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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. enter image description here

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.

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    \$\begingroup\$ 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: $$H(s)=\dfrac{\omega^2}{s^2+\dfrac{\omega}{Q}s+\omega^2}$$ \$\endgroup\$ May 15, 2021 at 13:54
  • \$\begingroup\$ Or just read Sallen and Key, since they discuss the relationships in their paper. \$\endgroup\$
    – jonk
    May 15, 2021 at 15:54
  • \$\begingroup\$ 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 \$\endgroup\$ May 15, 2021 at 18:04
  • \$\begingroup\$ 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 \$\endgroup\$ May 15, 2021 at 18:10

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When you compare the general form of the transfer function with the function derived from the circuit (given in your contribution) you can identify the pole frequency wp and the pole-Q (Qp).

The expression which appears as a factor at s² is 1/wp².

And the factor appearing at the s-element is 1/(wpQp).

From this comparison you can derive the formulas for Wp and Qp.

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