This note provides expressions for the overall f3db frequency f3db_net for multi-stage unity-gain low-pass (LP) cascaded filters.
N is the number of stages and each stage individually has the SAME -3db frequency f3db. The stages are assumed non-interacting.
The overall f3db frequency is the half-power point of the combined filter stages.
Results are given for first order filters (e.g. single pole RC filter) and 2nd order filters with the maximally-flat Butterworth response (Q = 1/√2).
x3db is the ratio f3db_net/f3db. f3db is the -3db frequency of each individual stage. N is the number of identical stages.
First Order Single-Pole LP Filters
For N stages, the half-power point is given by:
Some examples are:
N x3db
1 1.0
2 0.6436
3 0.5098
2nd Order Double-Pole LP Filters with Q=1/√2
For N stages, the half-power point is given by:
Some examples are:
N x3db
1 1.0
2 0.8022
3 0.7140
(Note that the results for the 2nd order double-pole case are exactly the square-root of the results for the single-pole case)
First Order and Second Order Filters
For the simple case of a single first-order filter cascaded with a single 2nd order filter (with Q=1/√2) with the same individual f3db frequency, the half-power point is given by:
