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Maxim > Design
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Circuits > APP 4525

Dec 21, 2010

Keywords: equalizers, low-noise low-distortion op
amps, low-frequency loudspeakers, bass response
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APPLICATION NOTE 4525

"Linkwitz Circuit" Equalizes a Closed Box

Abstract: The
Linkwitz equalizer, intended for complex-pole systems in which
Q > 0.5, neutralizes the original response by compensating
the original poles with zeros, and then creates a new highpass
response as specified by the designer. Like the original, the
new circuit provides a 2nd-order response.

Attend
this brief webcast by Maxim on
TechOnline

A
similar version of this article appeared in the May 6, 2008
issue of EE Times magazine.

The linearity and
high power-handling capability of modern low-frequency
loudspeakers have made it easy to electronically extend the
bass response. This extension is especially useful in
second-order systems like a closed box, where cone excursions
are limited by the box.

Many such circuits are
available, but the most popular today is the so-called
Linkwitz equalizer¹ intended for complex-pole systems in which
Q > 0.5. The circuit neutralizes the original response by
compensating the original poles with zeros, and then creates a
new highpass response as specified by the designer. Like the
original, the new circuit provides a 2nd-order
response.

Figure 1 shows the basic circuit for one
channel (two are required for stereo). The circuit must be
driven by a low-impedance source,
so you should provide a buffer stage in front of it. Because
the Figure 1 circuit inverts the signal, this buffer
should be an inverting type to maintain the original signal
phase.

Figure 1. This "Linkwitz
transform circuit" lets you neutralize the original response
of a bass circuit, and substitute new complex poles that
improve the bass response.

Following is the design
procedure suggested by Mr. Linkwitz:

First, specify
fo, Qo, fp, and
Qp:

fo and Qo are
determined from the frequency response of the original,
unequalized system. fo is the -3dB frequency, and
Qo is the system Q. Figure 2 helps to
determine the Qo value.

Figure
2. The amplitude response in Figure 1 varies with the system Q
as shown.

fo and Qo specify
the exact compensation for existing poles, and are determined
by the existing closed-box design. fp and
Qp are target specifications for the transformed
system. As an aid in finding the correct values for
fp and Qp, Figure 2 shows the response
for various Qo values.

Next, calculate the
constant "k," which should be positive to ensure realizable
equations modeling this circuit topology:

Choose C2. A good start is 470nF, which gives a
reasonably low impedance
level with low noise.

Calculate R1:

Calculate R2:

Calculate C1:

Calculate C3:

Calculate R3:

If the calculated resistor values are too large
(much over 100kΩ, for instance), you should increase the value
of capacitor C2 and then re-calculate the other component
values.

The frequency ratio fo/fc
sets the DC gain and low-frequency boost
for the circuit. It is recommended that boost not exceed 20dB,
because the increase in power and cone excursion at that level
becomes extreme. (For most music, on the other hand, the
energy level for signals below 40Hz is relatively
low.)

Because the op
amp selected in Figure 1 must handle the entire audio band,
it should have low noise, high slew rate, and low distortion.
Op amps
MAX4478 and MAX4495 are
suitable for low-voltage, single-supply designs. For higher
supply voltages, consider the dual MAX412 (to provide a
buffer on each channel, you will need two of these). Or, you
can use a single quad op amp such as the MAX4478 or
MAX4495.

An Example
To illustrate the procedure we equalize an
existing closed box, which transforms the existing
fo = 80Hz and Qo = 1.2 to fp
= 30Hz and Qp = 0.707 (a Butterworth response).
Going through the calculations above gives the following
component values, rounded to the closest standard values: R1 =
10kΩ, R2 = 15kΩ, R3 = 75kΩ, C1 = 0.82µF, and C3 =
0.12µF.

Figure 3 shows the original response,
the equalizer response, and the desired combination of those
two, which nicely matches the target. Figure 4 shows
the time delay before and after equalization. Although peak
delay is higher in the equalized system, the delay over those
frequencies that are important for music is
improved.

Figure
3. These curves illustrate how the equalizer improves an
existing system.

Figure
4. For most music, the equalized system also provides a more
favorable delay time.

¹S. H. Linkwitz,
"Loudspeaker system design," Wireless World, December
1978, p. 80.

Related parts

MAX4478

SOT23, Low-Noise,
Low-Distortion, Wide-Band, Rail-to-Rail Op Amps

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Download,
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APP 4525: Dec 21, 2010

AN4525, AN 4525, APP4525,
Appnote4525, Appnote 4525

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