Inverting amplifier

A voltage-feedback operational amplifier with its noninverting input grounded and input and feedback resistances $R_i,R_f$ has ideal gain $-R_f/R_i$. Virtual short and negligible input current give the result by node-current balance.

With the noninverting input grounded, a differential input impedance $R_{\mathrm{in}}$ draws current from the summing node. For zero output impedance, $V_o=-Av$ and Kirchhoff's current law gives $(v-V_i)/R_1+(v-V_o)/R_2+v/R_{\mathrm{in}}=0$. Eliminating $v$ yields the displayed voltage gain. The resistor-ratio result requires $|A|\gg 1+R_2/R_1+R_2/R_{\mathrm{in}}$ as well as stable, unsaturated operation. The same expression applies to small-signal phasors with complex $A(f)$.

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.

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.