# Cutoff frequency of transfer function at -6 dB

I'm trying to design a second order low-pass filter using the following transfer function:

$$\\ H(s)=\frac{f_c^2}{s^2 + 2 f_c s +f_c^2} \$$

with cutoff frequency fc = 3400 Hz

Whenever I plot this, the cutoff frequency is at -6 dB instead of -3 dB. I'm not really sure what I'm doing wrong. Here's my Matlab code:

fc = 3400;

s = 1i*logspace(0,6,1000);

H_d = fc^2 ./ (s.^2 + 2*fc*s + fc^2);

semilogx(abs(s), 20 * log10(abs(H_d)))

$$H(s) = \frac{f_c^2}{s^2+1.414f_cs+f_c^2}$$
note: for me, it's unusual to see the use of $$\f\$$ instead of $$\w\$$ in these formulas, but mathematically it should be equivalent, as long as you use $$\s=jf\$$ instead of $$\s=jw\$$
• A 2nd order low pass filter has a gain value at the resonant frequency equal to the circuit's Q. So when Q = 0.5, the gain at the resonant frequency is 0.5 or -6 dB. If Q is ten ($\zeta$ = 0.05) then gain is ten at resonance. – Andy aka Oct 16 '19 at 7:54