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The Crossover Design Cookbook
Chapter 2: How Components Work
by Mark Lawrence

## Combining basic components

Perhaps surprisingly, there are only a very few circuits which come up in cross overs. Even very complicated high-order cross overs are the result of combining several of these basic circuits. So, we'll look at each one. Below each circuit, I've written the complex impedance of the circuit. We'll see what this means in a couple of pages. Don't worry about the implied math - the important formulae are the first ones in the article, and the rest will be done by a computer program. We're just trying to get a general feel for these circuits. No one does this math by hand anymore. Hewlett-Packard pretty much put an end to hand mathematics in the '70s with the programmable HP35 calculator.

The mathematics of figuring out the combined impedance is straight forward, if tedious. To figure this stuff out just write "R" for a resistor, "1 / Cs" for a capacitor, and "Ls" for an inductor. Then, either add the components impedances if they are connected in series, or using the parallel resistor law if they are connected in parallel. With these rules you can write down the impedance. When first you write it down it will normally be in a more complicated form, so using standard high school algebra you simplify the equations. Or better yet let WolframAlpha.com simplify them for you. But you don't have to do all this grunge because I've already done it for you. For example, our first circuit is:

### Resistor and capacitor in series

 1 RCs 1 RCs + 1 Z = R + = + = Cs Cs Cs Cs

### Calculating a Resistor and Capacitor in Parallel

 1 1 1 1 1 R Z = R || = = = = = Cs ( 1/R + 1 / (1/Cs) ) (1/R + Cs) (1/R + RCs/R) ((1 + RCs) / R ) (1 + RCs)

### Resistor - Capacitor circuits

 RCs + 1 Z = Cs
 R Z = 1 + RCs

### Resistor - Inductor circuits

Z = R + Ls
 RLs Z = Ls + R

### Capacitor - Inductor circuits

 LCs² + 1 Z = Cs
 Ls Z = 1 + LCs²

Notice that the capicator - inductor circuits have an s² in the equation. These are second order circuits because their math comes from a second order polynomial.