A Model for the Simulation of Porous Materials in SPICE
A comparative analysis between melamine foam and polyurethane foam, supported by experimental measurements and by the proposal of a new simulation model for flow resistance, integrated into the SpicyTL software.
This article was published in issue no. 475 (May 2025) of AUDIOreview magazine
The main goal of this work was to explore the acoustic characteristics of melamine foam, a material that has so far been rarely used as damping inside loudspeakers. During the investigation, however, it became clear that a thorough revision of the simulation model for damping materials within SpicyTL was needed.
It had been on the “to-do” list for quite some time, and the initial motivation came from a set of measurements carried out on a small transmission line and three different materials (acrylic fiber, polyurethane foam, and melamine foam), shared by a user on a specialized forum. While the results for fiber and polyurethane foam could easily be replicated in SpicyTL, melamine exhibited very peculiar characteristics that did not fit the existing modeling parameters.
I was already partly aware of this limitation: although SpicyTL provides reasonably reliable results with the materials most commonly used inside transmission lines (and loudspeakers in general), I had noticed inconsistencies whenever the material properties diverged too much from those originally used to develop and calibrate the model. In fact, the existing model is somewhat rudimentary — it adjusts the flow resistance of the material as a simple linear function of frequency.
I therefore started working on a more refined model, designed to offer greater flexibility and accuracy while remaining simple enough to be calibrated for different materials.
The first results were very promising and, after obtaining a sheet of melamine foam, I reactivated one of my TL test systems for a more detailed investigation.
This article does not aim to be an academic study, but it strives to be accurate enough to support the development of a reliable simulation model suitable for integration into the software.
Melamine Foam
Expanded melamine foam (or melamine foam) is a sound-absorbing material based on thermosetting melamine resin that, through an expansion process, forms an open-cell structure similar to a lightweight foam. It has a very low density (typically between 8 and 12 kg/m³) and a finely porous surface, characteristics that give it excellent sound absorption properties, particularly at mid and high frequencies.
Although generally less dense than PUR (expanded polyurethane or polyurethane foam) — the material most commonly used inside loudspeakers, with a typical density of around 30 kg/m³ — it feels noticeably stiffer to the touch, yet also more fragile.
It offers high UV resistance, which makes it stable over time even in brightly lit environments, unlike polyurethane foams that tend to yellow, become brittle, and lose their mechanical properties relatively quickly when exposed to light.
Among its other interesting features, especially for industrial applications, are its high thermal resistance (it can withstand temperatures up to 200 °C, although this isn’t critical for loudspeaker use) and its inherent fire-retardant properties — it is self-extinguishing and does not emit toxic fumes.
Still, some care should be taken when handling or cutting it: it tends to crumble easily, so wearing a protective mask is recommended.
The most well-known commercial products are Basotect by BASF and Fonitek by the Sogimi Group.
Revision of the loudspeaker impedance model
A more rigorous modeling of the TL frequency response led me to first revise the loudspeaker impedance model as well. I chose to adopt the Leach model, which I will not discuss here in detail, referring instead to reference [1] for a complete treatment. This model introduces two parameters and describes the dispersive behavior of the voice coil inductance (lossy inductor) with a function of the following type:
where the parameters 𝐾 and 𝑛 represent, respectively, the amplitude and the slope of the impedance rise with frequency. The method describes an analytical procedure to estimate these parameters from impedance measurements (magnitude and phase), using a logarithmic regression that takes into account the signal phase. Although the rigorous calculation is rather laborious, good results can be obtained in just a few minutes with an empirical trial-and-error approach, especially if a graph of the measured impedance is available. Figure 1 shows the simulated impedance curves of the loudspeaker (Dayton Audio RS100-4) in free air, without (in red) and with (in green) the lossy inductor. For display consistency, I exported the graphs generated by LTspice as text files and converted them into a format readable by Clio using a Python script. In Figure 2, you can see how, in just a few minutes, it was possible to achieve an excellent match between the measurement (in red) and the simulation (in black). The graph is omitted for space reasons, but even the impedance phase curves show an almost perfect overlap.
