly be a material/engineering limitations. Indeed, one could theoretically make a driver with a conjugated impedance that varies with frequency just so that the efficiency is very high, but realistically one would end-up with unrealizable displacements, for one thing.

#### weltersys

The theoretical derivation of Hofmann's "iron law" can be found in the mathematical formulas of Thiele and Small, which work both in theory and practice.

At any rate, Brane's Repulse Attract Driver (R.A.D.) “magnetic negative spring” does not break Hofmann's iron law, it could be made even more efficient if a larger transducer and box were employed.

#### BKr0n

At any rate, Brane's Repulse Attract Driver (R.A.D.) “magnetic negative spring” does not break Hofmann's iron law, it could be made even more efficient if a larger transducer and box were employed.

#### Randy Bassinga

Getting low is for fidelity and a must, but size is a user factor. May I ask why does it need to be efficient too, when it's the conditions that result in inefficiency are the ones that allow for box to be reduced while maintaining extension?

My own unique case needs something small, low and efficient, but I know I have to 'push on one lever to be able to pull on another'

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#### BKr0n

Getting low is for fidelity and a must, but size is a user factor. May I ask why does it need to be efficient too, when it's the conditions that result in inefficiency are the ones that allow for box to be reduced while maintaining extension?
Decibels still translates to energy spent. Think linkwitz transform. Efficiency allows it to get loud quick, meaning less power for super loud, or being able to use that energy for something else.

Best I could do is the L22ROY2 from Seas. Great driver, but it's still 8" and sensitivity is still a little iffy.

#### Randy Bassinga

energy spent
I thought it was the energy being spent on what is making the system inefficient was also what was allowing the smaller box while maintaining that extension. That is not a loss but a feature that allows the small cab

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#### Randy Bassinga

When I was looking at drivers for extremely small and low subs, I checked the sensitivity rating against the Vas and Fs. If it looked too high, I passed on it as from experience, buying those drivers never worked out. Drivers rated in the mid 80s to me were more likely to work better in the smaller cabs which has been my experience time after time. If a datasheet showed high 80s or more that I take any claims in the description for very small boxes to be bs

#### BKr0n

I thought it was the energy being spent on what is making the system inefficient was also what was allowing the smaller box while maintaining that extension. That is not a loss but a feature that allows the small cab
Efficiency is how much energy does it take for a transducer to make that much noise at one watt at one meter. So lets say I have two different speakers; one with 92db rating and one with 88db. At 1 watt of power, the first speaker produces 92db of sound, as opposed to the second one, which at one watt, can only produce 88db.
When I was looking at drivers for extremely small and low subs, I checked the sensitivity rating against the Vas and Fs. If it looked too high, I passed on it as from experience, buying those drivers never worked out. Drivers rated in the mid 80s to me were more likely to work better in the smaller cabs which has been my experience time after time. If a datasheet showed high 80s or more that I take any claims in the description for very small boxes to be bs
That's hoffman's iron law for ya. There's a few things you can do to mitigate its effects, like MFB or isobaric loading, but physics is pretty tough to win against. That's why when I saw what @weltersys posted I was super surprised. Just eyeballing it, it may use some bending wave principle like tectonic, but that's purly speculation.

Best implementations I've seen of trying to beat iron law are from KEF and Grimm audio.

(btw the Seas extreme subs are quite good if you're looking for something with good specs that go low)

#### fers

The theoretical derivation of Hofmann's "iron law" can be found in the mathematical formulas of Thiele and Small, which work both in theory and practice.
Thank you for pointing out a good source which I must admit I did not read before. In hindsight, looking there first seems obvious. I just went and read the two-part original Thiele papers. I don't have issues for most assumption as it concerns vented enclosure (as long as we are not concerned with large enclosures or large drivers at high tuning frequencies.)

The constraints of vented box is more clearly derived as contained in the equation stating that efficiency n0 is:

n0 = 8 x 10^-12 fs^3 Vas/Qe,

however even then, the relation between this and box size completely hinges on an observed ratio that is approximately constant in the alignments Thiele explored where:

Vas fs^2 = sqrt(2) Vab f3^2, (*)

and this is where the meat of Hofmann's law would lay for simple vented enclosures, but there is no derivation for this observation (and thus with unknown applicability to say response shapes not associated with a common filter type).

