morsel
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Calling all hardcore math/ee geeks! What are the true equations for gain and 3dB points of a bass boost shelving filter, as described below? The Audio EQ Cookbook may provide some clues, but math is not my forté.
The bass boost circuit is a 6dB/octave low pass shelving filter. Bass response increases from the cutoff frequency down to the shelving frequency and levels off below the shelving frequency. Increasing R4 decreases the cutoff frequency and increases amp gain. Increasing Rbb decreases the shelving frequency and increases bass boost gain. Increasing Cbb decreases both cutoff and shelving frequencies. The graph shows Rbb = (46.6k, 30k, 18.3k, 10k, 4.1k, 0 Ohms).
Ao = overall gain for any conditions (should match the MicroCap graph)
Av = gain with no bass boost; Rbb = 0 Ohms
Abb = gain of bass boost; Xcbb >> Rbb; does not include Av
fs = shelving frequency; 3dB below Abb
fc = corner frequency; 3dB above Av
Ao = ?
Av = 1+R4/R3
Abb = 1+Rbb/(R3+R4)
fs ≅ 1/(2πRbbCbb)
fc ≅ 1/(2π((R3+R4)-(R3+R4)^2/Rbb+.707(R3+R4)((R3+R4)/Rbb)^1.532)Cbb)
The equations are approximations derived from numerical analysis of MicroCap AC modeling graphs and lose accuracy as bass boost gain drops towards 6dB. Below 6dB they are useless as fc and fs overlap.
What are the true equations for Ao, fs, fc?
The bass boost circuit is a 6dB/octave low pass shelving filter. Bass response increases from the cutoff frequency down to the shelving frequency and levels off below the shelving frequency. Increasing R4 decreases the cutoff frequency and increases amp gain. Increasing Rbb decreases the shelving frequency and increases bass boost gain. Increasing Cbb decreases both cutoff and shelving frequencies. The graph shows Rbb = (46.6k, 30k, 18.3k, 10k, 4.1k, 0 Ohms).
Ao = overall gain for any conditions (should match the MicroCap graph)
Av = gain with no bass boost; Rbb = 0 Ohms
Abb = gain of bass boost; Xcbb >> Rbb; does not include Av
fs = shelving frequency; 3dB below Abb
fc = corner frequency; 3dB above Av
Ao = ?
Av = 1+R4/R3
Abb = 1+Rbb/(R3+R4)
fs ≅ 1/(2πRbbCbb)
fc ≅ 1/(2π((R3+R4)-(R3+R4)^2/Rbb+.707(R3+R4)((R3+R4)/Rbb)^1.532)Cbb)
The equations are approximations derived from numerical analysis of MicroCap AC modeling graphs and lose accuracy as bass boost gain drops towards 6dB. Below 6dB they are useless as fc and fs overlap.
What are the true equations for Ao, fs, fc?