**Design of a high-accuracy passive loudspeaker system**

S. Kozaitis, Jan. 2008

**1 Introduction**

**2 Tweeter**

**3 Midrange-woofer**

**4 Crossover**

**5 Simulation**

**6 Enclosures and subwoofer**

**7 Construction**

**8 Finished speaker**

**Appendix**

Scan-Speak D2905/9700 Tweeter Data sheet

Scan-Speak 21W8554-00 midrange-woofer data sheet

Peerless 830500 subwoofer data sheet

Application notes for Peerless 830500 subwoofer

Amplifier review (similar amplifier)

**1 Introduction**

A high-performance loudspeaker system that outperforms many mass-market designs can be built using standard components. By selecting individual drivers and designing the associated circuitry and components, impressive results can be achieved. There are many drivers, circuits, and software packages that allow building a loudspeaker system for almost any budget and application. A three-way design that uses a passive crossover and an active subwoofer is presented here. It follows a logical design process that can be applied to other drivers to achieve a high-performance loudspeaker.

Individual high-quality components of a loudspeaker can usually be purchased for much less than a complete system. The design described here is for a high-performance (accurate), three-way system with drivers intended for use with transistor amplifiers. The drivers were chosen because they are highly regarded by several sources. They are the Scan-Speak D2905/9700 tweeter, Scan-Speak 21W8554-00 midrange-woofer, and Peerless 830500 subwoofer. There is a wide variety of subwoofers available. The subwoofer here was chosen because it has low distortion. The human ear is not sensitive to low frequencies (see the Fletcher-Munson loudness curves), and if there is a small amount of distortion present at very low frequencies, this distortion will appear at higher harmonics where the ear is much more sensitive. Although the harmonics are attenuated by a crossover filter, the distortion may still be audible due to the increased sensitivity of the ear. Using high-quality components, an impressive system can be achieved.

**2 Tweeter**

The tweeter section should provide a fairly flat frequency response in its region of operation. A graph of the frequency characteristics of the tweeter was taken from its data sheet and is shown below. The effect of the impedance peak at 500Hz may be noticeable, so a notch filter was designed for its equalization. In this way, the frequency response of the tweeter was made more constant.

**Frequency response of Scan-Speak
D2905/9700 tweeter.**

The peak in the frequency
response can be suppressed by a series *RLC*
circuit connected in parallel to the driver. The circuit is shown below and its
associated equations are

_{}, _{}, _{}, and _{}.
(1)

The resonant frequency is *f** _{s}*,

_{}

**Series notch filter.**

A value of *R** _{1}* = 4W was
initially used (labeled at

**3 Midrange-woofer**

The midrange-woofer reproduces sound at critical frequencies for music. The midrange driver has the frequency characteristics shown below.

**Frequency response of Scan-Speak
21W8554-00 midrange-woofer.**

A Zobel circuit is an *RC* circuit connected in parallel with
the driver to equalize the increase in resistance at increasing frequencies.
The equations for the Zobel circuit can be found from a variety of sources and
are

_{} and _{},
(2)

where *L** _{e}* and

A notch filter
circuit like that from the tweeter section can be used with the
midrange-woofer. A notch circuit was used here to flatten the frequency
characteristics of the driver. Initially, values of *a* = 9, *f** _{s}*
= 2.7 kHz, and

**4 Crossover**

There are a variety of possibilities for choosing a crossover and it is difficult to determine which is optimum. Probably the lowest order is best because of the simplicity, reduced sensitivity of component values, and increased overlap of driver responses. The drivers considered here do not seem capable of supporting a first-order crossover due to the lack of necessary frequency overlap, so a second-order crossover was used. The choice can be simplified further if we consider that different designs differ only by a damping factor. In this regard, the circuit diagrams for different crossover filters are the same, and the component values can be described by

_{} and _{}, (3)

where *R* is the resistance “seen” by the crossover, *f* is the crossover frequency, and *d* is the damping factor. The crossover frequency was chosen to be
1.7kHz. A value of *d* = 1.414
corresponds to a Butterworth alignment; however, a Butterworth has a “bump” in
amplitude in the crossover region, so a Linkwitz-Riley (*d* = 2) was considered. The resistance of the voice coil can be used
as a starting value for *R* in the
above equations. Using the damping factor as a parameter, second-order
crossover circuits can be seen as the same circuit to simplify design.

**5 Simulation**

Finding components with the calculated values is virtually impossible; however, substituting components with similar values can significantly change circuit characteristics. Therefore, care was used in substituting standard values, and then the system was simulated using PSPICE in an iterative fashion until the final circuit was realized. Using this trial-and-error approach allowed the desired response to be realized with standard value components.

The tweeter and midrange-woofer drivers were modeled in
PSPICE using the FTABLE part along with their resistance and impedance of their
voice coils. In the midrange circuit, the resistance seen by the
crossover was measured by the ratio of *V*
and *I*, and that value was used in the
crossover calculations. Then, values of components were changed to standard
values in incremental amounts to gain improvements in the circuit. The final
circuit for the crossover is shown below along with
simulation results for the crossover circuit only and the combined
driver-crossover response. The graphs show that the acoustic crossover
occurs at 1.72KHz and the electrical crossover point is at 1.42KHz. Using this
PSPICE simulation, the crossover circuit can be designed.

