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 The Crossover Design Cookbook Chapter 1: Simple Crossovers by Mark Lawrence 
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Now, we'll design a 2nd order cross over for the Watt V. We'll cross over at 2,000Hz, just as we did in the first order cross over. Second order cross overs are typical, and perfectly respectable. In fact, many people think second order cross overs are optimum. Since it's a second order crossover there will be two components, a capacitor and an inductor, for each driver. It's really that easy to recognize: one capacitor or inductor for each drive is a first order crossover; two components per driver is a second order crossover; three components per drive is a third order crossover; and so on. From third order up the math gets progressively more complicated, which is why there aren't a lot of books on crossovers.
Second order crossovers always have some distortion of the music. Always. The sum of the two crossover outputs will never equal the original music. It can be quite close, but never equal.
The power distribution from a second order crossover is a lot friendlier to the drivers. If your crossover is at 2,000Hz, then at 1,000Hz, one octave down, the tweeter signal is dropped by 12dB, which means only 6% of the music power is getting through; similarly at 4,000Hz only 6% of the music signal is getting through to your midrange driver, which probably does not produce very clean sound this high up.
The second order formulas above have an additional parameter "Q"; we'll set Q at .7. A Q of .7, as we will learn later, selects a Butterworth cross over which has optimal frequency response. If we select a Q of .58, we'll get a Bessel cross over, which has optimal transient response. A Q of .5 selects a crossover which is "critically damped." Critically damped means there is no overshoot or ring at any point in the output voltage. A Bessel filter and a Butterworth filter both have a small amount of overshoot  that means a square was comes out with a small hump in it. Critically damped second order filters are also called LinkwitzRiley second order filters 'cause Siegfried Linkwitz and Russ Riley first made them popular in their 1976 article on active crossovers. However, Isaac Newton knew in 1710 or so that a Q of .5 meant critically damped; Linkwitz and Riley had some interesting things to say about applying filter math to speakers, but they didn't invent any new math.
If you select a Q greater than .7 you have what's called a Chebyshev filter, and there will be a lot more overshoot and ringing. Don't do it. Just don't. If you think what you really need is a Chebyshev filter, then you should not be reading this book, 'cause I disagree as completely as you can imagine. Chebyshev filters are "maximally steep." This means you can design them to drop especially quickly in the first octave or so. But the faster they drop the more energy they store and release later, meaning the more overshoot and the more ringing. Chebyshev filters are often used in cheap car subwoofers to protect the woofer from burning up at very low frequencies. If you've ever been next to one of these at a red light, you know about the booming and overshoot and poorly controlled low bass that results from this bad engineering decision.


Remember, the ScanSpeak woofer has a resistence of 5.5 ohms and the focal tweeter has a resistence of 6 ohms.
For the woofer,
L = R / 2π f Q = 5.5 / (2π 2000 .7) = .00063 = .63 mH.
C = Q / 2π f R = .7 / (2π 2000 5.5) = 10E6 = 10µF.
For the tweeter,
L = R / 2π f Q = 6 / (2π 2000 .7) = .00068 = .68mH.
C = Q / 2π f R = .7 / (2π 2000 6) = 9.3µF.
This is what almost everyone else will tell you to use for a second order crossover. H.L.Mencken said, "For every complex problem there is an answer that is clear, simple, and wrong." This very popular solution is simple, clear, and wrong.
Butterworth was an engineer who first did this math in 1930. Although there was music recording and reproduction by then, there was absolutely nothing like high fidelity sound reproduction. Butterworth would find modern speakers and amplifiers to be magic. He was most certainly not thinking about crossovers when he did his work. So he never mentioned that butterworth low pass and high pass filters don't sum to one. In fact when you add them up you get a 3dB rise in the sum at the crossover frequency  in this case, 2,000Hz. In the diagram below we see what's called a Bode plot of the Butterworth and the LinkwitzRiley 2nd order filters. The LinkwitxRiley sums up to a beautiful flat line, but the Butterworth has a nasty hump in the middle. You cannot make Butterworth filters sum to a perfect flat line, but you can get pretty close. By looking at the plot below you can see that if were were to move the lowpass filter down a bit and the highpass filter up a bit we could minimize that nasty hump. You have to use a correction factor of 1.3 on the frequency, and you have to reverse the phase of the tweeter. 1.3 is the correction factor for a second order butterworth. For a second order bessel the correction factor is 1.445; for a second order LinkwitzRiley the correction factor is 1, that is no correction is needed. In all 2nd order crossovers the tweeter must be reversed.
