## Table of Content

- Design Considerations for Shallow Subwoofers
- Frequency Dependence of Damping and Compliance in Loudspeaker Suspensions
- Losses in Loudspeaker Enclosures
- Electrodynamic Transducer Model Incorporating Semi-Inductance and Means for Shorting AC Magnetization
- Frequency Dependence of the Loudspeaker Suspension (A Follow up)
- An Added-Mass Measurement Technique for Transducer Parameter Estimation
- A Contour Integral Method for Time-Domain Response Calculations

## Design Considerations for Shallow Subwoofers

The paper on Design Considerations for Shallow Subwoofers was a Convention paper presented at the 121 Convention, San Francisco 2006. I had barely started working at Tymphany when I was thrown into making this paper. Make a paper regarding the Shallow Subwoofer, was the order. I did my best in the few months time I had from about May until the beginning of August, when the paper had to be submitted to the AES.

## Frequency Dependence of Damping and Compliance in Loudspeaker Suspensions

The paper Frequency Dependence of Damping and Compliance in Loudspeaker Suspension was published in the Journal of the Audio Engineering Society, June 2010. Knud Thorborg was primary responsible and the editor of the paper.

During his work with Tymphany, Knud Thorborg came into contact with Andew Unruh, and together they were able to structure 50 years of experience into an early advanced model. The work was first published as an AES paper by Thorborg, Unruh and Struck. Later an article in the Journal by Thorborg and Unruh (September 2008) was published.

The model was challenged by Claus Futtrup (Scan-Speak), who found that the model was not sufficiently good to handle underhung designs, and especially showed some problems when working with short circuiting devices. The model was developed at the same time the Scan-Speak Illuminator woofers were being designed (late summer 2007 and onwards).

When Tymphany shut down their offices in Denmark, Scan-Speak decided to hire Knud Thorborg to continue his work on the model.

The advanced model presented here is the work of Thorborg primarily in cooperation with Claus Futtrup at Scan-Speak and the work was entirely funded by Scan-Speak.

The model expands the old Lumped Parameter model, as presented by A. N. Thiele and R. H. Small (P.S. The electrical equivalent circuit itself was well known prior to Thiele/Small). The new model expands the electrical side of the loudspeaker with additionally 3 components. Furthermore - the latest development of the model provides a simple mean of including damping in the suspension into the model, giving better agreement with reality on the velocity of the driver at low frequencies, as well as better agreement with the impedance, meaning the model is better to predict the low-frequency roll-off.

The model starts out like this:

+ Le -+ +- Cms -+ | | Bxl | | Sd +- Re - Le -+ Rss +-+|+- Rms - Mms -+ +--+|+- Ra - Xa -+ | | | ||| | | ||| | AMP + Ke -+ |G| + w Ams + |T| | | ||| ||| | +-------------------+|+------------------------+|+-----------+ electrical mechanical acoustical

ω·Ams is a mechanical admittance, which bypasses the suspension, Cms. For a given force, this allows the speaker to move a little further (representing visco-elastic creep), in particular at low frequencies.

The electrical equivalent (with things lumped together) looks like this:

+ Le -+ | | +- Re - Leb -+ Rss +-----+----+----+ | | | | | | | + Ke -+ | | Lces AMP Res Cmes | | | | Raes = w Rams = w Bl^2 Ams | | | | +------------------------+----+----+ Ze (blocked) Zem (motional)

It can be seen that the model contains two inductors, one is in parallel with the semi-inductor (as presented by Thorborg and Unruh) and further more in parallel with Rss (semi-inductor short-circuiting resistor), representing electrically conductive material in the magnet system.

The expansion on the mechanical side can also be seen, but why is Raes not simply lumped with Res? Answer: Raes is not a fixed value, but frequency-dependent. Although this slightly complicates our model, since it cannot be simulated as a simple ladder diagram of resistors, capacitors and inductors, this is what is minimum to provide a reasonable approximation to transducer damping. Although not perfect, is improving the modeling of cone velocity significantly, especially around Fs, hence improving the simulation of the frequency response roll-off significantly.

The location of ω·Rams in series with Lces is a result of the treatment found in the AES paper, where it is derived from a well known creep model by Knudsen and Jensen.

For details, see the AES paper.

Ams is a mechanical "admittance" - the opposite of a resistance - but more correct it is a specific admittance (since it is multiplied with frequency, ω, to provide the admittance value).

The model is shown on all Scan-Speak datasheets, with parameters for each transducer. Furthermore a box calculation software is published in the toolbox section. In the future, Scan-Speak might provide some additional information on their web site about this model.

## Losses in Loudspeaker Enclosures

The paper on Losses in Loudspeaker Enclosures was a Convention paper presented at the 130 Convention, London 2011. This paper takes the transducer model with frequency dependent damping one step further and explains the relationship between transducer model and box model.

