Linkwitz-Riley low-pass active filter

This uses standard Sallen & Key low-pass filters, with two identical Butterworth circuits in series. Each section looks like this:

Active LP circuit

This has the transfer function

where .

This corresponds to the required Butterworth low-pass response

if and .

If we impose C1 = C2 = C, and R2 = a2 R1, then

and .

I used K=1.5, so R1=2R2. At a crossover frequency w0 =2 pi x 400Hz, and using C1=C2 =5.6nF+10nF+22nF=37.6nF, this gives R1 = 14965ohms and R2 = 7482.7ohms I used 15kohms and 7.5kohms, which give w0 =2 pi x 399Hz.