Sealed Enclosure: Overview, Background
Andy Lewis rev 9/8/2011
Part 1: Concept: How and Why it Came to This!
The definitive analysis of the closed box loudspeaker system was done by Richard H. Small (of Thiele-Small fame), and was published in December 1972 in the Journal of the Audio Engineering Society. All modern texts describing this type of system are derivative of Small’s work. Earlier pioneers, such as Edgar Villchur, Henry Kloss, and Rudy Bozak, certainly understood the concepts, but Small literally "wrote the book" by inventing a way to precisely describe it mathematically, and to create models of real world systems.
(Small offered a detailed piece on building an actual loudspeaker based of his technology in the Jan/Feb, 1973 issue of the JAES. My methods are distillations of his analyses, informed by many years of practical application and careful measurements.)
On the first two pages of Small’s piece he describes, by my count, 52 variables. He examines every parameter of the driver, the enclosure, the voice coil, the magnetic field, the damping material, and in short, every imaginable aspect of the system in all its parts, all accounted for in a quantitative way. The man is a genius, and his work is a tour de force.
But in almost 40 years of relying upon his work to design my own systems, I have found that his work can be streamlined for the real world in two important ways.
The first is that a great deal of Small’s work is devoted to the interaction of the voice coil and magnetic field, and to efficiency and power considerations. This is all very important stuff, but much of it becomes superfluous after a driver is selected. If one assumes that an appropriate driver has been selected, and that its power considerations and excursion capabilities have been addressed, it becomes possible to ignore a great many of Small’s variables, and to concentrate only on the ones left over.
The second is that once the driver is actually mounted in the enclosure, the specific parameters describing the raw driver, such as fs, Qts, and Vas, are subsumed into the greater whole, and even these important numbers “fall away,” and are no longer necessary to predict the performance of the filter.
A sealed system is a second-order filter. It has a system resonance, a “Q” which defines the shape of its frequency response curve, and a sensitivity rating. These three parameters completely describe the performance of the system filter.
It is almost alarming to hear it expressed so bluntly and definitively, but it’s the truth:
Once a driver is mounted in an enclosure, and issues of power and excursion have been addressed, only three of Small’s 52 variables are left: system resonance (fc), system damping, (Qtc), and sensitivity (usually expressed in dB).
I have created a simple spreadsheet in Excel, which allows a person to manipulate these three variables as desired, to model an infinite variety of sealed systems. This most general of models does not address issues of enclosure size, woofer specifics, or other real-world considerations, and is not meant to be a design tool. It is a learning tool only, and a powerful illustration of the ultimate simplicity of the ubiquitous sealed loudspeaker. It’s a fun toy to play with, too!
Here it is for your enjoyment: Model General Sealed Enclosure
Part 2: How to Apply this to a Specific Woofer.
The second spreadsheet is a little more advanced, and allows a similar look at the performance of a simple sealed system when used with a specific woofer. It is closer to being a useful design tool, as opposed to a theoretical curiosity.
The above Model General Sealed Enclosure spreadsheet obviously can't be used to design a sealed-box enclosure, only to illustrate how they work. When using a specific woofer, there are far fewer possibilities. The characteristics and parameters of the driver dictate what can and can't be done with it.
Obviously, you can put a woofer in any sized box you want to. The response and damping will change depending on the effective size of the box.
As explained above, my approach is to consolidate as many variables as possible, and to manipulate what is left. In the case of the above Model General Sealed Enclosure, the relevant remaining variables are system resonance (fc), system damping, (Qtc), and sensitivity (usually expressed in dB). For every scenario, each of these parameters were inputted and affected the curve.
When creating a model then, obviously, the specifications of the woofer circumscribe how it can and must work in the real world. The information which needs to be inputted is specific to the driver, and the only variable remaining is enclosure size. But wait!
Enclosure size results in a precise in-box resonance, fc, which results in an easily calculable system damping Qtc.
Ultimately, enclosure size is not the most important thing. Performance is. So we return to Richard Small, who laid it out for us.
I believe the single most important equation in Small's seminal work describes a parameter not usually published nor emphasized in popular texts.
The precise relationship is described by Small in the above referenced article:
fs/Qts = fc/Qtc
where:
fs = driver resonant frequency
Qts = driver damping
fc = in-box resonance
Qtc = in-box system damping
In other words, there is a lockstep relationship between resonance and damping for a specific woofer. It is a constant regardless of enclosure size. Every change in resonance is accompanied by a proportional change in damping. This makes it very easy to design a model of the sealed enclosure for a specific woofer. You only need change one variable to know exactly what the response curve and damping will be.
I have written this sheet with Qtc as the missing variable. It could have been done just as easily with fc,or effective enclosure volume as the variable to be inputted, but I use Qtc because I'm comfortable with it. If you want to rewrite a similar spreadsheet using one of the others, feel free.
(Note: If you're more comfortable inputting enclosure volume, as opposed to system damping, will be able to do so when you advance to the Evaluate Woofer Sealed or Vented spreadsheet, which is a useful tool for evaluating woofers as you peruse your favorite catalog. But don't spend too much time on that now.)
But, in any event, here it is. See if you can find a way to use it to help you understand the sealed enclosure, of even to design one of your own. Just enter your driver parameters, then manipulate Qtc. The spreadsheet will calculate effective enclosure volume, and generate the correct response curve.