The electronic implementation of the filters with opamps

mfb_architecture.gif (4432 bytes)With the physics of amplitude and phase behavior (transfer functions) the result of crossovers can be calculated and simulated. In active filter design it is common to use op-amps and passive components to realize the result of the simulations in practice. Two configuration are used for this purpose: Sallen Key architecture (SK)and multiple feedback architecture (MF). SK architectures are slightly inferior in the high frequency domain to the MF but it is less sensitive to component variation. However, for audio crossovers this inferior behavior of SK is not audible (occurs at -30dB) and SK architectures are usually applied for audio crossovers.

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:

      and     

 

LP_SK_architectures.gif (4102 bytes)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 value.
HP_SK_architectures.gif (3664 bytes)

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.


 

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