Bassreflex enclosures improve the lowfrequency
response of loudspeaker systems. Bassreflex enclosures are also called
"ventedbox design" or "portedcabinet design". A
bassreflex enclosure includes a vent or port between the cabinet and the
ambient environment. This type of design, as one may observe by looking at
contemporary loudspeaker products, is still widely used today. Although the
construction of bassreflex enclosures is fairly simple, their design is not
simple, and requires proper tuning. This reference focuses on the technical
details of bassreflex design. General loudspeaker information can be found here.
Before discussing the bassreflex enclosure, it is
important to be familiar with the simpler sealed enclosure system performance.
As the name suggests, the sealed enclosure system attaches the loudspeaker to a
sealed enclosure (except for a small air leak included to equalize the ambient
pressure inside). Ideally, the enclosure would act as an acoustical compliance
element, as the air inside the enclosure is compressed and rarified. Often,
however, an acoustic material is added inside the box to reduce standing waves,
dissipate heat, and other reasons. This adds a resistive element to the
acoustical lumpedelement model. A nonideal model of the effect of the
enclosure actually adds an acoustical mass element to complete a series
lumpedelement circuit given in Figure 1. For more on sealed enclosure design,
see the Sealed Box Subwoofer Designpage.
Figure 1. Sealed
enclosure acoustic circuit.
In the case of a bassreflex enclosure, a port is
added to the construction. Typically, the port is cylindrical and is flanged on
the end pointing outside the enclosure. In a bassreflex enclosure, the amount
of acoustic material used is usually much less than in the sealed enclosure
case, often none at all. This allows air to flow freely through the port.
Instead, the larger losses come from the air leakage in the enclosure. With
this setup, a lumpedelement acoustical circuit has the following form.
Figure 2. Bassreflex enclosure acoustic circuit.
In this figure, Z_{RAD} represents the radiation impedance of the
outside environment on the loudspeaker diaphragm. The loading on the rear of
the diaphragm has changed when compared to the sealed enclosure case. If one
visualizes the movement of air within the enclosure, some of the air is
compressed and rarified by the compliance of the enclosure, some leaks out of
the enclosure, and some flows out of the port. This explains the parallel
combination of M_{AP}, C_{AB}, and R_{AL}. A truly realistic model would
incorporate a radiation impedance of the port in series with M_{AP}, but for now it is ignored. Finally, M_{AB}, the acoustical mass of the enclosure, is included as
discussed in the sealed enclosure case. The formulas which calculate the
enclosure parameters are listed in Appendix B.
It is important to note the parallel combination of M_{AP} and C_{AB}. This forms a Helmholtz resonator (click
here for more information). Physically, the port functions as the “neck” of the
resonator and the enclosure functions as the “cavity.” In this case, the
resonator is driven from the piston directly on the cavity instead of the
typical Helmholtz case where it is driven at the “neck.” However, the same
resonant behavior still occurs at the enclosure resonance frequency, f_{B}. At this frequency, the impedance seen by the
loudspeaker diaphragm is large (see Figure 3 below). Thus, the load on the
loudspeaker reduces the velocity flowing through its mechanical parameters,
causing an antiresonance condition where the displacement of the diaphragm is
a minimum. Instead, the majority of the volume velocity is actually emitted by
the port itself instead of the loudspeaker. When this impedance is reflected to
the electrical circuit, it is proportional to 1 / Z, thus a minimum in the impedance seen by the voice
coil is small. Figure 3 shows a plot of the impedance seen at the terminals of
the loudspeaker. In this example, f_{B} was found to be about 40 Hz, which corresponds
to the null in the voicecoil impedance.
Figure 3. Impedances seen by the loudspeaker diaphragm and voice coil.
The performance of the loudspeaker is first measured
by its velocity response, which can be found directly from the equivalent
circuit of the system. As the goal of most loudspeaker designs is to improve
the bass response (leaving highfrequency production to a tweeter), low
frequency approximations will be made as much as possible to simplify the
analysis. First, the inductance of the voice coil, L_{E}, can be ignored as long as . In a typical loudspeaker, L_{E} is of the order of 1 mH, while R_{E} is typically 8Ω, thus an upper frequency limit is
approximately 1 kHz for this approximation, which is certainly high enough for
the frequency range of interest.
Another approximation involves the radiation
impedance, Z_{RAD}. It can be shown [1] that this value is
given by the following equation (in acoustical ohms):
Where J_{1}(x) and H_{1}(x) are types of Bessel functions. For small values of ka,

