An exploration of the possibilities for product improvement focusing on modeling of transducers. As many engineering tasks require a common ground from which discussions among those active in a task can take place in a structured manner, this frequently involves a alleged model of the concept or product. This article discusses 1 specified model: the lumped component model. This article was originally published in audioXpress, March 2023.
Lumped component modeling involves representing continuum-physics specified as a loudspeaker with various three-dimensional parts as discrete or lumped zero-dimensional components and make alleged analogy circuits that to any degree capture the physics of the transducer involved. The crux of making these analogy circuits is to realize that there is an equivalence specified that the equations relating the primary and secondary variables in 1 physics are analogous to the equivalent equations in another physics.
The impedance in each physics domain gives you the “primary” or “independent” variable divided by “secondary” or “dependent” variable, which for the electrical domain is the voltage divided by the current. The electrical domain serves as a “parent” domain, since we want to take advantage of the well-established framework here with Kirchhoff’s current and voltage laws, establishing transfer functions from circuits, Bode plots for frequency responses, utilizing SPICE software, and so on. The downside of course being that an engineer trained purely in acoustics or mechanics might not be immediately comfortable with these circuits. We see in Table 1 how impedance is defined in the different domains and announcement the similarities.
Table 1: The impedance for electrical components and how analog expressions are found in mechanical and acoustical domains.
So, 1 can find an analogy between an electrical capacitance and a compliance, be it mechanical or acoustical, and similarly, between an electrical inductance and a mass. We will see any complex circuits later in this article utilizing this mapping, but first imagine the situation shown in Figure 1 with 2 springs being compressed by a common force. It should be easy adequate to realize that the full stiffness will be the sum of the stiffness of each of the springs on their own. As stiffness is the inverse of compliance, we can thus calculate the full compliance as:
Figure 1: 2 springs mechanically in parallel
And here we see an issue. From electrical engineering we admit this as the way to add capacitance erstwhile in series in an electrical circuit, whereas for the mechanical setup we would immediately say that the 2 springs are in parallel. Figure 1 illustrates a well-known issue being that you will not automatically have a “conform” depiction erstwhile mapping from 1 domain (such as mechanical) to another (electrical).
To have congruence with respect to series vs. parallel connections between 2 non-conforming domains, 1 can alternatively adapt the alleged Admittance or Mobility strategy, where impedances in 1 domain have admittance (inverse of impedance) analogs in the other, so that the primary variable in 1 domain maps to the secondary variable in the another domain, and vice versa.
The admittance analogy will guarantee that series and parallel connections are retained in the electrical domain, but as with anything, there are pros and cons to both the impedance analogy approach and the admittance ditto. I typically work consistently with the impedance analogy, but sometimes the analogy circuits are much easier to make in the admittance analogy first, and then subsequently I convert to the impedance analogy. The choice will besides be affected by the transductance rule for the transducer, meaning how energy is converted between domains, which is different between a standard “electrodynamic” driver and for an electrostatic transducer as examples, and we may contact more upon this in a future article. Finally, it should be noted that peculiar cases be for which only 1 of the 2 approaches is possible.
Degrees of Freedom
One crucial aspect of a lumped model is that it has a limited number of degrees of freedom, so there is only so much of the continuum physics aspects of a real transducer that are captured. Oftentimes it is not the lumped aspect itself that limits the information that can be extracted from a model, but alternatively the number of components that were put into the model in the first place.
For example, looking at the typical lumped circuit for a loudspeaker, as shown in Figure 2, utilizing the Impedance analogy strategy throughout, it is seen that all the moving mass of the loudspeaker resides in only a single inductor, and the full stiffness from primarily the spider and the environment sits in a single capacitor. This limits the effects that would be seen in a more elaborate lumped model, where the basket was included as possibly 2 added compliances between the magnet and the spider, and the spider and the surround/enclosure, respectively, with each their associated mass, to capture via the lumped model how the reaction force in the magnet systems displaces the individual parts of the driver differently.
Figure 2: Analogy circuit for loudspeaker driver in a closed and mobile enclosure.
Note however, that in Figure 2 a component has been added for the mass of the cabinet and magnet system. The reaction force will displace the magnet in the other direction of the voice coil, and while the effect is tiny for a conventional loudspeaker, we will see later that for any applications it should not be ignored.
Balanced Armature Receivers
One could go consecutive to focusing on having a very rigid basket, specified as has been sought by, for example, Børrensen [1] utilizing topology optimization (a subject for a later article), see Figure 3, along with a dense and stiff cabinet to have the physical setup match the simpler conventional lumped model for a loudspeaker. But I think it is simply a good thought to have more components in the first model, and then from there estimation where to stiffen the basket, or in general analyse the applicable variables in a model complex adequate that it does not lump respective of them together.
Figure 3: Børrensen’s mechanical topology optimized basket for the M1 bass/midrange unit. [1]
While the basket and the cabinet of a loudspeaker surely does flex and decision due to reaction forces, there are transducers for which this effect is much more important, and 1 specified transducer is the Balanced Armature Receiver (BAR). In Figure 4 a cross-section of the BAR is shown for a alleged Single version since it has a single membrane. The armature is magnetized by an AC current in the coil, and as the armature is suspended between magnets, a displacement can be obtained via this “moving iron” setup.
Figure 4: The interior build of Single BAR with a U-shaped armature.
