Inverting two-pole band-pass filter (source code)

= Inverting two-pole band-pass filter
{title2=$H(s)=-K s\tau_1/[(1+s\tau_1)(1+s\tau_2)]$}

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 $\tau_i=R_iC_i$. 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.