**PSPICE circuit used for simulation.**

**Magnitude response of crossover circuit
in dB (red) - tweeter circuit, (blue) – midrange circuit, and (green) –
combined.**

**Magnitude response of combined driver – crossover
circuit in dB, (red) – tweeter circuit, (blue) – midrange circuit, and (green)
– combined.**

The group delay is related to the frequency response in a complex way. The group delays for the tweeter and midrange circuits are shown below in the crossover region and are compared to the group delays of the Linkwitz-Riley and Butterworth crossovers at 1.42kHz with an 8W load. The tweeter circuit showed a group delay that was neither like the Linkwitz-Riley or the Butterworth circuits. Its response was less linear than the Linkwitz-Riley circuit, and its overall deviation from linearity was similar to the Butterworth circuit. In addition, the overall change in the group delay was about half that of the Linkwitz-Riley circuit.

**Group delay response of tweeter circuit
(green) – actual circuit, (red) – Linkwitz-Riley crossover, and (blue) –
Butterworth crossover.**

The midrange circuit also showed a group delay that was neither like the Linkwitz-Riley or the Butterworth circuits as shown below. Its response was more similar to the Butterworth circuit with about twice the overall change in group delay. The actual crossovers are also shown below.

**Group delay response of woofer circuit
(green) – actual circuit, (red) – Linkwitz-Riley crossover, and (blue) –
Butterworth crossover.**

.

**Crossover circuit.**

**6 Enclosures and
subwoofer**

The
midrange-woofer will be in a sealed box with a maximally-flat response (*Q** _{tB}* = 0.707) so it can be integrated
easily with the subwoofer. The equations for the driver in a sealed box are

_{} and _{},
(4)

where *Q** _{t}*,

A vented system was considered for the subwoofer to extend the low-end of the frequency response of the system and to limit the size of the enclosure. The frequency response of the Peerless driver is shown below.

**Frequency response of Peerless 830500 subwoofer.**

A common alignment is that of a Butterworth fourth-order vented subwoofer. The equations for that enclosure are

_{}, _{}, and _{}, (5)

where *f** _{B }*is the resonant or tuning frequency. Using

**Frequency response of Peerless 830500
subwoofer in Butterworth fourth-order vented alignment.**

To extend the low-frequency response, a 40 liter enclosure was considered with a tuning frequency of 25 Hz. The low tuning frequency was used to limit the “boominess” of the bass response. An amplifier with a nominal 6db (4.65db actual) boost at 30 Hz compensated for the roll-off of the bass response. Therefore, the subwoofer should exhibit a flat frequency response down to approximately 30 Hz. Below that, the system’s response drops off rapidly. The frequency response of the driver in the enclosure is shown below, and the frequency response of the amplifier follows. Using the vented subwoofer-amplifier configuration allows the low-frequency response to be extended.

**Frequency response of Peerless 830500
subwoofer in 40 liter enclosure ***f*_{B}** = 25Hz.**

**Frequency response of amplifier with
various crossover frequencies.**

Based on the power required and
tuning frequency, the port size of the vented cabinet can be calculated. The
relationship between the minimum radius of the port *r* in mm and the vent air speed is

_{} , (6)

where *P* is the peak power. The variable *S* is air speed in the vent as a fraction of the speed of sound and
should be kept at less than 0.1 (10%) to reduce port noise. Flared ports also
help. The value of the power used should be the maximum power, which is twice
the RMS power. Here, 480W was used because the amplifier was rated at 240W. The
calculated value in this case was *r* =
27.8 mm. Increasing *r* may help at the
expense of a longer port. Using standard values of PVC pipe yielded the data
below. Therefore, there are different possibilities for size and length of the
ports.

**Possible vent dimensions for subwoofer**

Number of ports |
Diameter of port |
Length of port |
% of speed of sound |

1 |
2 in |
8.28 in |
10.2 |

2 |
2 in |
18 in |
6.0 |

1 |
3 in |
20.6 in |
5.3 |

**7 Construction**

The speaker cabinets came from a pair of Wharfedale 70 speakers from the early 1960’s. They have an interesting look, are constructed very well, and are in excellent shape. The picture below shows the inside of the cabinet with back removed. The drivers are installed with the tweeter inside the wooden box. Wood was added to make the two chambers for the midrange and subwoofer. The original mounting for the 2 inch port was used. Using these cabinets avoided the need to build new ones.

Acoustic filling often helps by damping reflected waves inside a chamber. Additional acoustic filling compared to the original amount was used in the sealed chamber and also glued to the sides of the ported chamber as shown below. The crossover circuit can be seen mounted to the bottom of panel as can the subwoofer amplifier in the back panel of the cabinet.

Modifying the boost of the amplifier turned out to be difficult, so a “flat” response of the amplifier was retained. The crossover frequency and gain of the amplifier were adjusted for the best listening experience. By mounting the subwoofer as close as possible to the floor, gain due to the room could compensate for any weakness at low frequencies.

**8 Finished speaker**

A front view of the completed loudspeaker is shown below. Subjective listening tests indicate that the system reproduces a natural sound quite accurately and can handle even the most demanding power levels.