 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 highorder 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. HewlettPackard 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






Resistor  Inductor circuits




Z = R + Ls


Capacitor  Inductor circuits






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.

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

