There are two main steps in building a speaker that sounds good: choosing the right driver and designing an optimal enclosure. There are tradeoffs to consider in both steps, including loudness, bass response, size, weight, cost, and many more. Our goal is to make the best speaker possible while allowing for portability, and keeping cost at a reasonable level.
Before going on to the actual prototyping and experimenting, it will be helpful to explain some of the theory behind speakers and subwoofer enclosures. In this log we will discuss our research into the theory of speaker design.
Speakers are electrical, mechanical, and acoustic devices, meaning they transduce electrical energy into mechanical vibration, which in turn creates air pressure waves (sound). The mathematics behind this transformation can be described by a set of physical specifications known as the Thiele and Small parameters. These parameters are named after a pair of professors who pioneered the electric-acoustic modeling of speaker enclosures.
Breakdown of a speaker. Source: https://en.wikipedia.org/wiki/File:Speaker-cross-section.svg
Basically, a speaker can be described as a sort of damped harmonic oscillator. Or in other words, the mass hanging from the end of a spring. The Thiele and Small parameters relate the speakers to this mass spring system. Both the speaker and the mass/spring will have a natural resonant frequency.
For a more in depth resource on the topic, see: https://en.wikibooks.org/wiki/Engineering_Acoustics
Image source: http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html
Why do speakers need enclosures? Well, when the diaphragm of a speaker pushes forward, it generates positive pressure in front of it and negative pressure behind it. Without an enclosure the pressure waves will fan-out around the speaker and cancel themselves out.
Image source: https://soundphysics.ius.edu/?page_id=1343
The simplest type of enclosure is a sealed enclosure, which completely encloses the back side of the speaker and stops the backwards generated waves from escaping. More complex designs, such as ported enclosures, allow some of these backward waves to reinforce the forward waveform, thereby making the speaker louder. If the system is properly tuned (just like a damped harmonic oscillator) it can even help the speaker produce lower frequencies.
There are many varieties of ported enclosures, the most common being a bass-reflex or vented system, in which a hole (or vent) is cut into the cabinet. A passive radiator design exhibits similar properties as a bass-reflex system, but has a few benefits that fit well with our goals. It requires less volume, is closed off to the outside, and is easier to tune. Instead of cutting a new hole (or making a new enclosure) as is required to tune a vented design, you can simply add or remove mass from the passive radiator. The lack of a hole also makes it much simpler to protect the electronics and driver when used outside.
Back in Thiele and Small's days the frequency response of this speaker/enclosure system would be determined by comparing it to set of precomputed response (or "Alignments"). Nowadays, speaker modeling software allows one to quickly compute the frequency response for changes to the enclosure (e.g. adding weight to the passive radiators).
For electrical engineers like myself, can be helpful to view the transformation of electricity into acoustic energy for a passive radiator speaker enclosure as the following circuit (Warning - PDF).
One thing to note is that resonance of the speaker (and that of the passive radiator) will show up as electrical impedance maximums or minimums at the input to the speaker. This gives us a simple method to experimentally determine the resonant frequency of our DIY passive radiators (and verify speaker manufacturer’s stated values).
The resonant frequency together with the weight and surface area of the passive radiator allows us to almost completely characterize the passive radiator as a dampened tuned mass. However, the dampening factor cannot be determined solely by this method. (If only we had a working laser interferometer to measure the cone displacement of the passive radiator for given sound pressure level). Nonetheless, it does give us enough information to verify our speaker simulation models.