The short version of the story is: When Richard H. Small made his paper on Vented Boxes in 1973, it was necessary to cut some corners. These simplifications were wisely made and this is the reason why the Thiele/Small model has been used unchanged and essentially unchallenged for so many years. The simplifications on both transducer side and box model side means that the model does not explain what is physically going on. Anyway, when the transducer model is expanded, so must the box model or the quality of the simulation would be reduced, not improved. The changes on the box model side means we can no longer use the conclusions by Richard Small (Ql = 7, ignore Qa). The paper concludes that instead the box model must use absorption, which is also in reality the most significant loss factor in real loudspeaker systems.

This paper moves on to establish an absorption loss model. It is attempted to be made as simple as possible. The model is based on the model of fiber material established by Marshall Leach (1989), with improvements by Putland (1994 - 1996) and with an improvement by Tarnow (1996 - 2002) on aerodynamic flow resistance through a fibrous material (the most significant damping when damping material is applied, according to Putland) and it is 2 - 3 times larger than the tentative results presented by Bradbury (which is widely used). Furthermore the paper outlines the calculation of a partially filled box, which is the normal situation in real world loudspeakers.

The paper does not attempt to explain large-signal behavior and as such is a simple description, but the conclusions can be transposed onto other models of loudspeakers, including large-signal models. Whether a transducer model includes some sort of frequency dependent damping or not is a guideline for choosing the most suitable box model.

When applying the transducer model with frequency dependent damping and the box model with absorption, the simulation of a vented box system will correctly show two impedance peaks of different heights. The height of the peaks (especially the peak at the higher frequency) depends on the amount of damping material (as in real life).

The simulation of the predicted frequency response is more correct, not only at the resonance frequency but also at the other frequencies around the resonance frequency and all they way up to somewhere around Fmin (where minimum impedance Zmin is found), or until directivity becomes noticeable. The response is no longer a simple filter function and the assessment of damping cannot be described by a single Q-value for the roll-off. For the simulation software published at Scan-Speak, found in the toolbox section, the assessment of damping (or quality of impulse response) is instead provided by a Group Delay plot, with a home-made limit guideline.

### Addendum / errata / comments to this paper

Errata: Equation 10 does not contain the B-factor, which it should. See W. Marshall Leach's original AES article, eq. 5.

The paper describes an advanced equation for determining the resonant frequency, which includes the influence of Rams (equation 1). This equation is not an exact expression derived from the equivalent circuit, but an approximation (an exact expression proved difficult to derive analytically and unnecessarily complicated). The equation provides a complex number as the resonance frequency, which is not physically possible, but it is an effect of the approximation. The entire paper only treats this as an approximation and only considers the REAL part of the expression. I wish I had been more clear about this in the paper.

The paper describes the resonance frequency of the complete system as a result of the combined electrical and mechanical circuits. As a mechanical engineer it makes more sense to me to exclude the effects from the electrical side and consider only the "natural vibration mode" of the mechanical system. The choice in the paper to include the electrical side in the considerations was made to accommodate the common practice that electrical engineers (according to Knud Thorborg) consider the peak of the impedance to be the resonant frequency of the complete system.

The content of this paper was presented (one week later) as a lecture at the High End show in Munich (MOC). You can see the slide show (PDF download) and hear an audio recording (30 MB) of the lecture, originally available at the High End Society homepage.

## Electrodynamic Transducer Model Incorporating Semi-Inductance and Means for Shorting AC Magnetization

The paper on Inductance with shorting devices was published in the Journal of the Audio Engineering Society, September 2011.

This inductance model was already presented previously in the paper about frequency dependent damping, but the focus was on the mechanical damping properties. Here the inductance side of the model is in focus. It is shown how a fairly large inductance model is simplified to the 5-parameter model of the electrical equivalent circuit and by example, it is shown that this model is suitable for anything from subwoofers to dome-tweeters for the entire audio frequency range (beyond the applicable frequency range of the transducers).

Due to the simplifications, the model is not perfect - which is most noticeable when the voice coil is overhung by a large factor - and maybe even with short circuit devices located outside the air gap (i.e. not a copper cap, which lines the wall of the air gap, but rings of copper and/or aluminium, etc.). The model is significantly better than any other known model (compared with the models of Small, Vanderkooy, Leach, Wright and Klippel - only the last 3 are compared directly) and is considered generally applicable.

Potential applications are mentioned in the paper, from the study of cone vibration modes and passive crossover design, to improved box simulations and maybe even detection of room modes.

As a side-note, this paper on inductance also mentions (in appendix 2) the possibility to re-locate the frequency dependent damping. Although scientifically not proven to be more or less correct this alternative location seems to be more practical.

In future work the specific admittance "Ams" might be replaced by the Greek Lambda, Λms, which is the Greek L, symbolizing the relationship with Lces. Λms is the imaginary component of suspension compliance and has the same unit as Cms. At the same time, by coincidence, Λ looks similar to A. This new nomenclature is supposed to prevent the admittance from being interpreted as a resistance and hereby attempt to prevent potential confusion. In the electrical equivalent circuit, Lces is paired with Λes.

Added 2018-10-27: You can download a spreadsheet by Stephen T Bolser for calculating the semi-inductance model parameters from the DIY Subwoofers pages of Brian Steele. I haven't verified any of this, so use at your own risk: Semi-Inductance Spreadsheet (zoom to the bottom for the download link).

## Frequency Dependence of the Loudspeaker Suspension (A Follow up)

The paper on Frequency Dependent Compliance + Damping with the LOG model was published in the Journal of the Audio Engineering Society, October 2013.

This paper changes the layout of the equivalent circuit so that Ams becomes Λms and Rams becomes Λes.

+ Le -+ | | +- Re - Leb -+ Rss +-----+----+----+----+ | | | | | | | | + Ke -+ | | | | AMP Cmes Res Lces ω·Λes | | | | | | | | | | +------------------------+----+----+----+ Ze (blocked) Zem (motional)

It is not that this model is better at describing what is actually happening in the transducer, but it is easier for a computerized optimizer to curve-fit the model with the measured impedance because the components do not mutually influence each other (Lces vs. Λes).

In this paper Lces and Λes are not constants, but frequency dependent (as described by the modified LOG model). Hence the model is named the SI-LOG model (Semi-Inductance + LOG compliance and damping model). The introduction of the frequency dependent Lces and Λes as described by the LOG model adds another variable to the model, λ (lower case lambda).

With the improved curve fitting, the most important part of this paper shows how each component can be "derived" from the measurement, validating that the found value is good, at least within a suitable frequency range. Only mechanical parameters are shown, but of course the mixing technique of measurement and model can also be applied to electrical (blocked impedance) parameters.

Be aware that although this validation process can rule out some bad fits (when model parameters and derived parameters are detected as unrelated), a validated parameter is not a guarantee that you have found the only solution (although the found solution seems good, a better one may exist). The reason why a validated parameter cannot be guaranteed is due to the mixing of model and measurement. Although this compromises your findings, it is a nice validation anyway; as it rules out some bad fits (it narrows your field of search for a good set of parameters).

The paper also describes how the admittance can be described as a loss with a hysteresis loop and how this connects the two versions of admittance (series Rams with parallel Λes) as: Λes = Lces^2 / Rams. With this relationship you can convert back and forth between the two models (with reasonable approximation).

Prior to the release of this AES paper, Knud Thorborg participated in the Baltic-Nordic Acoustics Meeting in 2012 (BNAM2012), where he presented two papers. The first one titled "Traditional and Advanced Models for the Dynamic Loudspeaker" and the second one titled "Measurement of the Advanced Loudspeaker Parameters using Curve-Fitting Method".

## An Added-Mass Measurement Technique for Transducer Parameter Estimation

The paper titled An Added-Mass Measurement Technique for Transducer Parameter Estimation was published in the Journal of the Audio Engineering Society, December 2017. I co-authored the paper with a friend of mine, Jeff Candy.

This paper describes a novel measurement procedure, where a total of three impedance measurements conducted in free-air, two with added masses and one without, makes it possible to extract the lumped parameters of the transducer, including viscoelasticity. In this paper we use the Knudsen LOG model, but other models can be applied as well. The result can be verified by simulating the impedances (with the chosen viscoelastic model) against the measured impedances, and although it's not perfect, the interesting part is that the extraction of the motional impedance and subsequently Bl is model-free, and only minor curve fitting is required (to identify the mechanical parameters, like moving mass, Mms). Besides, the extracted data can be analyzed to see the quality of the measurement data (and the Mms fit), which we call the "mass consistency" diagnostics information.

It is quite an improvement over the old-fashioned added-mass approach which was widely used in the loudspeaker industry in the past. As something rather unique in the history of the AES, AFAIK, the quality of the proposed measurement technique was verified with an ANOVA Gage R&R measurement system analysis, and it shows that the method (at least for certain suitable transducers) is able to robustly determine the parameters with high precision.

## A Contour Integral Method for Time-Domain Response Calculations

The paper titled A Contour Integral Method for Time-Domain Response Calculations was published in the Journal of the Audio Engineering Society, May 2018. This is the second paper co-authored with Jeff Candy.

This paper is essentially of mathematical character, as it describes a way to calculate the time (step) response of some tricky functions for visco-elasticity and semi-inductance in loudspeakers by using contour integration of the equivalent frequency response function.

Added 2018-10-30: The AES article contains a Scilab script for readers to easily see how the algorithms are applied in its entirety. Here's a Scilab script, which is expanded quite a bit when compared to the script in the AES article: stepresponse.sce

This script specifies a loudspeaker unit in a 10 liter box, with port tuning at 40 Hz (in this case the L16RNX woofer, but feel free to modify as you like). The normalization is shown, the Weideman Contour integration is performed, and the result is calculated. Besides, the script "checks" the result with an FFT on the calculated time response and compares with the specified response function, so you can see that the time response is correct. Since the FFT requires quite a few datapoints, the script takes some time to run. As soon as you feel confident in our method, you can omit the FFT and cut down on the time response calculations to benefit from the speed of the calculation.

Please note that the Weideman method may show instability at sufficiently long time because of the need to limit mu (based on avoiding pole crossing). In the present example, the standard choice mu = max([1 h]) leads to instability at t > 40. However, this setting is conservative and for the present example, mu_c = 0.5 * max([1 h]) is appropriate and allow accurate integration up to t = 50 and greater. Ultimately, one should check to see that a pole has not been crossed when applying this method.