# Engineering Acoustics/Sealed Box Subwoofer Design

## Introduction

A sealed or closed box baffle is the most basic but often the cleanest
sounding subwoofer box design. The subwoofer 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 subwoofer 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 subwoofer
box and driver combination.

## Closed Baffle Circuit

The sealed box enclosure for subwoofers can be modeled as a lumped
element system if the dimensions of the box are significantly shorter
than the shortest wavelength reproduced by the subwoofer. Most subwoofer
applications are crossed over around 80 to 100 Hz. A 100 Hz wave in air
has a wavelength of about 11 feet. Subwoofers 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 subwoofer enclosure system.

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

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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 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 the setup for finding resonance:

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Where voltage V1 is held constant and the variable frequency source is
vaied
until V2 is a maximum. The frequency where V2 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:

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

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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:

C~AS~ = C~MS~S~D~^2^

From the driver acoustic compliance, the box acoustic compliance is
found. This is where the final application of the subwoofer 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 more flattened response is
desire 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 compliance ratio

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

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## 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~γ

#### 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. The face of the box
which the driver is set in should not be a square. If the driver were
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 either be mounted offset on a
square face, or the face should be rectangular, with the driver closer
to one edge.

The ratios between internal height, width and depth should never be
integer (2:1, 3:1 etc.), as this would encourage the formation of
standing waves inside the box. Some have suggested the use of the
Golden ratio and others the
third root of 2, both of
which are close to each other and close to the
IEC-recommended
ratios for room dimensions (which conform to the same acoustical
requirements). In practice most manufacturers formulate their boxes
based on aesthetic and cost considerations, while ensuring, through
testing, that no major box resonances appear. In high-quality units this
entails the extensive use of rigid in-box bracing, sound absorption
material, sophisticated alloys or polymers, complex geometrical shapes,
including curves, etc.
