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The Crossover Design Cookbook
Chapter 3: Speaker Motors and Crossovers
by Mark Lawrence

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The simplest way to turn down the tweeter response is to place a resistor between the amplifier and the tweeter. The resistor will waste some power, and the tweeter will make less sound. Of course, if we did this, we'd have to redesign the tweeter cross over from scratch: the tweeter cross over was designed based on the tweeter's DC resistance, and we would be changing that. This would mean ordering new inductors, waiting a week for them to come, and then finding out if we have used the correct resistor. If we haven't, it's back to square one. We'll use a trick to get around this: a pair of resistors wasting power is called an L-pad. An L-pad works by placing a resistor in parallel with the tweeter, which reduces the resistance seen by the cross over, then placing another resistor in series with the tweeter, which raises the resistance seen by the cross over back up to the correct level. ### A tweeter with an L-pad attached

L-pads are reputed to be very difficult to design, requiring a computer program. Hah!

We'll have three resistances - the resistor in parallel with the speaker will be called Rp. The resistor in series with the speaker will be called Rs. The effective resistance of the speaker will be called Re. The math works like this: we want the total resistance of Rs + Rp || Re to equal Re. (that previous math is read "Rs plus Rp parallel Re". The "||" sign means use the parallel resistor law.) This will guarantee that our cross over still works. Next, we want (Rp || Re) / (Rs + Rp || Re) to equal the power decrease necessary to correctly pad our tweeter. This math can be solved easily, and here's the solution.
 K Rp = Re Rs = Re (1 - K) Re = Re 1-K Watts = (K - K²) Watts = (1-K) Watts = K²

where we look up K from the table below:

dBK dBK
1.891 7.447
2.794 8.398
3.708 9.356
4.63110.316
5.56211.282
6.50112.251

For our Focal tweeter, we need (94 - 89)dB of pad, which is 5dB. Looking at our table, this means we need a K of .562. Now, we take the DC resistance of the Focal tweeter, 6Ω, and find that we need Rp = 6 * (.562) / (1 - .562) = 7.7Ω, and Rs = 6 * (1-.562) = 2.6Ω.

These resistors must be able to dissipate a fair amount of power. The "Watts" formula above tell us how to calculate this. We'll design this L-pad to handle 40 watts. Rp will take 40 watts * (K - K²) = 40 * (.562 - .562²) = 10 watts, and Rs will take 40 watts * (1- K) = 40 * (1-.562) = 20 watts. The tweeter will take 40 watts * K² = 40 * .562² = 13 watts. This is just about exactly the power rating of the Focal tweeter, so we're in good shape, as long as we don't turn up our Krell KSA 1,000 to full blast.

When we did our L-pad math, we assumed the tweeter was a resistor. We now know that this is not so: the tweeter does not have a Zobel compensator, so the tweeter has a raising impedance as the frequency goes up due to the inductance of the voice coil. This raise in the impedance will cause the tweeter to draw more power out of the crossover at the highest frequencies, and therefore sound a bit bright. This means our L-pad will not work very well past about 5,000Hz, and the tweeter will be too bright in the top octave or two. We'll use a Zobel network to compensate for this, just as we did for the woofer. Referring to our Zobel formula, we find the Focal tweeter has .1 mH inductance, so the Zobel resistor is 6Ω and the Zobel capacitor is .0001 / 36 = 2.8µF. ### An R-L-C resonance compensated driver   ### Driver impedance, with resonance compensator, with resonance and Zobel compensators

If we find that the tweeter is not correctly balanced with the woofer, we can now change the L-pad resistor values without changing any of the surrounding cross over components.

There are commercial L-pad potentiometers available which could be substituted for our two resistors, but these L-pad potentiometers are designed for 8Ω or 4Ω speakers, not for our 6Ω speaker, so they won't work quite right in our speaker. Also, the potentiometers themselves are imperfect and will introduce frequency dependant errors. I don't use them.