But let's say I take this for granted (and I would not be surprised bounds could be rigorously derived for the ratio), this still does not support the applicability of Hofmann's law to other enclosure types, such as bandpass, horns, tapped horns, transmission lines, paraflex and QW whatevers. As a passing observation, I see something pointing in that direction as the observed ratio I just mentioned allows for slightly smaller enclosures in the case of high initial roll-off response shapes, such as Chebyshev with high ripple.

In any case, I still see people refer to Hofmann's for these more complex enclosures, usually making use of the total internal volume of the enclosure. For a reasonable construction where the target response is a wideband and mostly flat, I can see most of the argument going through and deriving only the ratio (*) above for these enclosures would be needed. Is this a case of mostly empirical observations that mostly fit with the law's predictions?

A more general expression and derivation would be needed for the case of restricted bandwidth enclosures. All the considerations with regard to efficiency and the associated constraints are based on the midband theoretical efficiency of the driver which is somewhat irrelevant in these enclosure types. In fact, this would be true of any non-flat, wideband response. Finally, as mentioned briefly in the beginning, for very large enclosures and/or drivers, the law might not hold as the radiation impedance will differ substantially from the assumed simple form in the Thiele paper. This might actually be a clue to bring proper front-loaded horns under the same umbrella.

#### weltersys

From the reviews, Brane's Repulse Attract Driver (R.A.D.) is good for it's application, a small battery powered "boom box".

#### weltersys

In any case, I still see people refer to Hofmann's for these more complex enclosures, usually making use of the total internal volume of the enclosure. For a reasonable construction where the target response is a wideband and mostly flat, I can see most of the argument going through and deriving only the ratio (*) above for these enclosures would be needed. Is this a case of mostly empirical observations that mostly fit with the law's predictions?

Horn theory goes back further than Thiele and Small, Bjørn Kolbrek's articles are a good start:
https://www.grc.com/acoustics/an-introduction-to-horn-theory.pdf
Empirical observations have guided modifications to theorys.

#### olsond3

For me the interesting characteristics of a woofer system are the low frequency extension and the undistorted SPL max. Gary Gesellchen of Vanatoo published an excellent paper with AES and AudioXpress July 2023 "Bass Reflex Performance Envelope". If you have sufficient power / power handling, Xmax and equalization, the TS parameters and box volume are a secondary concern. As amplifier power and equalization are now cheap, this leaves woofer power handling and Xmax the primary parameters to consider. The paper can be found on his website https://vanatoo.com/ under support/downloads at the bottom of the page. As links tend to change over the years I will attach the file here. This paper is exceptional in my view and right up there with the work of Thiel and Small, as with one clever design constraint he distills the performance of a woofer system down to a few easy to understand graphs.

#### Attachments

• Vanatoo - Bass Reflex Performance Envelope.zip
1.9 MB · Views: 26

#### fers

Horn theory goes back further than Thiele and Small, Bjørn Kolbrek's articles are a good start:
I have Kolbrek's book! While there are full sets of constraints, including efficiency and whatnot, it is not aimed at relating performance to total enclosure size. As a start, I derived the expression for the total volume of a full size exponential horn as a function of all the other parameters. A partial dependence on 1/fc^3 where fc is the cutoff frequency does appear, but it still does not look like something of an Hofmann's law as there are many constraints with horns that relate to bandwidth so I would need a systematic way to account for that.

#### fers

If you have sufficient power / power handling, Xmax and equalization, the TS parameters and box volume are a secondary concern.
While true in some settings, this statement is not universal. My main concrete interest is mobile/portable/battery powered rigs outdoor, aka the worst possible conditions for bass. In this case efficiency is high up in the list of desirable features, but so is small size and extension which makes you constrained head first by Hofmann's law.

On another note, I am also attracted to the challenge and not only the results and generally find efficient designs, energy wise, aesthetically pleasing in some kind of abstract engineering sense of aestheticism, if that makes any sense. Examples of such designs are distribution transformers, some electrical motors, free piston Stirling engines, zeppelins and class-d amplifiers.

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#### BP1Fanatic

But let's say I take this for granted (and I would not be surprised bounds could be rigorously derived for the ratio), this still does not support the applicability of Hofmann's law to other enclosure types, such as bandpass, horns, tapped horns, transmission lines, paraflex and QW whatevers.