The Leach model was implemented in LTspice through a voltage-controlled current source (VCCS), whose response is defined by a function in the Laplace domain that replicates its behavior. I assure you it’s simpler than it might sound, and I’ll explain it in more detail in the next section, since I used the same technique to model the behavior of the damping material.
Revision of the damping material model
Several theoretical models have been developed to describe in detail the acoustic behavior of poroelastic materials. Among the best known are:
Delany–Bazley model: one of the most widely used for its empirical simplicity; it describes the resistance and reactance of a porous material as functions of frequency and flow resistivity. It is valid for homogeneous structures and within a certain frequency range.
Miki model: a variant of the Delany–Bazley model with corrected coefficients for improved accuracy, especially in the low-frequency region.
Johnson–Champoux–Allard (JCA) model: a more comprehensive physical approach that considers wave propagation in air-saturated porous media and accounts for tortuosity, porosity factor, air viscosity, and material permeability. In practice, it takes into account the microstructure of the material, providing a much more accurate description of its behavior in a three-dimensional context. It is highly accurate, but also quite complex to implement, as it requires FEM simulations and parameters that are difficult to measure precisely.
The model developed in this work is not intended to replace or improve upon the existing ones, but rather to provide a much simpler solution that can be easily integrated into circuit simulation environments such as LTspice, while still maintaining good adherence to real-world behavior. It is specifically designed to work well in the most relevant range — mid to low frequencies — requires only three parameters (which can be empirically adjusted), and allows simulation of the material’s effect directly as a variable flow resistance.
To define the damping function, I started by observing the behavior of the absorption coefficient of several porous materials, which tends to increase rapidly with frequency until reaching a saturation region. The acoustic absorption coefficient (often denoted as α) represents the fraction of sound energy absorbed by a material when an acoustic wave hits it, and is to some extent related to the material’s flow resistivity. It is clear that this is only true within a certain frequency range: for instance, when the flow resistivity becomes very high, preventing the sound wave from penetrating the material, the material behaves almost like a reflective surface, with the absorption coefficient decreasing again, particularly at low frequencies.
However, within the range of interest — that is, for typical flow resistivity values of porous materials used in audio applications — we can expect a good correlation between the behavior of the absorption coefficient and the flow resistance itself. Based on this qualitative observation, I explored several mathematical forms that could reproduce this behavior. The exponential function proved particularly suitable, as it can describe a rapidly increasing trend that gradually stabilizes beyond a certain frequency. After various empirical tests, I arrived at the following formula:
where:
RI is the initial value of the flow resistance, i.e., the low-frequency value, and SpicyTL derives it directly from the material’s density;
k is the maximum growth factor relative to RI;
f₀ is the transition frequency, defining the range where the increase becomes significant;
N is an exponent that controls the slope of the curve.
As mentioned in the previous section, to implement this function in SpicyTL I adopted the same approach used for modeling the loudspeaker impedance — namely, the use of a voltage-controlled current source (VCCS) called G. The schematic of how this circuit is implemented in LTspice is shown in Figure 3.
The G device generates an output current Iout proportional to the voltage difference across its control terminals. By connecting the control terminals to the same output nodes, the control voltage coincides with the voltage across the generator itself. In this way, the generated current is directly proportional to the applied voltage, and the generator behaves like a resistor.
As mentioned above, to obtain a resistance whose value varies with frequency, we can use the Laplace function, which is natively supported in LTspice, and apply it directly to G. According to the software manual, the transfer function of this element must be expressed as a function of the complex variable s. Since s = jω = j2πf, we can rewrite our formula as follows:
When rewriting the formula as a function of s, I used a small mathematical “trick” to eliminate the imaginary part (since resistance a real quantity). In practice, I used the magnitude of the variable s, that is |s|, which corresponds to 2πf. The correct syntax in the SPICE environment is: abs(s).
Another thing to keep in mind when using the Laplace function is that resistance must be defined as conductance — that is, the inverse of resistance (1/R). Therefore, the directive to be entered in LTspice becomes:
Laplace=1/(RI+(k-1)*RI*(1-exp(-((abs(s)/(2*pi*f_0))^N))))
A small note for LTspice tinkerers: due to the way the circuit solver handles hierarchies, it was not straightforward to transfer the parameters used in the Laplace function (which is located inside a subcircuit) to the main schematic, where it is much more convenient for SpicyTL users to set them.
