TA Speaker Topics - Designing a Linkwitz Transform

Topic No. 13

Designing a Linkwitz Transform Circuit
by
John L. Murphy
Physicist/Audio Engineer

I first became fascinated with the possibility of "transforming" the response of an existing loudspeaker system into a "better" response when I read Siegfried Linkwitz' three part article in the very first issues of Speaker Builder magazine in 1980. The article appeared in Speaker Builder issues 2/80, 3/80 and 4/80. On page 16, Figure 25 of issue 4/80 Linkwitz outlines a method of designing a simple 1 op amp circuit that performs two critical changes to the frequency response of the loudspeaker with which it is used.

First it "neutralizes" the response of the existing speaker by precisely "undoing" the 2nd order high-pass response of a specified closed box speaker. Second, the circuit creates a new 2nd order high-pass response as specified by the designer. On the face of it it seems to be a method, "trick" if you will, for changing the response of a closed box to any other response desired. Say you're not happy with your too small speaker with F(sc) of 100 Hz and Q(tc) of 2. You just design up a simple one op amp circuit, connect it in series with your stereo preamp and BOOM your funky little speaker is magically transformed into a 35 Hz Q(tc) = .707 KILLER speaker system. Sound too good to be true? It's not. Read on.

The circuit is really quite simple. Here is what it looks like:

For details on calculating the component values you can see the original article or you can download my Linkwitz Spreadsheet. This is an M/S Excel file which calculates the component values based on your input of the F(sc) and Q(tc) of the EXISTING speaker and the F(sc) and Q(tc) for the NEW RESPONSE. It includes the schematic and design notes.

Here's the link for the spreadsheet download:

Download the Linkwitz Transform Spreadsheet for Excel (Windows)

18Jun09: NOW includes a 2nd order high-pass filter to reduce boost at infrasonic frequencies. Thanks to Witold Waldman!

(You may need to right-click and select "Save Target As..." so your browser doesn't try to open the sheet over the net.)

How accurate is the Linkwitz Transform?

I've verified my Linkwitz transform designs using various SPICE packages and usually find that the simulation matches the intended design quite nicely. Then as I move from paper to the circuit board I begin to see small errors creep in, mostly by way of small (5 to 10%) errors in component values.

The original design formulas Siegfried Linkwitz gave for calculating the component values (Speaker Builder issue 4/80, p. 16, fig. 25) seem to me to be very precise. The Excel spreadsheet I give away at my site accurately implements the Linkwitz design formulas.

Personally, I've found the largest source of error in the whole process of designing a Linkwitz Transform to be the accurate determination of the starting F(sc) and Q(tc) of the unassisted closed box system. When I actually measure the response achieved by my prototype circuits I see good agreement with the targeted response. The sanity check I use is to first model the closed box in software and print that response. Then I measure the response of the circuit as built.

I then manually plot the new combined response right on the closed box simulation page by taking the dB level for the EQ and adding it to the dB level for the closed box, point by point. If everything is correct you will plot out the intended new target response. Any error will be apparent in the new response. I usually get a good enough match that no further tweaking is necessary. But if you want to tighten up the match the place to start is with your component tolerances. The most common film capacitors are of the 5, 10 or 20% variety. If you can measure your capacitors and resistors you can select your components for 1% or better match to the target values. This usually means carefully creating composite capacitors by combining several smaller values in parallel. I can usually find 1% accurate resistors just by fishing around the 5% parts bin with my Ohm meter. But on occasion I have to gang up resistors to get a target value. You can readily purchase 1% R's if necessary.

All-in-all I think the Linkwitz Transform is a powerful way to customize the response of a closed box loudspeaker.

Here is a link to a related discussion of group delay and the improvement that can be obtained by using a Linkwitz Transform equalizer with your speaker system.

Visit Siegfried's web site at: www.linkwitzlab.com

jlm

Spreadsheet Revision History:

18Jun09
Witold Waldman added a 2nd order high-pass response to reduce infrasonic boost.

11Jul02
John L. Murphy and Peter Rettweiler revised the Phase Response to reflect the op amp's polarity inversion.

23Feb02
John Murphy revised the frequency scale, plot scales, colors and did other cleanup.

22Feb02
Peter Rettweiler added an improved ARCTAN function to remove the discontinuities in the Phase Response.

8-9Sep99
Luc Henderieckx revised the spreadsheet to include frequency and phase response plots of the closed box, the transform circuit response and the combination of closed box and transform. Thanks Luc!

circa 1987
John L. Murphy created the "Linkwitz Transform" spreadsheet for EXCEL on the Macintosh and began distributing it to speaker designers.