A 3rd-order Butterworth filter is a maximally flat filter with -60dB/decade roll-off. This implementation is derived from a cascade of a 2nd-order active filter and a 1st-order passive filter.

The 2nd-order Butterworth filter can be derived as follows: Since ω must equal zero, ζ = 1/√2.

The Sallen-Key topology is a means of realizing a 2nd-order active filter with the following
structure:

The generic derivation of the Sallen-Key transfer function is:

This transfer function allows a variety of filters to be implemented simply by changing the
types of the components. In this instance, Z_{1} and Z_{2} are chosen to be
resistors and Z_{3} and Z_{4} are chosen to be capacitors. This arrangement
creates a 2nd-order low-pass filter. The characteristics of this filter can then be matched
to a Butterworth filter of the desired cut-off frequency.

The components of the Sallen-Key filter can be calculated for C_{1} = kC_{2} and R_{1}
= R_{2}. k(RC)^{2} = 1/(ω_{o})^{2}, 2kRC = 1/ω_{o}. For a
100kHz cut-off frequency, let R = 10kΩ and the other component values become equal to C = C_{1} = 318.31pF
and C_{2} = 79.577pF.

There is now a maximally-flat 2nd-order filter. To produce a maximally-flat, 3rd-order filter, another
1st-order stage can be cascaded with the output of the 2nd-order filter. This stage is a simple RC low-pass
filter. For a cut-off frequency of 100kHz, the component values for this filter can be found with ω_{o}
= 1/(R_{3}C_{3}). Letting R_{3} = 10kΩ gives C_{3} = 159.1549pF.

The transfer function for this filter: produces the pole-zero plot: Since all of the poles are in the left-half plane, the filter is stable. It can also be noted that the angle separating the poles can be described by 180°/N where N is the order of the filter.

The maximally-flat behavior can be observed in the pass-band with the -60db/decade roll-off after 100kHz.

The attenuation at 100kHz is slightly less than -3dB. This can be attributed to variance in the components used to assemble the filter from the values used in the design phase.