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 The Crossover Design Cookbook Chapter 2: How Components Work by Mark Lawrence 
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In order to understand cross overs better, we need to understand resistors, capacitors, and inductors. What we're going to learn is that capacitors are like resistors, except their resistance goes down as you go up in frequency. Inductors are like resistors except their resistance goes up as you go up in frequency. Even though we didn't use any resistors in our Watt V example, we'll start at the beginning, which is resistors.
Resistors resist the flow of electricity. For example, if you have a garden hose with a spray valve on the end, by turning the valve more open or closed, you can increase or reduce the flow of water through the hose. This valve is functioning exactly like a variable resistor, like a volume knob. If you were trying to water your indoor plants with such a spray valve attached to a 4" fire hose, it would be important to have it adjusted just right. Resistors must also typically be "just right".
Unlike your garden hose valve, which is water cooled, resistors get hot. In the process of resisting the flow of electricity, resistors convert some of the electrical power to heat. Thus resistors are specified by 2 numbers: their resistance, given in units called "ohms", and their power rating, typically 1/8 watt to 25 watts. If you try to dissipate more than the rated power in a resistor, it will melt.
The resistance of a typical resistor in a cross over will be between 1 and 100 ohms, rated between 5 watts and 50 watts. In a preamplifier, a typical resistor will be between 1,000 and 100,000 ohms, rated at 1/4 watt. In a power amplifier, a typical resistor will be between 1/4 ohm and 10,000 ohms, rated between 1/4 watt and 5 watts. A couple feet of wire typically has a resistance of 1/100 to 1/10 ohms, so it is very difficult to make connections with less than 1/100 ohms.
Ohm figured out how resistors work, and also gave us Ohm's Law: V = IR.
If you connect two resistors end to end in a line, the resistances add. So, 50 ohms connected to 100 ohms equals 150 ohms. If you connect two resistors side by side, the new resistance is calculated by the Parallel Resistor Law. This Parallel Resistor Law, plus a little algebra, is the key to calculating cross over component values. It gets used a lot when deriving crossover formulas.
R = R1 + R2 
 
Series  Parallel 
Whenever you combine equal resistors, whether in series or in parallel, the watt ratings of the resistors simply add. Two 10 watt 10Ω resistors in series acts like a 20 watt 20Ω resistor. Three 5 watt 5Ω resistors in parallel act like a 15 watt 1.7Ω resistor. If the resistors are unequal, the rule for power is much more complicated.
The resistance of two resistors in parallel is smaller than either of the two resistors taken alone. This is quite intuitive, if you think about it: two hoses side by side flow more water (offer less water resistance) than either hose by itself. If there is a 4 inch fire hose (low resistance) next to a 1/8 inch diameter hose (high resistance), the overall effect is only slightly different than using just the 4 inch hose, but there's still more water flow. Of course, two 4 inch hoses flow exactly twice as much water as one 4 inch hose, and two 10 ohm resistors in parallel equals 10*10 / (10+10) = 100 / 20 = 5 ohms. 5 ohms flows exactly twice as much electric current as 10 ohms at a given voltage.
If you connect two resistors in series to a voltage, the voltage is divided between the two.

In the US, we write resistor values like this: "3.3k" (3,300 ohms) and "5.1Ω" (5.1 ohms). In Europe, these same resistors would be written as "3k3" and "5r1". I think they do this just to be different.
Resistors basically come in three different construction types: carbon, metal film, and wire wound. Carbon resistors are the cheapest and the noisiest. I mostly don't use them. Metal film resistors have the most precise values, and are comparatively quiet, but are usually limited to about 2 watts or less. Wire wound resistors are pretty much what you can use in a cross over. When you buy wire wound resistors, it's critically important to make sure that they are "non inductive wound". An inductively wound resistor acts like a combination of a resistor and an inductor, and will completely screw up your cross over.
Resistors also have a small amount of inductance, so at very high frequencies they stop flowing current. However, this does not happen until you get up to many millions of hertz, so we never worry about it.
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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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