A rule lumped model can now be presented here that captures the applicable physics; the electrical behavior, the mechanical behavior, and of course the most crucial of them, the acoustical outputs, in the respective domains represented by the lumped components. The model considers the effect of reaction forces possibly displacing the full receiver housing and how this modifies the acoustic pressures. The sequential coupling from electrical to mechanical is two-way, and the same goes for the mechanical-acoustical coupling. This means that 1 could analyse the induced voltage and acoustic force resulting from vibrating the housing, and that could be of interest to proceeding aid manufacturers and earphone companies. The lumped model is shown in Figure 5 with the individual components for each applicable part of the Single BAR.
Figure 5: rule lumped analogy circuit for the Single BAR.
The lumped circuit for the Single was made in Spring 2022 and components were fitted up against an existing finite component of the Single BAR made by COMSOL and can be found in its Application Library [2]. Note that all fitting here was “physical” in that component values were found via geometry and material parameters, alternatively of simply doing a numerical fitting.
The agreement between the lumped model and the finite component model is good across all active physics up to a frequency where lumped modeling requires more degrees of freedom to capture the applicable effects, with more details on my own blog post on the model [3], and with the sound force levels for the 2 models compared in Figure 6.
Figure 6: Sound force level in the coupler for the lumped model and the finite component model.
With specified a model, the output from more complex setups can be evaluated quickly, specified as for the Shure SE535 earphone shown in Figure 7. The earphone utilizes 2 BARs — a Single and a Dual — and it incorporates a crossover filter between the two. Their combined output into, for example, a front volume, a tube, and a coupler, can now be evaluated in a substance of a fewer seconds, and the model will encapsulate aspects specified as complex electrical impedance and complex velocities for each of the masses. plan changes are easy made via the available parameters for materials and geometries, and so a parameter sweep or even parameter optimization can be utilized for exploring the solution space.
Figure 7: Exploded view of the Shure SE535 earphone. Thanks to Gojko Obradović for suggesting this Shure earphone as an example.
With the combination of a Single and a Dual, 1 now of course has the option to optimize each of them for their intended output with respect to their individual frequency responses, alternatively of having the usual compromises found for a lonely receiver, Single or Dual, found for example in a proceeding aid. With 2 different receivers in the earphone, the Single BAR can now be modified to extend further up in frequency, whereas the Dual BAR can handle the lower frequencies on its own. The block diagram shown in Figure 8 outlines how the different lumped sections can be connected for evaluating the coupler force resulting from an applied input voltage.
Figure 8: Block diagram for connecting the lumped component circuits.
Circuit Calculations
The transfer functions for the crossovers could be something like a second order Linkwitz-Riley kind (of course I have no thought what kind of filter is being used), and so mathematically it would be expressed as:
and this could be incorporated purely mathematically into the overall circuit calculations, which would be equivalent to having a alleged active crossover, or it could be realized via a passive implementation, so that each filter section is realized with explicit components that then connect to the components for the Single and the Dual lumped circuits, respectively. The second setup would correctly capture the effect of the receivers loading the filter section, and so the designers could easy effort out different filter configurations.
For the Dual BAR the lumped circuit is seemingly missing, and so 1 could go by way of creating a full finite component model in COMSOL Multiphysics as shown in Figure 9. However, I have besides made the Dual BAR lumped circuit, which is presented here for the first time in Figure 10. Again, there is very good agreement between the 2 models, and it is my hope that my fresh lumped models will benefit transducer and proceeding aid engineers in their improvement work.
Figure 9: Finite component model for the Dual BAR with an exaggerated view of the displacement.
Figure 10: The rule lumped model for the Dual Balanced Armature Receiver.
The presented circuits have complete couplings between each intersection, and possibly that can lead to fresh utilization ideas, specified as counting steps based on the induced voltage from the housing moving. These circuits are very powerful for transducer engineers, since erstwhile they are established, a lumped model runs in about 1 second, while the more complex finite component models can have computation times that can be three, four, or even 5 orders of magnitude longer. That means that 1 should start out with the lumped model for the overall estimates and parameter studies, and then proceed with the finite component model for the refinements at higher frequencies as needed.
Concluding Remarks
I am a large proponent of modeling-based engineering. Not any 1 modeling kind in particular; it could be analytical expressions, lumped models, transmission line models, finite component models, or hybrid models that combine the aforementioned types. For me, it is very hard to discuss designs and interior workings of a product without any kind of model characteristics. The lumped modeling method described in this article can frequently give engineers a very good representation of the product that they are working on, and it helps immensely in meetings to have something concrete to present and discuss. The lumped model type, like any other, has its limitations, but it is frequently the best place to start erstwhile starting out on a fresh design, and as has been demonstrated, you can cover any complex designs and inactive have an highly fast plan change cycle.
In 1 of my next articles, we will look at a client case where a transducer setup was modelled utilizing the lumped approach, and how the client can do any of their plan explorations via delivered analogy circuits.
Author Acknowledgements: Thanks to Gojko Obradović (Principal Hardware Architect Acoustics, Oticon A/S) for suggesting the Shure earphone as an example.
Resources
[1] “Børrenson M1: fresh Flagship Loudspeaker from the Danish Company,” Fidelity Magazine, www.fidelity-magazine.com/borresen-m1
[2] “Balanced Armature Transducer, Comsol, www.comsol.com/model/balanced-armature-transducer-61741.
[3] R. Christensen, “#40: Balanced Armature Receiver – Lumped Parameter Modeling (Preprint, Part 1), June 11, 2022,
www.acculution.com/single-post/040-balanced-armarture-receiver-lumped-parameter-modelling-preprint-part-1
This article was originally published in audioXpress, March 2023
