Inverting two-pole band-pass filter ($H(s)=-Ks{\tau }_1/[(1+s{\tau }_1)(1+s{\tau }_2)]$)

An ideal inverting operational amplifier with series $R_1,C_1$ input and parallel $R_2,C_2$ feedback has the displayed transfer function, where $K=R_2/R_1$ and ${\tau }_i=R_i{C}_i$. Its peak is at ${\omega }_0=1/\sqrt{{\tau }_1{\tau }_2}$ with voltage gain $K{\tau }_1/({\tau }_1+{\tau }_2)$. A wide input-high-pass/feedback-low-pass separation produces a broad plateau, not a narrow resonance. Validity requires stable negative feedback, sufficient loop gain and linear output operation.

For a response proportional to $s/[(1+s{\tau }_1)(1+s{\tau }_2)]$ with positive time constants, its half-power width is $1/{\tau }_1+1/{\tau }_2$. Combining with the geometric-mean centre frequency gives the displayed quality factor; the arithmetic-geometric mean inequality proves the bound. Equality requires matched time constants and gives half-power frequencies $(\sqrt{2}\pm 1){\omega }_0$. Adjusting gains does not make this topology a high-Q resonant filter.