Inverting low-pass amplifier ($G(\omega )=-R_f/[R_i(1+i\omega R_{f}C)]$)

An ideal operational amplifier with a grounded noninverting input, input resistor $R_i$, and parallel feedback resistor $R_f$ and capacitor $C$ implements a low-pass filter. The virtual short and Kirchhoff's current law give $V_{\mathrm{out}}/V_{\mathrm{in}}=-Z_f/R_i$, where $Z_f=R_f/(1+i\omega R_{f}C)$. The DC gain magnitude is $R_f/R_i$ and the cutoff frequency is $1/(2\pi R_{f}C)$. The capacitance reduces high-frequency feedback electrical impedance, causing the gain magnitude to fall.