To solve the problem, I rewrote the function as follows:
Laplace = 1 / (|Rf(s)|)
and linked it to a .text directive:
.text Rf(s) = "(RI+(k-1)*RI*(1-exp(-((abs(s)/(2*pi*f_0))^N))))/S_damp*Delta_z"
This directive was then saved in an .inc file (for example, param.inc) and called within the subcircuit using:
.inc param.inc
Unlike a .text directive written directly inside the body of the subcircuit (which would cause an error), the trick — discovered almost by chance — is that if the .text is inserted into the subcircuit via an .inc file, LTspice still processes it before expanding the subcircuit and treats it as a global definition. The parameters defined in the main schematic (k, f₀, N) are therefore already available at the time of substitution.
Edit 04/01/2026
With the latest LTspice update, the syntax for the Laplace function has slightly changed: it has become less permissive, but also more elegant.
The modulus symbol || is no longer allowed; however, since the modulus is already implicitly handled by the function, its use was in any case redundant.
In addition, it is now possible (and actually required) to use the .func directive to import a function saved in a .inc file.
The correct syntax therefore becomes:
Laplace = 1 / (Rf(s))
.func Rf(s) (RI+(k-1)RI(1-exp(-((abs(s)/(2pif_0))^N))))/S_damp*Delta_z
Let’s now see how the model responds to real poroelastic materials. I created six copies of the circuit shown in Figure 3 and set the parameters to replicate the absorption coefficient graph of Basotect G+ measured at different thicknesses. Despite its relative simplicity, the model appears to reproduce quite well the behavior of melamine resin (Fig. 4). It should be noted that the simulation graph represents the material’s flow resistance, while the graph for the real material represents its absorption coefficient. Therefore, the parameters cannot be directly transferred to SpicyTL but must be recalibrated empirically, taking into account the measurements made on the transmission line test setup. The parameter that varies the most with material thickness (and this can also be intuitively observed from the graph) is the transition frequency, which ranges from about 250 Hz for 100 mm thickness down to 2500 Hz for 20 mm.
Let’s finally apply the new model to SpicyTL and see first how it behaves with polyurethane foam (PUR). To get a broader picture, both simulations and measurements were extended up to 2 kHz, though they can be considered reliable only slightly above 1 kHz. It should be remembered that LTspice operates in the lumped domain — that is, using a one-dimensional lumped-parameter model. This approach is simple and effective for simulating low-frequency behavior, but it does not account for three-dimensional propagation, directivity, or the radiation of the acoustic wave into space. The test TL used is the one with the Dayton Audio RS100-4 loudspeaker, already described in [2]. A smooth foam strip 19 mm thick was inserted along the entire length of the TL, corresponding to a 50% fill of the cross-section.
In Figure 5 we can see how the model very closely reproduces the measured response at the line’s output — the most challenging part to match. The levels are not normalized: both the measurement and the simulation were taken 2 mm from the opening. Such an overlap would have been unthinkable with the old model based on a simple linear proportionality between frequency and flow resistance.
In Figure 6, we can see the same simulation performed with melamine foam; in this case as well, I cut a strip from a 19 mm sheet. The parameters used differ significantly from those of PUR, particularly in the frequency at which the material begins to act effectively, which is noticeably higher for melamine foam. The f₀ value for PUR is set at 500 Hz, compared to 900 Hz for the melamine foam.
If we directly compare the measurements for the two materials (Fig. 7), we can see that melamine allows a higher low-frequency output from the opening, along with better attenuation above roughly 300 Hz. Unfortunately, there is also a price to pay: SpicyTL provides quite accurate feedback on how the material affects the speed of sound within the transmission line. Estimating sound velocity from the impulse response is neither simple nor advisable [2, 3], but we can make a fairly accurate evaluation based on the position of the resonance peaks observed in the SPL measurement at the output. To achieve a perfect match, the speed of sound must be set to 200 m/s for PUR and 220 m/s for melamine. From this data, we can expect melamine to have a less pronounced effect on the virtual volume reduction caused by the damping material.
