# Acoustics/Sealed Box Subwoofer Design

A sealed or closed box baffle is the most basic but often the cleanest
sounding sub-woofer box design. The sub-woofer box in its most simple
form, serves to isolate the back of the speaker from the front, much
like the theoretical infinite baffle. The sealed box provides simple
construction and controlled response for most sub-woofer applications.
The slow low end roll-off provides a clean transition into the extreme
frequency range. Unlike ported boxes, the cone excursion is reduced
below the resonant frequency of the box and driver due to the added
stiffness provided by the sealed box baffle.

Closed baffle boxes are typically constructed of a very rigid material
such as MDF (medium density fiber board) or plywood .75 to 1 inch thick.
Depending on the size of the box and material used, internal bracing may
be necessary to maintain a rigid box. A rigid box is important to design
in order to prevent unwanted box resonance.

As with any acoustics application, the box must be matched to the
loudspeaker driver for maximum performance. The following will outline
the procedure to tune the box or maximize the output of the sub-woofer
box and driver combination.

## Closed baffle circuit

The sealed box enclosure for sub-woofers can be modelled as a lumped
element system if the dimensions of the box are significantly shorter
than the shortest wavelength reproduced by the sub-woofer. Most
sub-woofer applications are crossed over around 80 to 100 Hz. A 100 Hz
wave in air has a wavelength of about 11 feet. Sub-woofers typically
have all dimensions much shorter than this wavelength, thus the lumped
element system analysis is accurate. Using this analysis, the following
circuit represents a sub-woofer enclosure system.

![](Circuit_schema.jpg "Circuit_schema.jpg")

where all of the following parameters are in the mechanical mobility
analog

:   $V_e$ - voltage supply
:   $R_e$ - electrical resistance
:   $M_m$ - driver mass
:   $C_m$ - driver compliance
:   $R_m$ - resistance
:   $R_{Ar}$ - rear cone radiation resistance into the air
:   $X_{Af}$ - front cone radiation reactance into the air
:   $R_{Br}$ - rear cone radiation resistance into the box
:   $X_{Br}$ - rear cone radiation reactance into the box

## Driver parameters

In order to tune a sealed box to a driver, the driver parameters must be
known. Some of the parameters are provided by the manufacturer, some are
found experimentally, and some are found from general tables. For ease
of calculations, all parameters will be represented in the SI units
meter/kilogram/second. The parameters that must be known to determine
the size of the box are as follows:

:   $f_0$ - driver free-air resonance
:   $C_{MS}$ - mechanical compliance of the driver
:   $S_D$ - effective area of the driver

#### Resonance of the driver

The resonance of the driver is usually either provided by the
manufacturer or must be found experimentally. It is a good idea to
measure the resonance frequency even if it is provided by the
manufacturer to account for inconsistent manufacturing processes.

The following diagram shows verga and the setup for finding resonance:

Where voltage $V_1$ is held constant and the variable frequency source
is varied until $V_2$ is a maximum. The frequency where $V_2$ is a
maximum is the resonance frequency for the driver.

#### Mechanical compliance

By definition compliance is the inverse of stiffness or what is commonly
referred to as the spring constant. The compliance of a driver can be
found by measuring the displacement of the cone when known masses are
place on the cone when the driver is facing up. The compliance would
then be the displacement of the cone in meters divided by the added
weight in Newtons.

#### Effective area of the driver

The physical diameter of the driver does not lead to the effective area
of the driver. The effective diameter can be found using the following
diagram:

![](Effective_area.jpg "Effective_area.jpg")

From this diameter, the area is found from the basic area of a circle
equation.

## Acoustic compliance

From the known mechanical compliance of the cone, the acoustic
compliance can be found from the following equation:

$V_{as} = P C^2 C_{ms} S_d^2$

Where $P$ is air density and $C$ the speed of sound at a given
temperature and pressure.

From the driver acoustic compliance, the box acoustic compliance is
found. This is where the final application of the sub-woofer is
considered. The acoustic compliance of the box will determine the
percent shift upwards of the resonant frequency. If a large shift is
desire for high SPL applications, then a large ratio of driver to box
acoustic compliance would be required. If a flat response is desired for
high fidelity applications, then a lower ratio of driver to box acoustic
compliance would be required. Specifically, the ratios can be found in
the following figure using line (b) as reference.

$C_{AS} = C_{AB}r$

$r$ - driver to box acoustic compliant ratio

![](Compliance.jpg "Compliance.jpg")

## Sealed box design

#### Volume of box

The volume of the sealed box can now be found from the box acoustic
compliance. The following equation is used to calculate the box volume

V~B~= C~AB~&gam

#### Box dimensions

From the calculated box volume, the dimensions of the box can then be
designed. There is no set formula for finding the dimensions of the box,
but there are general guidelines to be followed. If the driver was
mounted in the center of a square face, the waves generated by the cone
would reach the edges of the box at the same time, thus when combined
would create a strong diffracted wave in the listening space. In order
to best prevent this, the driver should be either be mounted offset of a
square face, or the face should be rectangular.

The face of the box which the driver is set in should not be a square.

