A 3rd-order Butterworth filter with a Sallen-Key and 1st-order cascade.

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Concept:

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.

Circuit Diagram:

Butterworth Derivation:

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

Sallen-Key Topology:

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, Z₁ and Z₂ are chosen to be resistors and Z₃ and Z₄ 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.

Picking components:

The components of the Sallen-Key filter can be calculated for C₁ = kC₂ and R₁ = R₂. 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 = C₁ = 318.31pF and C₂ = 79.577pF.

1st-order Stage:

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/(R₃C₃). Letting R₃ = 10kΩ gives C₃ = 159.1549pF.

Poles and Zeros:

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.

Bode Plot:

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

Testing:

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.