Open-loop gain

The open-loop gain $A$ is the gain of an op-amp with no feedback applied — the raw differential gain of the bare chip:

$v_O=A(v_2-v_1)$

where $v_2$ is the non-inverting input, $v_1$ the inverting input, and $v_O$ the output, all measured to ground. For a real op-amp $A$ is enormous, typically $10^5$ to $10^6$ ($100$–$120\text{dB}$).

Why it is useless on its own

A gain of $10^5$ sounds wonderful until you use it. An input difference of just $100\mu \text{V}$ produces $v_O=10^5\times 100\mu \text{V}=10\text{V}$. Thermal noise, a stray thermocouple at a solder joint, a few millivolts of offset — anything — instantly drives the output hard against a supply rail. Open-loop, the op-amp is a comparator, not a linear amplifier. To get controllable, predictable gain we wrap it in Negative feedback and let the huge $A$ work for us: it is precisely because $A$ is so large that the virtual short holds and the Closed-loop gain collapses to a clean resistor ratio.

Frequency dependence

The ideal model pretends $A$ is constant at all frequencies. It is not. $A$ is large only near DC; above a corner frequency $f_b$ — often just a few hertz — it rolls off at $-20\text{dB/decade}$ (a single-pole roll-off, like an RC lowpass filter). Above the corner the product of gain and frequency is roughly constant:

$A(f)\cdot f\approx f_t=\text{constant}$

$f_t$ is the gain-bandwidth product: the frequency at which $|A|$ has fallen all the way to unity ($0\text{dB}$). For a 741 $f_t\approx 1\text{MHz}$; modern parts reach hundreds of MHz.

$A(f)$ large only at DC; −20 dB/decade above $f_b$; unity at the gain-bandwidth product $f_t$.

The practical payoff is the closed-loop bandwidth. Because $A\cdot f$ is constant, an amplifier with closed-loop gain magnitude $|G|$ has a $-3\text{dB}$ bandwidth of approximately

$f_{-3\text{dB}}\approx \frac{f_t}{|G|}$

So a $\times 100$ inverting amplifier built from a 741 ($f_t=1\text{MHz}$) is flat only to about $10\text{kHz}$; push it to $100\text{kHz}$ and the output is attenuated and phase-shifted. You trade gain for bandwidth one-for-one. This finite-and-frequency-dependent $A$ is one of the Real op-amp imperfections, and its effect on gain accuracy is folded into Closed-loop gain. Note this small-signal bandwidth is independent of Slew rate limiting — check both.