A further comment on Figure 7: earlier I stated that polyurethane foam starts to act significantly at a lower frequency than melamine foam, yet the graph seems to suggest the opposite. The explanation can be found in Figure 9, which also shows the parameter values used for simulating the two materials. The higher final value of flow resistance (determined by k) and the gentler slope (determined by N) cause melamine to “overtake” polyurethane as early as 200 Hz.
The situation is reversed when the filling ratio is increased to around 90% of the TL cross-section (Fig. 8). In this case, melamine excessively attenuates low frequencies, to the point that the output from the opening becomes almost negligible in the system’s overall response. At higher frequencies, it remains effective, but the foam catches up, resulting in a performance practically equivalent to that of melamine.
It remains to be clarified to what extent the observed behavior is determined by the actual thickness of the material and how much depends instead on the filling percentage of the line. At low filling levels, at least up to about 50%, experience suggests that it still makes sense to talk about thickness: the material is distributed along the walls, and the acoustic wave interacts with a relatively defined layer. However, as the filling percentage increases, the very concept of thickness tends to lose meaning, overlapping with that of the distance the wave must travel through the material. Under these conditions, melamine (probably due to its extremely fine porosity) no longer behaves as an absorbing layer but as a mass of material that the wave is forced to pass through entirely. And in this passage, rather than being absorbed, the wave is largely blocked or reflected.
This regime shift highlights how, in the design of TLs (and more generally in damped acoustic loads), it is not enough to simply choose the right material — it is equally essential to dose and position it correctly.
During the measurements carried out with high filling percentages, I also had a very clear visual confirmation: the pistonic motion of the loudspeaker caused the melamine to move visibly, which — unlike the foam — behaved almost like a plug under those conditions. This dynamic effect, in addition to further limiting the passage of the acoustic wave, can introduce unwanted and, above all, unpredictable reactive behaviors, since the simulation model assumes that the material remains static. To avoid this, it is important that the melamine be firmly fixed to the inner walls of the transmission line, even under normal operating conditions.
Conclusions
This work was originally conceived to explore the acoustic properties of expanded melamine foam as a damping material in transmission line loudspeakers, but it soon evolved into a broader analysis that required revising the SpicyTL simulation model — both for the loudspeaker impedance and for the modeling of the variable flow resistance of porous materials.
A new, simple yet flexible approach was developed to describe the variation of resistance as a function of frequency, based on a parametric function easily integrable into LTspice. The model was then compared with real measurements on the most common porous materials (PUR and melamine), showing excellent agreement.
Experimental observations clearly show that flow resistivity is strongly influenced by the actual thickness of the material, and that this parameter has a significant impact — particularly on the transition frequency f₀.
Excessive TL filling — especially with expanded melamine — introduces unwanted behaviors that negatively affect the system’s performance. When used in the right amount, melamine offers excellent performance across the entire audio frequency range of interest, but it requires larger volumes compared to polyurethane. This makes it particularly suitable for two-way designs with midwoofers, where limiting mid-frequency emission from the opening is important.
For subwoofers, which do not radiate in the mid-to-high spectrum, polyurethane foam may instead be preferable, as it offers well-established performance and allows a significant reduction in enclosure volume thanks to its strong influence on the speed of sound.
Some tests performed with melamine in the first third of the TL and PUR in the remaining section suggest that mixed configurations deserve further investigation.
The experience gained indicates that an optimal filling ratio is around 50% of the TL cross-section, providing a good balance between high-frequency damping and low-frequency performance. However, it would be useful to continue the investigation systematically, to better understand how the material thickness affects the TL response. This would help build a reference parameter base, useful for future applications and for easier model calibration.
References
[1] W. Marshall Leach, Loudspeaker Voice-Coil Inductance Losses: Circuit Models, Parameter Estimation, and Effect on Frequency Response (JAES, Vol. 50, No. 6, June 2002)
[2] Andrea Rubino, Designing a Transmission Line Using SPICE (AUDIOreview No. 408, April 2019; No. 409, May 2019; No. 410, June 2019)
[3] Joseph D’Appolito, Testing Loudspeakers (Audio Amateur Press, 1998)