### 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: 4^{e}-order
filters consist of two 2^{e}-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 **2 ^{e}-order
filters** are given by:

and

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 R_{1 }=_{
}R_{2 }, C_{2 }=
2C_{1 }for **low pass filters**,
then we can derive the following equation:

in which R_{1
}= 5-10kohm

Analogous for **high pass filters**.
Take C_{1}=_{ }C_{2
}, R_{1 }= 2R_{2}
and derive:

in which C_{1
}= 5-10nF.

Note that these simplifications only are effective for
Q=0.707. For example a 4^{e}-order BW filter must
be calculated from two different Q values. Each section has its own Q
value.

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 R_{1 }=_{ }R_{2
}= R, C_{2 }= C_{1 }=
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.