Ideal operational-amplifier gain

The input reaches the inverting terminal through Rin, and feedback returns output through Rf. In the ideal linear model, the amplifier keeps the input difference near zero, so current through Rin also flows through Rf. Output must have the opposite sign, giving Av=−Rf/Rin. These gains apply in linear closed-loop operation with negative feedback. Check supply, bandwidth and output range for a real device. Application: Design introductory signal-amplifier stages and active analog circuits. The numerical example Inverting: Rf=20 kΩ, Rin=10 kΩ and Vin=0.10 V give Av=−2 and Vout=−0.20 V. Non-inverting with Rf=20 kΩ and Rg=10 kΩ gives Av=3 and Vout=0.30 V, assuming both remain within the linear supply range anchors the idea in a concrete measurement case.

In the non-inverting circuit the signal is applied to the plus input. Rf and Rg feed a fraction of output back to the minus input. The virtual-short assumption makes that fraction approximately Vin, yielding Vout/Vin=1+Rf/Rg. With positive resistors in this topology, gain is at least one. These gains apply in linear closed-loop operation with negative feedback. Check supply, bandwidth and output range for a real device. The corresponding units are Av dimensionless; resistors in Ω; input and output in V. Inverting: Rf=20 kΩ, Rin=10 kΩ and Vin=0.10 V give Av=−2 and Vout=−0.20 V. Non-inverting with Rf=20 kΩ and Rg=10 kΩ gives Av=3 and Vout=0.30 V, assuming both remain within the linear supply range. This checks that the reference quantities are compatible.

A real op amp cannot drive output voltage arbitrarily. It stays within, often short of, the supply rails; output current, slew rate and frequency response also limit the signal. If the calculated output exceeds these limits, the amplifier saturates and the linear gain equation no longer describes it. These gains apply in linear closed-loop operation with negative feedback. Check supply, bandwidth and output range for a real device. Scope: Assume ideal linear operation, negligible input current, negative feedback and output within supply rails. Inverting topology references the non-inverting input; non-inverting topology uses Rf and Rg as a divider. Real bandwidth, slew rate, offset, common-mode range and output current limit operation. A resistor ratio is dimensionless. A practical application is Design introductory signal-amplifier stages and active analog circuits.

With very large open-loop gain and negative feedback, the output adjusts so the input difference is nearly zero in linear operation. The inverting input is not physically connected to ground; it only sits at nearly the same potential. Without feedback or when the output saturates, the approximation fails. These gains apply in linear closed-loop operation with negative feedback. Check supply, bandwidth and output range for a real device. Check the calculation with this case: Inverting: Rf=20 kΩ, Rin=10 kΩ and Vin=0.10 V give Av=−2 and Vout=−0.20 V. Non-inverting with Rf=20 kΩ and Rg=10 kΩ gives Av=3 and Vout=0.30 V, assuming both remain within the linear supply range. Keep the model limits and units in view.

The ideal resistor ratio assumes the output can deliver the required voltage and current. Real op amps have finite bandwidth and gain-bandwidth product, slew rate, input offset, bias currents, and common-mode and output-swing limits. Check supply rails, load and frequency range in the datasheet. Correct DC gain does not guarantee an undistorted high-frequency signal. These gains apply in linear closed-loop operation with negative feedback. Check supply, bandwidth and output range for a real device. The relation describes With ideal op amps under negative feedback, resistor ratios set inverting and non-inverting voltage gain. This helps distinguish its target quantity from neighbouring quantities.