Suppose we have
$$ H(s) = \dfrac{ k \frac{\omega_o}{Q}s}{s^2 + \frac{\omega_o}{Q}s + \omega_n^2}$$
What is the definition of the quality factor? My understanding is that it is the gain at the cut-off frequency. I know how to find this from a bode plot by extending a line from the band pass gain and extending a line when there is a drop off in gain and finding their intersection. This gives us the cut-off frequency and hence we can find the gain at the cut-off frequency, which is the quality factor.
Also, if I had a transfer function as above, how would I find the cut-off frequency mathematically without a bode plot? If I set \$ |{H(\omega_o j)}| = \dfrac{1}{\sqrt{2}} \$, I can find the cut off frequency, but doesn't that assume that the quality factor is \$\dfrac{1}{\sqrt{2}}\$? Also, to do this, doesn't the gain have to be 1?
Thanks.