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The closed-loop gain is the input-to-output transfer after closing the feedback connection. In the elementary noninverting voltage model it tends to $1/\beta$ for large stable loop gain, rather than retaining the uncertain open-loop gain $A$. A frequency-dependent transfer must be checked for stability before that approximation is used.

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.

The loop gain is the product of the forward amplifier response and the return-network fraction, with the summing-junction minus sign kept explicit. Closed-loop poles are determined by $1+L=0$ in this convention. Large low-frequency magnitude controls gain error, while frequency-dependent phase affects stability.

The closed-loop gain from an equivalent differential input-voltage disturbance to the output is the noise gain. For an ideal-input inverting amplifier, $\beta =R_1/(R_1+R_2)$ and $G_{\mathrm{n}}=1+R_2/R_1$, although the voltage gain is $-R_2/R_1$. A dominant-pole amplifier has approximate closed-loop bandwidth $f_T/G_{\mathrm{n}}$ when loop gain is large at low frequency and the circuit is stable.

A non-inverting operational amplifier applies the signal to its non-inverting input and returns part of the output to its inverting input through negative feedback. Connect resistor $R_f$ from the output to the inverting input, and resistor $R_g$ from that input to a reference circuit node at electric potential $V_{\mathrm{ref}}$.

In stable, unsaturated ideal operation, the inputs draw no electric current and their electric potentials are equal. Writing the signal as $V_{\mathrm{in}}$, Kirchhoff's current law at the inverting input givessoFor a grounded reference circuit node, the voltage gain is $1+R_f/R_g$. For example, $R_f=20\mathrm{k\Omega }$ and $R_g=10\mathrm{k\Omega }$ give gain three. The equal-input-potential approximation requires sufficiently large loop gain; amplifier saturation or unstable feedback invalidates it.^[ref]

Return a fraction of the output to the inverting input so that it opposes the input difference. Large stable loop gain makes the closed-loop gain depend mainly on the external feedback ratio rather than the uncertain open-loop gain. Frequency-dependent phase lag can destabilize the loop; saturation and finite slew rate also limit the approximation.^[ref]

In an operational-amplifier negative feedback loop, an operational amplifier has large open-loop gain $A$, with $V_o=A(V_{+}-V_{-})$. Feeding a fraction $\beta V_o$ back to the inverting input opposes an output increase. For a non-inverting signal input,Its sensitivity to open-loop gain changes is reduced by $d\ln A_{\mathrm{cl}}/d\ln A=1/(1+A\beta )$. Thus resistor ratios can set a reproducible gain despite device variation; feedback also suppresses internal distortion and reduces output impedance. Gain is traded for usable bandwidth, subject to the amplifier's frequency response.

For the non-inverting connection, place $R_f$ between output and inverting input and $R_g$ between that input and ground. Then $\beta =R_g/(R_f+R_g)$, giving gain $1+R_f/R_g$. For an inverting connection, ground the non-inverting input, inject the signal through $R_{\mathrm{in}}$ and return $R_f$ from output to the inverting node. Small differential input and negligible input current then give gain $-R_f/R_{\mathrm{in}}$. The near-zero node is a virtual ground, not a physical ground connection.

The feedback must remain stable at the frequencies where loop gain matters. Extra phase lag can turn the intended opposition into oscillation; adequate phase margin, supply headroom, input common-mode range, output-current capability and slew rate are required. Feedback neither prevents clipping nor removes all input noise.^[ref]