Active Linkwitz-Riley 4th Order Crossover
There are several varieties of active crossovers. My choice is a 4th order Linkwitz-Riley. The basic building block of this crossover is shown here [2.7kb]. Kits are available from John Pomann and from Rod Elliot. As illustrated, the output is equivalent to a 2nd order Butterworth. Two identical filters are cascaded to give the 4th order Linkwitz-Riley response. The values of R and C are the same in all cases - but note that some component values are equal to 2R or 2C. The crossover frequency is
The outputs for one filter stage are equal to
where ω =2πf.
Note that cascading two passive 2nd-order Butterworth filters does not create a 4th order Linkwitz-Riley, as can be seen from the component values [21kb] for the two designs. The reason cascading works here is that the op-amp clamps the output voltage equal to its input voltage, essentially independent of the impedance that follows the op-amp. This is not true for passive filters.