That means our 2,000Hz butterworth filter must actually use a low pass frequency of 2,000/1.3 = 1540Hz, and high pass frequency of 2,000*1.3 = 2,600Hz. And you have to hook up the tweeter with the leads reversed, the positive lead goes to ground and the negative lead goes to the crossover output.
For the woofer,
L = R / 2π f Q = 5.5 / (2π (2000/1.3) .7) = .00081 = .81 mH.
C = Q / 2π f R = .7 / (2π (2000/1.3) 5.5) = 1.3 E5 = 13µF.
For the tweeter,
L = R / 2π f Q = 6 / (2π (2000*1.3) .7) = .000525 = .525mH.
C = Q / 2π f R = .7 / (2π (2000*1.3) 6) = 7.1 E6 = 7.1µF.
That's all there is to it. These cross overs will work.
In the second order formulas, if you're not sure what to use for Q, use .7. If you're not sure what to use for R, use 8. The resulting cross overs will work reasonably well. Later, we'll see what Q and R should really be, and how to make the cross overs work really well. "f" is the cross over frequency, 2,000Hz. "π" is 3.14159, "low pass" is the cross over for the woofer, and "high pass" is the cross over for the tweeter.
Later we'll have a discussion of this parameter Q. Q is not a part of first order systems, a system only has a Q if it's second order or higher. The discussion on Q is long and complicated, but the result is simple. If you use Q = .5, we say that the speakers are critically damped. That means the system stores the least possible amount of energy. If you use Q = .577, that's a bessel filter. That means the system has the flatest possible time response, which means it comes as close as possible to a simple time delay. If you use Q = .707 that's a butterworth filter and has the flatest possible frequency response. If you use Q > .707 that's a chebyshev filter and it's a really poor choice for an audio system. You'll have ripples in the frequency response, you'll have peaks and valleys in the phase response, your crossovers will store energy and ring like a bell, and generally speaking you're going to have weird boomy sound. Don't be tempted by the dark side.
Again, you might have trouble finding exact inductor values. I look through my catalogs and I find a .63mH inductor and a .7mH inductor, but no .68. I would just use the .7 and not worry about it. If you're really berserk, you can get a DVM which measures inductance and unwind the .7mH inductor a little bit.
For capacitors, I find a 10µF capacitor easily, and also a 6.8µF and a 2.7µF, which add up to 9.5µF, or a 6.8µF and two 1.5µF, which add up to 9.8µF. Inductors usually measure quite close to the rated value. Capacitors are usually off by anywhere from 5% to 15%, so unless you're going to measure each component, don't bother trying to get too precise.
Of course, there's more to cross overs than this. If this were the whole story, it would have been written down long ago. For now, however, we're going to ignore the other complications and see how these cross overs work. Later, we'll develop significantly better cross overs for the ScanSpeak  Focal two way system.
By the way, if you want to build yourself a set of Watt V alike speakers, the box should have a net internal volume of .26 cubic feet, and a 2 inch vent which is 9.5 inches long. I believe that one of the keys to the superlative sound of the Watt V is the extraordinarily stiff box (about 3 inch thick walls of phenolic), so if you try to build one, make a good box. Wilson Audio sprays damping material on their woofer  if you want to try this, try automotive undercoating spray. Personally, I would be real slow to spray goop on a $135 driver. Maybe you're a braver man than I.
Mount all power resistors with at least 1/4" of free air space on all sides. This means use the leads of the resistor to stand the resistor off the board, and allow air to flow under the resistor too. Make sure the resistors are noninductively wound.
Orient inductors as shown below. Inductors have strong magnetic fields and will couple together if you let them. This is how transformers work: they are just two strongly coupled inductors. Remember the speaker motor coils when considering orientation. Use air core, or 300 to 500 watt bobbin (iron core) inductors. The inductors should have a DC resistance of no more than about 1/2Ω.
Use capacitors rated for at least 100V AC, or 200V DC. Mylar or Polypropylene capacitors are the best for cross overs. If you must use electrolytics, use film capacitors for at least 40% of the total capacitance value.
This is a series crossover, there's a seperate circuit for each driver and the music must flow through the crossover to get to the drivers.
Well, we've come a long way. If all you want is to build some simple quick and dirty 2nd order cross overs, you know all you need to know right now. You can stop reading here. For better crossovers that have much more accurate frequency response, you'll need to read the next couple of chapters.
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Copyright © 20022021 Mark Lawrence. All rights reserved. Reproduction is strictly prohibited.
Email me, mark@calsci.com, with suggestions, additions, broken links. Revised Saturday, 26Jun2021 01:24:15 UTC 
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