Lumped Element Modeling of Transducers

Many engineering tasks require a common ground from which discussions among those involved in a project can take place in a structured manner. This will often involve a so-called model of the concept or product, and this article discusses one such model: the lumped element model.

- René Christensen

Lumped element modeling involves representing continuum-physics such as a loudspeaker with various three-dimensional parts as discrete or lumped zero-dimensional components and make so-called analogy circuits that to some degree capture the physics of the transducer involved. The crux of making these analogy circuits is to realize that there is an equivalence such that the equations relating the primary and secondary variables in one physics are analogous to the equivalent equations in other 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, using 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 notice the similarities.

So, one 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 some complex circuits later in this article utilizing this mapping, but first imagine the situation shown in Figure 1 with two springs being compressed by a common force. It should be easy enough to realize that the total 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 total compliance as: