Fundamentals and Port-Modeling of Loudspeakers
An example of multiphysics interaction
A loudspeaker is an ideal example to demonstrate multiphysics modeling that covers the domain of electric-magnetic, mechanic, and acoustic. Starting from introducing the structure of a voice-coil loudspeaker, the dynamics will be derived utilizing port-modeling that captures the bilateral interactions between each domain. The takeaway of this article is the basics of loudspeakers including the relations between physical parameters and the performances (e.g. bandwidth and the driving efficiency/sensitivity). As an extra, the design methodology of the speaker enclosure will also be briefly discussed based on those relations.
Voice-coil loudspeaker
Voice-coil is one of the most popular actuators used as an acoustic source, as the name implies, it was composed of an electric coil actuator and a suspended diaphragm each detailed and shown in the figure below.
Voice-coil actuator
Similar to the electric motor (specifically a linear motor), the coil was driven by an external amplifier, resulting in an alternating current, which eventually pushes the coil back and forth through the electromagnetic force. Bilaterally, while creating mechanical forces, the inductive voltage caused by the moving coil will be applied to the circuit. As you may guess, the reversed operation of the voice-coil is essentially a microphone that transforms acoustic vibration back into electric signals.
Diaphragm and the suspension
Suspended by the frame and the pocket neck (also called the “spider”), the diaphragm was limited to an axle motion and will sit stably at the equilibrium point without any applied signal. As the interface from the mechanical motion to the acoustic environment, the cone surface is expected to be stiff or light enough such that the motion of each point is in phase. In other words, in the interested frequency range, no internal oscillation should occur. Although violating this assumption is not critical in the dynamics since the acoustic load is negligible in most cases, it does affect the discussion later in the section “Baffled piston model”.
Modeling and transfer functions
The modeling of each domain will be carried out separately, then linked through the multiphysics port such as transformer or gyrator depending on the perspective.
Voice-coil dynamic
According to its construction, a resistive inductor with additional inductive voltage covers the story well.
Diaphragm dynamic
The dynamic equation of a spring-loaded mass (the diaphragm) can be analogously described as an RLC electric circuit where the effort was chosen as the speed of the object for this case. Alternatively, it is also possible to select force as the effort, which will result in a series-connected RLC circuit.
Acoustic load
With a vibrating diaphragm, the pressure and volume rate applied between the cone surface and the atmosphere effectively creates a mechanical load. This interaction can also be realized as a port connection where the speed and the force (with opposite signs) were shared at the contact. Unlike the lumped models already performed above, an acoustic system was in nature a distributed (infinite-order) system and theoretically leads to a complicated expression.
Luckily, this load is negligible in most cases and the mechanical dynamics were dominated by the diaphragm and its suspension. Following such an assumption, the load could be interpreted as an open circuit in a speed-effort realization or a short circuit in a force-effort realization. Either way, the interaction between the diaphragm and the atmosphere degenerated from a bi-lateral to a uni-directional relation (passing the speed without any reaction force).
Baffled piston model
Viewed as multiple sources, the acoustic emission of the diaphragm can be well modeled as a baffled piston in the limitation of a synchronized surface motion. Referring to [1,2], the amplitude of the far-field pressure can be approximated through the following equation. The blue-colored directivity is out of the scope and readers may focus only on the orange-colored part, which is in proportion to the speed of the diaphragm “Q∝u_cone” and the frequency “ω”.
Interaction port, gyrator, and the transformer
The interaction between physical domains can be modeled either by the gyrator (anti-symmetric flow and effort), or the transformer (symmetric flow and effort) and is always related to some form of energy exchanging. Inheriting the interpretation used in the model provided above, the “transformer” gain was the motor constant “K_em” and roughly the surface area of the diaphragm “A_dp” correspondingly.
Total impedance and the output transfer
With each domain modeled with ports and interaction as transformers, a combinational result of the whole system can be derived following the procedure shown below.
The numerical result can be carried out using the parameters from the commercial speaker unit (FE103NV) and depicted in the figure below. As one may observe, the effect of the mechanical resonance around 100Hz was reflected in the electrical port as expected. It might sound a bit weird to have a higher impedance at the resonance but remind that higher impedance implies a lower power transfer (higher resistance is “easier” to drive).
The transfer function from the amplifier input “V_amp” to the far-field pressure “P” was also straightforward as shown.
The bandwidth of the speaker started around the mechanical resonance and stops at the RL cutoff frequency.
The theoretical results can be compared with the measurement from the manufacturer.
Remarks and extras
From the analysis performed above, some useful relations can be claimed.
Sensitivity/efficiency
The gain at the flat region from the input voltage to the sound pressure is roughly in proportion to both the motor constant “K_em” and the diaphragm area “A_dp”. Furthermore, a lighter diaphragm and coil (moving parts) also increase the efficiency.
Cutoff frequency
A speaker can barely produce output lower than its natural frequency, thus always check this specification before purchasing a unit. On the other hand, the upper limit is bounded by the inductive part. Notice that the model provided above neglected any internal oscillation which breaks the baffle piston model and is the cause of the strong variation in the mid to higher frequency region.
Enclosure design
Unfortunately, a free operating speaker consists of two opposite phased baffle pistons on two sides of the diaphragm and degrades the performance. The central idea of a speaker enclosure is to hide the unwanted emission from the back of the side. The most ideal setup might be an infinite baffle board or practically a large-sized one. Other approaches use a nearly enclosed box and result in a compact appearance but also induce new issues.
Enclosure induced acoustic load
In the theoretical model provided above, the acoustic load “Za” was assumed to be small and negligible, this assumption breaks with a finite-sized enclosure. For instance, a fully sealed enclosure acts as an additional spring to the diaphragm and potentially raised the natural frequency and kills the bass. Even worse, the distributed system of the fluid in the enclosure (i.e. the air) has its dynamics and resonant modes. Similar to the mechanical modes of the suspended diaphragm, those modes from internal space also affect the total impedance and create variations in the output response.
The internal design affects the output transfer through its effective acoustic load. You can “hear” the shape of the enclosure!!
Playing with the design
Knowing that the acoustic load induced by the enclosure affects the response, one can avoid or even utilize it. A trivial one is to fill the internal space with cotton or similar material expecting to suppress the internal modes, which is popular in the audio community.
Some designs purposely induced a resonance by the Helmholtz resonator placed a bit lower than the natural frequency that extends the lower frequency range. With the same speaker unit (FE103NV) used above, the total response with the following enclosure was provided in the next figure. As expected, the enclosed structure raised the natural frequency (92 to 110Hz) and the impedance of the acoustic load (the resonator) was reflected as the additional peak on the electric side.
The article ends here and I hope you find out interesting.
References
[1] Cohen, Abraham B, Hi-Fi Loudspeakers and Enclosures, Hayden, 1968
[2] Morse, P.M. & Ingard, K. U., Theoretical Acoustics, McGraw-Hill, 1968
[3] https://diyaudioprojects.com/Speakers/Seas-Granite-Speakers/