Mark K's Speaker Pages

The Linkwitz transform is a well known term, but perhaps not as well understood as it should be. Most folks understand it as a "way to get more bass out of a woofer." A number of misconceptions surround it. Part of the problem is that the actual concept isn't fully grasped. When engineers start talking about poles and zeros, non-engineers start convulsing. OK, well, maybe it doesn't get to the level of a full blown seizure, but you get the picture.

What I'll attempt to do is show how the Linkwitz transform works purely by showing graphs. I will start with a sealed box woofer of any Q and Fc and then use two equalization curves to show how this arbitrary sealed box woofer can have it's Q and Fc transformed into any other Q and Fc.

Let's start with a graph of an arbitrary sealed box woofer. I randomly chose a driver with a somewhat high Q and a roll off of around 50 Hz. The curve was generated in Soundeasy v17 and imported into Praxis given it's flexibility in curve handling and frequency domain math.

In the curve below, we have the desired response overlaid with the original response. In words, we have a high Q, higher Fc woofer box combination and want to get a lower Q, lower Fc response. Note the exact numbers don't matter. We could do the opposite, turn a low Fc woofer into a higher one. (More on that later.)

The question is, how do we get from the black curve to the red one? We'll get there by applying two equalization curves.

Consider the curve below. The red curve is just a "flipped" version of the driver curve for the low end only.

Let's call this red curve "equalization curve 1." Now what would you get if you made an electrical equalization-a notch and boost circuit that looked like the red curve. Forget for a minute that making this kind of circuit would be somewhat challenging since the gain goes up towards low frequencies in a fashion that would make this circuit hard to actually implement. You would need a woofer with infinite Xmax, gain, etc. A challenge let's say.

Nonetheless, say you harnessed the energy of a collapsing star and held a patent on the "infinite Xmax woofer technology." This would allow you to take the black woofer response and filter it with the red equalizer to get the curve below.

So here you have an equalized response that is flat to DC. That would be one outstanding product. Again, this isn't really practical. Remember, the curve we wanted was the red curve below, not to be able to get "flat to DC." So let's multiply this flat black curve by another equalization curve in red.

Let's call this red curve "equalization curve 2."  Note again, that we took our raw driver response, multiplied it by equalization curve one then two to get the target response. (In this case equalization curve 2 and goal curve look the same. And that's because the initial (black) response is flat.)

Let's look at equalization curves one and two by themselves. Equalization curve one is in black, and two is in red.

OK, so equalization curve 1 is not really practical to build, but let's multiply equalization curve 1 by equalization curve two and look at the result.

Now this is where it gets interesting. This curve is just a notch and shelving circuit combined. In the graph below, you see the original woofer response in black, equalization curves 1 and 2 multiplied together in red, and the final result in aqua.

In words, you've taken a sealed box woofer of arbitrary Q and Fc, and by applying an equalization curve that is basically a combined notch/boost and shelving filter, you can "transform" the woofer into some other arbitrary Q and Fc.

That's it. It's fairly straightforward to make an electrical circuit that combines the shelving function and boost/notch circuit. The circuits are already out there. so you don't even have to do this. You can find the actual circuit on SL's page. Easier still, you can just have SE "drop in" the circuit into your schematic. You just have to put in the values for the in box Q and Fc and the Q and Fc that you want to transform the original ones to. SE will calculate the values of all the resistors and caps for you.

Now you can use the same process to raise the cutoff frequency instead of lowering it. In the curve below, I've modeled Zaph's ZA14 driver. In a closed box, it has an Fc of 115 and a Q of around 0.5 as denoted by the black curve. By adding a shelving filter, shown in red, you can get a resultant curve that turns the driver/box into a high pass filter with a Q of 0.7 and and Fc of 200. I'll explain how this can be useful in part two.

Probably the cleanest LT transform spreadsheet is found on the Trueaudio website. You can also calculate the passive values from SL's site, or SE. I haven't looked, but I'm sure you can handle this in lspCAD as well.