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 The Crossover Design Cookbook Chapter 2: How Components Work by Mark Lawrence 
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Cross overs are built out of three basic components: resistors, capacitors, and inductors. The reason for this is, these components are pretty much all we have. Engineers can easily imagine other sorts of components which would have interesting uses, but mostly, these imaginary components can't be built out of wire. They can be simulated with amplifiers and other components, but that requires a power input to work.
Briefly, here are the results of this section:
Resistors turn power into heat. This is useful when we have excess power and want to get rid of some. In most other cases, it's bad.
Capacitors block low frequencies and let high frequencies get through. This property makes them a natural for usage in a cross over. At a given frequency "f", the "resistance" of a capacitor is 1 / (2π f C), so, connected to an 8 ohm speaker, a 10µF capacitor blocks frequencies below 2,000Hz and lets frequencies through above 2,000Hz. Why 10µF for an 8Ω speaker at 2,000Hz? The effective resistance of our 10µF capacitor at 2,000 hz is
1 / (2π f C) = 1 / (6.26 2000 10x10^{6}) = 1 / .125 = 8Ω
At 2,000Hz, the 10µF capacitor has the same impedance as the 8Ω speaker, so the power is divided evenly between the capacitor and the speaker. At lower frequencies the capacitor takes more of the power, leaving less and less for the tweeter. At higher frequencies the capacitor takes less and less of the power, leaving more and more for the tweeter.
Inductors block high frequencies, and let low frequencies get through. This property makes them a natural for usage in a cross over. At a given frequency "f", the "resistance" of an inductor is 2π f L, so, connected to an 8 ohm speaker, a .6mH inductor blocks frequencies below 2,000Hz and lets frequencies through above 2,000Hz. Why .6mH? At 2,000Hz, the effective resistance of a .6mH inductor is
2π f L = 6.26 2000 .6x10^{3}) = 6.26 * 2 * .6 = 8Ω
At 2,000Hz, the .6mH inductor has the same impedance as the 8Ω speaker, so the power is divided evenly between the inductor and the speaker. At higher frequencies the inductor takes more of the power, leaving less and less for the woofer.
Now, we can see how the cross overs above work. In the first order cross over, we have a (roughly) .6mH inductor attached to the woofer, which is letting through frequencies below 2,000 Hz and blocking frequencies above 2,000Hz. For the tweeter, we have a (roughly) 10µF capacitor which is blocking frequencies below 2,000Hz and letting through frequencies above 2,000Hz. At precisely 2000Hz the power is split evenly between the tweeter and the woofer.
In the second order cross over, for the woofer we have a (roughly) .6mH inductor which is letting through frequencies below 2,000Hz. We also have a (roughly) 10µF capacitor which is shorting the speaker out to ground at frequencies above 2,000Hz. For the tweeter, we have a (roughly) 10µF capacitor which is blocking frequencies below 2,000Hz, and a (roughly) .6mH inductor which is shorting out the tweeter to ground at frequencies below 2,000Hz.
The component values are off a little bit for two reasons: the speakers are not precisely 8 ohms, but rather 5.5Ω and 6Ω, and we haven't yet accounted for the parameter "Q".
Why did we make two different cross overs for the same speaker? The first order cross over has 6dB per octave performance in the stop band, and the second order cross over has 12dB per octave performance in the stop band. Here's what this all means. "dB" stands for deciBell, and is a way to measure power. The Bell is named after Alexander Graham, and deci means tenth. "6dB" means twice as much power. "Octave" means a doubling of frequency, for example from 1,000 Hz to 2,000Hz. On a piano, the key pattern repeats each octave, and the corresponding notes have the same name, e.g. "C". Since there are eight notes "A B C D E F G A" in this doubling, it's called an octave, from the Latin for eight. "Hz" stands for Hertz, which means one cycle per second, so 2,000Hz means 2,000 cycles per second. Hertz was the first guy to observe electromagnetic waves, and would undoubtedly have gotten a Nobel Prize for it but for the unfortunate fact that Mr. Nobel was a teenager at the time. Middle "C" on the piano is 262Hz. Middle C is a moderately high note for a man to sing, and a medium note for a woman.
So, a 6dB per octave roll off below 2,000Hz means that the power is halved at 1,000Hz, halved again at 500Hz, and halved again at 250Hz, which is about middle C. Our cross over frequency of 2,000Hz corresponds to about the C three octaves above middle C. With the 6dB per octave roll off, the tweeter is down 6dB at 1,000Hz, 12dB at 500Hz and 18dB at 250Hz. The woofer is down 6dB at 4,000Hz, 12dB at 8,000Hz, and 18dB at 16,000Hz.
A driver must be about 15dB to 20dB down before we don't hear it. Thus, with our 6dB per octave first order cross over, the tweeter becomes inaudible at about middle C, three octaves below the cross over frequency, and the woofer becomes inaudible at 16,000Hz, three octaves above the cross over frequency. This makes a first order cross over inappropriate for this system: the Scan Speak woofer cannot go above 6,000Hz, and only produces heat and awful noises when you tell it to try. The Focal tweeter cannot go below 1,000Hz, and will likely burn up if you tell it to try. So, although the first order cross over was trivial to design and exceedingly simple to build, it is basically useless for this application.
Our second order cross over rolls off at 12dB per octave, twice as fast as our first order cross over. This means the drivers are active for about one and a half octaves beyond the cross over frequency. The tweeter will be driven down to about 700Hz, and the woofer up to about 5,600Hz. This is marginal, but acceptable. So, although we had to use slightly more complicated math (that "Q" factor), and twice as many inductors and capacitors, we got a more useful result.
At this point, you may be thinking "well, if second order is better than first order, why not fiftieth order?". We'll see later on that first order is trivial, second order is easy, third order is complicated, fourth order pretty much requires computer software, and fifth order or higher is exceedingly difficult. It's also more or less impossible to build an actual fifth or higher order filter, as you need precision in the components that simply isn't available. Also, pretty much whenever we use second order or higher, we get some kind of time and phase distortion: the higher the order, the worse the distortion. There's no free lunch.
Low Pass (Woofer)  High Pass (Tweeter) 
Above, we see what is called a Bode plot of these cross overs. When drawn by hand, Bode plots are usually drawn using only straight lines, because this makes them incredibly simple to draw: just go straight until you hit the cross over frequency, then turn at 6 or 12dB per octave. The actual frequency response of these cross overs does not have the sharp corner at 2,000Hz, but rather is rounded off a little. Sharp corners require infinite energy, so we won't have any of those. 'Cause infinite energy has an infinite gravitational field, and your stereo, living room, and house would disappear into the resulting black hole. Imagine all life as you know it stopping instantaneously. That's bad.
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Copyright © 20022019 Mark Lawrence. All rights reserved. Reproduction is strictly prohibited.
Email me, mark@calsci.com, with suggestions, additions, broken links. Revised Thursday, 15Aug2019 09:30:53 CDT 
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