Gain-Bandwidth Product: Gain and Speed Are a Trade
Lower the closed-loop gain and the band widens
The faint line sloping down is the open-loop gain, which falls as frequency rises. The horizontal axis is logarithmic in frequency, and the vertical axis is in decibels. The bold line is the closed-loop gain set by feedback: it runs flat, then bends where it meets the open-loop line and falls along with it. The frequency at that bend is the bandwidth, and lowering the closed-loop gain moves the bend to the right.
Infinite gain is a fiction: gain falls off
You learned that an ideal op-amp has infinite gain, but a real device is different. At low frequency the open-loop gain is very large (often around a hundred thousand), but above a low frequency called the dominant pole, the gain drops by -20 dB, to 1/10, every time the frequency rises 10 times. Most op-amps place a compensation capacitor inside to set this pole low on purpose, so that they do not oscillate. On a Bode plot with a logarithmic horizontal axis, the open-loop gain is flat and then becomes a line falling at a constant slope, and the frequency where this line meets a gain of 1, or 0 dB, is the unity-gain frequency.
Lower the closed-loop gain, widen the band
With feedback, the closed-loop gain stays flat at the value you chose. But as frequency rises and that flat line reaches the sloping open-loop line, the device has no margin left to support that gain, so from there it falls along with the open-loop gain. The frequency of that meeting point is the bandwidth of the closed-loop amplifier. So when you lower the closed-loop gain, the flat line sits lower and meets the open-loop line farther right, at a higher frequency, and the bandwidth widens by that much. The graph is an approximation drawn with two straight lines; the real curve bends smoothly near this point.
Gain × band = constant = GBW
Because the open-loop line has a constant slope, the product of the closed-loop gain and its bandwidth is the same wherever you cut it. That value is the unity-gain frequency, where the open-loop gain falls to 1, and it is called the gain-bandwidth product, GBW. This constant product holds for op-amps whose response is dominated by a single pole. GBW is a specification the device is born with, so if you want large gain you must accept a narrow band, and if you want a wide band you must accept small gain. Strictly speaking, what you divide by is the noise gain of the feedback network: in a non-inverting amplifier it equals the closed-loop gain, and in an inverting amplifier it is the gain magnitude plus one. One stage cannot raise both, so if you need more, split the gain over several stages or use a device with a larger GBW.
Back to the first screen
With the closed-loop gain set high, the flat line met the open-loop line early and bent at a narrow band. As you lowered the gain, the bend slid to the right and the band widened. Throughout, the product of gain and bandwidth did not change at all, because the open-loop gain falls at a constant slope. That fixed product is GBW, the unity-gain frequency where the gain falls to 1. The ideal of infinite gain holds only roughly at low frequencies; in practice, gain must be traded for speed. This is the first real-world limit of the op-amp.
What comes next
GBW was the limit on how high a frequency a small signal can follow. When the signal is large, another wall appears. The next unit takes up the slew rate, the maximum speed at which the output voltage can change per unit time, and the voltage limit that keeps the output from crossing the supply rails. The small-signal limit and the large-signal limit have different causes.