closed-loop gain
Closed-loop gain is the gain you actually get and actually design for — the gain of the whole circuit once you have wrapped feedback around the op-amp and closed the loop. It is the honest, useful number: turn the input up by 1, the output goes up by the closed-loop gain. Unlike the raw open-loop gain (which is enormous and unreliable), the closed-loop gain is modest, steady, and chosen by you.
The formula is Acl = A / (1 + A times beta), where A is the raw open-loop gain and beta is the feedback fraction. When the loop gain A times beta is huge, this simplifies to the clean result Acl = 1/beta, set by the feedback resistors alone. For a non-inverting amplifier the gain is 1 + Rf/Rg; for an inverting amplifier it is -Rf/Rin. Example: Rf = 90 kΩ and Rg = 10 kΩ give a non-inverting gain of 1 + 90/10 = 10. Swap them around and you can dial any gain you like with two resistors.
Why this matters: because the gain depends on a resistor ratio rather than on the op-amp, it is precise, stable over temperature, and the same from one chip to the next. Feedback also cuts distortion and raises bandwidth at this gain. There is a deep tradeoff to remember, though: bandwidth equals the gain-bandwidth product divided by the closed-loop gain. Ask for a gain of 100 from a 1 MHz op-amp and you only get 10 kHz of bandwidth. Higher gain always costs speed.
A 1 MHz gain-bandwidth op-amp set for a closed-loop gain of 10 gives a bandwidth of 1 MHz / 10 = 100 kHz; set for a gain of 100 it gives only 10 kHz. Same chip, different gain, very different speed.
Bandwidth = gain-bandwidth product / closed-loop gain — gain and speed trade off.
The simple Acl = 1/beta holds only while the loop gain is large. Near the op-amp's bandwidth the loop gain shrinks, Acl sags below its ideal value, and distortion creeps back up.