Easy.

Tapped Horns are BP6S or 6th order series tuned bandpass enclosures where the whole enclosure is the horn, vent, or port.

Positive flare TH = high efficiency, big enclosure, plays low.
Negative flare TH = low efficiency, small enclosure, plays low.
Straight flare TH = is in between the 2 types above. No angles, so ease of build.

There's your example of Hofmann's Iron Law...efficient, small, low...pick 2.

#### BP1Fanatic

Transmission Lines (TL) follow the same laws as TH's.

TL = the driver is mounted on the outside of the enclosure and has bass reflex (BR) properties.
TH = the driver is mounted inside the TL enclosure and has BP6S properties.

#### fers

Easy.

Tapped Horns are BP6S or 6th order series tuned bandpass enclosures where the whole enclosure is the horn, vent, or port.

Positive flare TH = high efficiency, big enclosure, plays low.
Negative flare TH = low efficiency, small enclosure, plays low.
Straight flare TH = is in between the 2 types above. No angles, so ease of build.

There's your example of Hofmann's Iron Law...efficient, small, low...pick 2.
I understand that, but my concern regards a theoretical confirmation of such an observation. If one can translate this idea in an equation derived from first principles that brings all enclosures under the same umbrella, it can, on the one hand, be used to better understand the source of this limitation and on the other hand, be a really useful tool to help design decisions! I'd wager some configurations will have an edge over others but this has not been empirically sorted out at this point, at least not to the level I'd wish.

To give a concrete example, I know you can conceive of all these enclosure types (except maybe true horns) as BP configurations, but I'd like to fully characterize the trade-offs involved when going along the continuum between QW and Helmholtz resonators. Returning to the case of true horns, what differences, if any, can there be between a full horn and, say a BR of the same total volume? One could say I can just model all of these and compare, which I can, but it can be difficult to make these comparisons rigorously in general sense as many parameters will need to be adjusted for a fair comparison it can be complex to do these adjustments. Moreover, doing it this way, rather then finding the physical constraints as equations does not allow one to grasp the fundamental trade-offs.

At the end of the day, speakers are fair and square within the established physics and if such a constraint exist an expression should not be too hard to derive from general principles. My interest for the need of such an expression may be biased from my background as a mathematical physicist, but I operate well with these tools.

#### BP1Fanatic

Like a said before, model a positive flare TL and a negative flare TL of the same volume and you'll see a prime example of Hofmann's Iron Law.

A TL is a BR enclosure where the whole enclosure is the port.

#### bolserst

I see people often speak of Hofmann's law when referring to a fundamental tradeoff between the size, the low frequency extension and the efficiency of an enclosure design. It is often mentionned that all enclosure types are subjected to the same tradeoff. After seeing references to this "iron law" being broken with some new driver technology by Brane Audio, I was puzzled as what they proposed (successfully or not, this is beyond the point of this thread) would only solve the problem of a driver having to work against a reducing compliance in smaller and smaller enclosures. This made me think that I misjudged what is referred to as "Hofmann's law", as I thought the theoretical basis for such a general law had to be concerned with the radiation impedance available at a certain size, irrespective of driver technology. The only "derivation" I could find for it by Henry Kloss is concerned only with the limitation tentatively addressed by the Brane Audio approach (while not a completely rigorous derivation, I believe the details can be rigorously filled in for that setting).

This leaves me unsatisfied with the way it is broadly applied to any enclosure types with, to my knowledge, little theoretical basis. I must admit I did not attempt to seriously derive anything of the sort as I assumed previous work might already have been done on the topic, hence this thread. I am looking for something as close as could be to a theoretical derivation of something akin to Hofmann's law, maybe beginning with the case of simple vented enclosure. It probably needs to be more subtle than the way it is usually applied: It is possible to make small, loud and efficient enclosure over a very restricted bandwidth, so bandwidth probably needs to enter the equation somehow. Is it really all about dealing with the decreased compliance of a smaller enclosure or does the radiation impedance associated to a certain size has anything to do with it?

Your might check out Richard Small's take on this issue in Reference (1) & (2)
He noted that the efficiency for any direct radiator loudspeaker system configuration can be written in the form:
ηo = kη x f3^3 x VB
Where kη is an efficiency constant determined by the system configuration (ie enclosure type and frequency response shape)

With kη set, the familiar "Hofmann's Law" tradeoffs between efficiency, size, and bass extension are apparent.
However, kη can/will be different for different enclosure types or frequency response shapes.

