Linkwitz-Riley high-pass active filter

This uses cascaded Sallen & Key Butterworth high-pass filters, as in the low-pass filter. Each section looked like this:

This has the transfer function

Equation 1

where Equation 2.

This reproduces the required Butterworth HP response

Equation 3

if Equation 4 and Equation 5.

If we impose C1 = C2 = C, and R2 = aR1 then Equation 6 and

Equation 7.

I used K=1, so R1=2R2.

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