Use a "Linkwitz circuit" to equalize a closed-box audio loudspeaker

Many such circuits are available, but the most popular today is the so-called Linkwitz equalizer (Reference 1) 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 high-pass response as specified by the designer. Like the original, the new circuit provides a 2^nd-order response.

Figure 1 shows the basic circuit for one channel (two are required for stereo).

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.
(Click on image to enlarge)

The circuit must be driven by a low-impedance source, so you should provide a buffer stage in front of it. Because the circuit of Figure 1 inverts the signal, this buffer should be an inverting type to maintain the original signal phase.

Following is the design procedure suggested by Mr. Linkwitz:

First, specify f_o, Q_o, f_p, and Q_p: F_o and Q_o are determined from the frequency response of the original, unequalized system. F_o is the -3 dB frequency, and Q_o is the system Q. Figure 2 helps to determine the Q_o value.

Figure 2: The amplitude response in Figure 1 varies with the system Q as shown.
(Click on image to enlarge)

F_o and Q_o specify the exact compensation for existing poles, and are determined by the existing closed-box design. F_p and Q_p are target specifications for the transformed system. As an aid in finding the correct values for f_p and Q_p, Figure 2 shows the response for various Q_o values.

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

Choose C₂. A good start is 470 nF, 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 100 kΩ, for instance), you should increase the value of capacitor C₂ and then re-calculate the other component values.

The frequency ratio f₀/f_c sets the dc gain and low-frequency boost for the circuit. It is recommended that boost not exceed 20 dB, 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 40 Hz 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 such as the 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.

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

Figure 3 shows the original response, the equalizer response, and the desired combination of those two, which nicely matches the target.

Figure 3: These curves illustrate how the equalizer improves an existing system.
(Click on image to enlarge)

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 4: For most music, the equalized system also provides a more favorable delay time.
(Click on image to enlarge)

Reference
1. S. H. Linkwitz, "Loudspeaker system design," Wireless World, December 1978, p. 80.
(Editor's note: If you are interested in more on the Linkwitz circuit, Google the phrase to see many interesting articles and other implementations.)

About the author
Leo Sahlsten, Senior Field Applications Engineer, joined Maxim Integrated Products in 2000. Before joining Maxim, he worked for Nokia Networks in Finland. He studied electronic engineering at the Helsinki Technical College, and has a B Sc degree in electronics engineering.