As an example from Reference (1), (2), & (3) comparing sealed and ported enclosures with B2 and B4 response shapes respectively and minimal losses, kη for the vented system is twice that of the sealed. So, it is possible for a vented enclosure to be twice as efficient(ie +3dB louder) as the same size sealed enclosure with the same -3dB rolloff point. Reference (2) also derives kη for response shapes other than B4.

You may find it interesting that in Reference (1) Small mentions two methods for increasing kη for a sealed enclosure:
“…There are two known methods of circumventing the physical limitation imposed by (36) or Fig. 8. One is the stabilized negative-spring principle [15] which enables VAT to be made much larger than VB but requires additional design complexity. The other is the use of amplifier assistance which extends response with the aid of equalization…”

The first is what Brane Audio is doing, increasing kη by roughly a factor 10. If you take a look at [15], Reference (4), this approach was evaluated back in 1971! A pneumatic servo mechanism was used to stabilize the mechanical negative spring. Brane uses a magnetic spring instead of mechanical and a similar pneumatic approach for stabilization.

The second approach has been discussed in Reference (5), where the combination of large increases in BL and electrical equalization to retain desired frequency response shape can net an efficiency increase of 4x or so. The downside is the need for a special amplifier that can output higher voltage than is typical for consumer products.
"…To conclude, if your design can accommodate equalization before the power amplifier and the power amplifier can provide higher voltage swing, then raise your driver’s Bl product to the highest possible value consistent with material and economic constraints! This will result in the highest efficiency design."
https://www.diyaudio.com/community/...ency-is-mostly-irrelevant.358153/post-6300579

If based on radiation impedance considerations, I would expect such a "law" to not necessarily be as hard as touted as it seems to mostly be a material/engineering limitations. Indeed, one could theoretically make a driver with a conjugated impedance that varies with frequency just so that the efficiency is very high, but realistically one would end-up with unrealizable displacements, for one thing.
From what I can tell, the Hoffman's Law releationship was derived with the assumption that the radiation impedance is insignificant compared to the mechanical mass of the cone/voice coil.
From Reference (3):
“…These efficiency and response relationships confirm that for each type of system, specifications of size, response, and efficiency are not independent; if two are specified, the third is determined and may be calculated. They also reveal that system small-signal performance does not depend on the diameter of the driver. The choice of driver size may be decided on the basis of cost or large-signal performance specifications…”

If interested, see References (6) and (7), regarding the issue of cone size and how it relates to efficiency.

References:
(1) Closed-Box Loudspeaker Systems-Part 1: Analysis, Small, Richard H. 1972 AES
Available from: https://aes2.org/publications/elibrary-page/?id=2022

(2) Vented-Box Loudspeaker Systems--Part 1: Small-Signal Analysis, Small, Richard H. 1973 AES
Available from: https://aes2.org/publications/elibrary-page/?id=1967

(3) Efficiency of Direct-Radiator Loudspeaker Systems, Small, Richard H. 1971 AES
Available from: https://aes2.org/publications/elibrary-page/?id=2120

(4) Improvement of Low‐Frequency Response in Small Loudspeaker Systems by Means of the Stabilized Negative‐Spring Principle, Terrance Matzuk 1971 JASA
Available from: https://doi.org/10.1121/1.1912510

(5) Comparison of Direct-Radiator Loudspeaker System Nominal Power Efficiency vs. True Efficiency with High-Bl Drivers, Keele, Jr., D. B. (Don); 2003 AES
Available from: https://aes2.org/publications/elibrary-page/?id=12453
Free from: https://www.xlrtechs.com/dbkeele.com/papers.htm

(6) Efficiency Does Not Depend On Cone Area, Ashley, J. Robert; 1971 AES
Available from: https://aes2.org/publications/elibrary-page/?id=2119

(7) Maximum Efficiency of Direct-Radiator Loudspeakers, Keele, Jr., D. B. (Don); 1991 AES
Available from: https://aes2.org/publications/elibrary-page/?id=5523
Free from: https://www.xlrtechs.com/dbkeele.com/papers.htm

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