The electronic implementation of the filters with opamps
Here the formulas for calculating the component values will be given for Butterworth filters and Linkwitz-Riley filters using SK architectures. Just like the transfer functions the "rules" that apply for BW and LR can be applied here. This is: 4e-order filters consist of two 2e-order stages in which LR has equal stages and BW different stages. It is common to use unity gain buffers in crossover designs. The formulas for 2e-order filters are given by:
For BW, FSF=1 and Q=0.707. A common simplification is to set filter components as ratios and the gain to unity (k=1). Take R1 = R2 , C2 = 2C1 for low pass filters, then we can derive the following equation:
in which R1 = 5-10kohm
Analogous for high pass filters. Take C1= C2 , R1 = 2R2 and derive:
in which C1 = 5-10nF.
Note that these simplifications only are effective for
Q=0.707. For example a 4e-order BW filter must
be calculated from two different Q values. Each section has its own Q
For LR, FSF=1 and Q=0.5. A common simplification for this type of filter is to set the gain to unity (k=1) and take R1 = R2 = R, C2 = C1 = C. With the above formulas we can derive the following equation:
in which C is preferably chosen by the designer such that R does not become too large. The equation is the same for LP and HP filters.