and 


Hence, the lowfrequency impedance on the loudspeaker
is represented with an acoustic mass M_{A}_{1} [1]. For a simple analysis, R_{E}, M_{MD}, C_{MS}, and R_{MS} (the transducer parameters, or ThieleSmall parameters) are converted to their
acoustical equivalents. All conversions for all parameters are given in
Appendix A. Then, the series masses, M_{AD}, M_{A}_{1}, and M_{AB}, are lumped together to create M_{AC}. This new circuit is shown below.
Figure 4. LowFrequency Equivalent Acoustic Circuit
Unlike sealed enclosure analysis, there are multiple
sources of volume velocity that radiate to the outside environment. Hence, the
diaphragm volume velocity, U_{D}, is not analyzed but rather U_{0} = U_{D} + U_{P} + U_{L}. This essentially draws a “bubble” around the
enclosure and treats the system as a source with volume velocity U_{0}. This “lumped” approach will only be valid for low
frequencies, but previous approximations have already limited the analysis to
such frequencies anyway. It can be seen from the circuit that the volume
velocity flowing into the
enclosure, U_{B} = − U_{0}, compresses the air inside the enclosure. Thus, the
circuit model of Figure 3 is valid and the relationship relating input voltage, V_{IN} to U_{0} may be computed.
In order to make the equations easier to understand,
several parameters are combined to form other parameter names. First, ω_{B} and ω_{S}, the enclosure and loudspeaker resonance frequencies, respectively,
are:


Based on the nature of the derivation, it is
convenient to define the parameters ω_{0} and h,
the Helmholtz tuning ratio:


A parameter known as the compliance ratio or volume
ratio, α, is given by:
Other parameters are combined to form what are known
as quality factors:


This notation allows for a simpler expression for the
resulting transfer function [1]:
where



It can be shown [2] that for ka < 1 / 2, a loudspeaker behaves as a spherical source. Here, a represents the radius of the
loudspeaker. For a 15” diameter loudspeaker in air, this low frequency limit is
about 150 Hz. For smaller loudspeakers, this limit increases. This limit
dominates the limit which ignores L_{E}, and is consistent with the limit that models Z_{RAD} by M_{A}_{1}.
Within this limit, the loudspeaker emits a volume
velocity U_{0}, as determined in the previous section. For a simple
spherical source with volume velocity U_{0}, the farfield pressure is given by [1]:
It is possible to simply let r = 1 for this analysis without loss of
generality because distance is only a function of the surroundings, not the
loudspeaker. Also, because the transfer function magnitude is of primary
interest, the exponential term, which has a unity magnitude, is omitted. Hence,
the pressure response of the system is given by [1]:
Where H(s) = sG(s). In the following sections, design
methods will focus on  H(s)  ^{2} rather than H(s), which is given by:


This also implicitly ignores the constants in front of  H(s)  since they simply scale the response and do not affect
the shape of the frequency response curve.
A popular way to determine the ideal parameters has
been through the use of alignments. The concept of alignments is based upon
filter theory. Filter development is a method of selecting the poles (and
possibly zeros) of a transfer function to meet a particular design criterion.
The criteria are the desired properties of a magnitudesquared transfer
function, which in this case is  H(s)  ^{2}. From any of the design criteria, the poles (and
possibly zeros) of  H(s)  ^{2} are found, which can then be used to
calculate the numerator and denominator. This is the “optimal” transfer
function, which has coefficients that are matched to the parameters of  H(s)  ^{2} to compute the appropriate values that
will yield a design that meets the criteria.
There are many different types of filter designs, each
which have tradeoffs associated with them. However, this design is limited
because of the structure of  H(s)  ^{2}. In particular, it has the structure of a
fourthorder highpass filter with all zeros at s = 0. Therefore, only those filter
design methods which produce a lowpass filter with only poles will be
acceptable methods to use. From the traditional set of algorithms, only
Butterworth and Chebyshev lowpass filters have only poles. In addition,
another type of filter called a quasiButterworth filter can also be used,
which has similar properties to a Butterworth filter. These three algorithms
are fairly simple, thus they are the most popular. When these lowpass filters
are converted to highpass filters, the transformation produces s^{8} in the numerator.
More details regarding filter theory and these
relationships can be found in numerous resources, including [5].
The Butterworth algorithm is designed to have a maximally flat pass band. Since the slope of a
function corresponds to its derivatives, a flat function will have derivatives
equal to zero. Since as flat of a pass band as possible is optimal, the ideal
function will have as many derivatives equal to zero as possible at s = 0. Of course, if all derivatives
were equal to zero, then the function would be a constant, which performs no
filtering.
Often, it is better to examine what is called the loss function. Loss is the
reciprocal of gain, thus
The loss function can be used to achieve the desired
properties, then the desired gain function is recovered from the loss function.
Now, applying the desired Butterworth property of
maximal passband flatness, the loss function is simply a polynomial with
derivatives equal to zero at s = 0. At the same time, the original
polynomial must be of degree eight (yielding a fourthorder function). However,
derivatives one through seven can be equal to zero if [3]
With the highpass transformation ,
It is convenient to define Ω = ω / ω_{3}_{dB}, since or 3 dB. This defintion allows the
matching of coefficients for the  H(s)  ^{2} describing the loudspeaker response when ω_{3}_{dB} = ω_{0}. From this matching, the following design equations
are obtained [1]:


The quasiButterworth alignments do not have as
welldefined of an algorithm when compared to the Butterworth alignment. The
name “quasiButterworth” comes from the fact that the transfer functions for
these responses appear similar to the Butterworth ones, with (in general) the
addition of terms in the denominator. This will be illustrated below. While
there are many types of quasiButterworth alignments, the simplest and most
popular is the 3rd order alignment (QB3). The comparison of the QB3
magnitudesquared response against the 4th order Butterworth is shown below.


Notice that the case B = 0 is the Butterworth alignment. The reason
that this QB alignment is called 3rd order is due to the fact that as B increases, the slope approaches 3
dec/dec instead of 4 dec/dec, as in 4th order Butterworth. This phenomenon can
be seen in Figure 5.
Figure 5: 3rdOrder QuasiButterworth Response for
Equating the system response  H(s)  ^{2} with  H_{QB}_{3}(s)  ^{2}, the equations guiding the design can be
found [1]:




The Chebyshev algorithm is an alternative to the
Butterworth algorithm. For the Chebyshev response, the maximallyflat passband
restriction is abandoned. Now, a ripple,
or fluctuation is allowed in the pass band. This allows a steeper transition or
rolloff to occur. In this type of application, the lowfrequency response of
the loudspeaker can be extended beyond what can be achieved by Butterworthtype
filters. An example plot of a Chebyshev highpass response with 0.5 dB of
ripple against a Butterworth highpass response for the sameω_{3}_{dB} is shown below.
Figure 6: Chebyshev vs. Butterworth HighPass Response.
The Chebyshev response is defined by [4]:
C_{n}(Ω) is called the Chebyshev polynomial and is defined by [4]:

cos[ncos ^{− 1}(Ω)] 
 Ω  < 1 
cosh[ncosh ^{− 1}(Ω)] 
 Ω  > 1 
Fortunately, Chebyshev polynomials satisfy a simple
recursion formula [4]:
C_{0}(x) = 1 
C_{1}(x) = x 
C_{n}(x) = 2xC_{n}_{ }_{− 1} − C_{n}_{ }_{− 2} 
For more information on Chebyshev polynomials, see the Wolfram Mathworld: Chebyshev Polynomials page.
When applying the highpass transformation to the 4th
order form of , the desired response has the form [1]:
The parameter ε determines the ripple. In particular, the
magnitude of the ripple is 10log[1 + ε^{2}] dB and can be chosen by the designer,
similar to B in the quasiButterworth case. Using
the recursion formula for C_{n}(x),
Applying this equation to  H(jΩ)  ^{2} [1],




Thus, the design equations become [1]:






With all the equations that have already been
presented, the question naturally arises, “Which one should I choose?” Notice
that the coefficients a_{1}, a_{2}, and a_{3} are not simply related to the parameters
of the system response. Certain combinations of parameters may indeed
invalidate one or more of the alignments because they cannot realize the
necessary coefficients. With this in mind, general guidelines have been
developed to guide the selection of the appropriate alignment. This is very
useful if one is designing an enclosure to suit a particular transducer that
cannot be changed.
The general guideline for the Butterworth alignment
focuses on Q_{L} and Q_{TS}. Since the three coefficients a_{1}, a_{2}, and a_{3} are a function of Q_{L}, Q_{TS}, h,
and α, fixing one of these parameters yields
three equations that uniquely determine the other three. In the case where a
particular transducer is already given, Q_{TS} is essentially fixed. If the desired
parameters of the enclosure are already known, then Q_{L} is a better starting point.
In the case that the rigid requirements of the
Butterworth alignment cannot be satisfied, the quasiButterworth alignment is
often applied when Q_{TS} is not large enough.. The addition of
another parameter, B,
allows more flexibility in the design.
For Q_{TS} values that are too large for the
Butterworth alignment, the Chebyshev alignment is typically chosen. However,
the steep transition of the Chebyshev alignment may also be utilized to attempt
to extend the bass response of the loudspeaker in the case where the transducer
properties can be changed.
In addition to these three popular alignments,
research continues in the area of developing new algorithms that can manipulate
the lowfrequency response of the bassreflex enclosure. For example, a 5th
order quasiButterworth alignment has been developed [6]. Another example [7]
applies rootlocus techniques to achieve results. In the modern age of
highpowered computing, other researchers have focused their efforts in
creating computerized optimization algorithms that can be modified to achieve a
flatter response with sharp rolloff or introduce quasiripples which provide a
boost in subbass frequencies [8].
[1] Leach, W. Marshall, Jr. Introduction to Electroacoustics
and Audio Amplifier Design. 2nd ed. Kendall/Hunt,
Dubuque, IA. 2001.
[2] Beranek, L. L. Acoustics.
2nd ed. Acoustical Society of America, Woodbridge, NY. 1993.
[3] DeCarlo, Raymond A. “The Butterworth
Approximation.” Notes from ECE 445. Purdue University. 2004.
[4] DeCarlo, Raymond A. “The Chebyshev Approximation.”
Notes from ECE 445. Purdue University. 2004.
[5] VanValkenburg, M. E. Analog Filter Design. Holt,
Rinehart and Winston, Inc. Chicago, IL. 1982.
[6] Kreutz, Joseph and Panzer, Joerg. "Derivation
of the QuasiButterworth 5 Alignments." Journal
of the Audio Engineering Society. Vol. 42, No. 5, May 1994.
[7] Rutt, Thomas E. "RootLocus Technique for
VentedBox Loudspeaker Design." Journal
of the Audio Engineering Society. Vol. 33, No. 9, September 1985.
[8]
Simeonov, Lubomir B. and ShopovaSimeonova, Elena. "PassiveRadiator Loudspeaker System Design
Software Including Optimization Algorithm." Journal of the Audio Engineering
Society. Vol. 47, No. 4, April 1999.
Name 
Electrical Equivalent 
Mechanical Equivalent 
Acoustical Equivalent 
VoiceCoil Resistance 
R_{E} 


Driver (Speaker) Mass 
See C_{MEC} 
M_{MD} 

Driver (Speaker) Suspension Compliance 
L_{CES} = (Bl)^{2}C_{MS} 
C_{MS} 

Driver (Speaker) Suspension Resistance 

R_{MS} 

Enclosure Compliance 


C_{AB} 
Enclosure AirLeak Losses 


R_{AL} 
Acoustic Mass of Port 


M_{AP} 
Enclosure Mass Load 
See C_{MEC} 
See M_{MC} 
M_{AB} 
LowFrequency Radiation Mass Load 
See C_{MEC} 
See M_{MC} 
M_{A}_{1} 
Combination Mass Load 


M_{AC} = M_{AD} + M_{AB} + M_{A}_{1} 
Figure 7: Important dimensions of bassreflex enclosure.
Based on these dimensions [1],






V_{B} = hwd (inside enclosure volume) 
S_{B} = wh (inside area of the side the speaker is
mounted on) 
c_{air} = specific heat of air at constant volume 
c_{fill} = specific heat of filling at constant volume (V_{fill}_{ }_{ing}) 
ρ_{0} = mean density of air (about 1.3 kg/m^{3}) 
ρ_{fill} = density of filling 
γ = ratio of specific heats for air (1.4) 
c_{0} = speed of sound in air (about 344 m/s) 
ρ_{eff} = effective density of enclosure. If little or no filling (acceptable
assumption in a bassreflex system but not for sealed enclosures), 