# Q of Sallen key low pass filter

During i study Sallen-key active low pass filter, i wondered why formula of Q is 1/(3-A). A is Voltage Gain. So i searched from web, and i found this.

When R1=R2 and C1=C2, the Q be 1/(3-A). OK. But, i don't know why the formula of Q's defined as a formula in the picture above.

Why the formula of Q's defined as a formula in the picture above? I'd really appreciate it if you could derive this formula.

• There's nothing stopping you from doing it, yourself: replace $R_1=R_2=R$ and $C_1=C_2=C$ in all the formula for the transfer function, normalize to the denominator's $s^2$ term, then compare with the generic formula for a second order transfer function: $$H(s)=\dfrac{\omega^2}{s^2+\dfrac{\omega}{Q}s+\omega^2}$$ May 15, 2021 at 13:54
• Or just read Sallen and Key, since they discuss the relationships in their paper.
– jonk
May 15, 2021 at 15:54
• From this simulation you can confirm the effects of K on Q with limits when dominates the denominator (no pun intended) and then reduces the pole frequency and Q sharply when K is too high. tinyurl.com/4b6femxv. Also remember in future that GBW depends on the square of Q at the resonant frequency not just the gain x Q May 15, 2021 at 18:04
• There you find max Q of about 100 with a DC gain of 9 to 10 dB thus 39dB to 40 dB gain at peak May 15, 2021 at 18:10