The gain that fades as you speed up
The last guide chased DC errors — offset, bias current, drift — flaws that sit there even when nothing moves. Now we ask the opposite question: what changes when the signal is fast? The first surprise is that the op-amp's open-loop gain is not the fixed, enormous number the golden rules pretend it is. It is enormous only at DC and low frequencies. As the signal speeds up, that raw gain quietly melts away.
Why does it melt? On purpose. Inside almost every general-purpose op-amp the designer plants one deliberate, dominant slow point (a tiny on-chip capacitor) so the whole chip behaves like a single gentle low-pass filter. The open-loop gain starts around 100,000 to 1,000,000 (100 to 120 dB) at DC, holds flat up to a very low corner of perhaps 10 Hz, then falls along a steady slope of 20 dB per decade — gain divided by 10 for every factor-of-10 rise in frequency. That tidy, single-slope roll-off is the secret behind every number in this guide.
Gain-bandwidth product: pick gain or speed, not both
Here is the gift hidden in that constant 20 dB-per-decade slope: along it, gain times frequency stays roughly constant. That product is the gain-bandwidth product (GBW), and it equals the frequency where the open-loop gain has fallen all the way to 1 — the unity-gain frequency. If a part is rated GBW = 1 MHz, then at 1 MHz the chip has no spare gain left at all.
The closed-loop consequence is the headline of this whole guide: your closed-loop bandwidth equals GBW divided by your closed-loop gain. Set the gain to 10 and a 1 MHz part gives you 100 kHz of bandwidth; set gain 100 and you get only 10 kHz; a unity-gain buffer keeps the full 1 MHz. The product is fixed — you spend it on gain or on speed, never both. This is the AC face of the trade you already know: feedback swaps raw, untrustworthy gain for good behaviour, and bandwidth is part of the bill.
For an op-amp with GBW = 1 MHz:
closed-loop gain closed-loop bandwidth
---------------- ---------------------
1 (buffer) 1 MHz
10 100 kHz
100 10 kHz
1000 1 kHz
rule: bandwidth = GBW / closed-loop gain
gain x bandwidth = GBW (constant)Picture it on a Bode plot. The open-loop gain is a single falling diagonal; your chosen closed-loop gain is a flat horizontal line at, say, 20 dB (a gain of 10). They cross at one frequency — that crossing is your bandwidth. To the left, the diagonal sits well above your line, so the loop has spare gain and the response is flat and faithful. To the right, the line would need gain the chip no longer has, so the closed-loop response gives up and rolls off down the diagonal. In plain words: your amplifier is also a low-pass filter, and you set its corner by choosing the gain.
Slew rate: a different kind of speed limit
Now a twist that catches almost everyone. Bandwidth is a small-signal limit — it describes tiny wiggles, where the chip stays linear. Slew rate is a large-signal limit: the maximum rate, in volts per microsecond, that the output can physically move, no matter how hard the feedback shouts. They are two separate ceilings; a part can have generous bandwidth yet a stingy slew rate, or the reverse.
Where does this ceiling come from? Inside, a fixed bias current has to charge that one deliberate compensation capacitor. Current into a capacitor sets its rate of change — the rechargeable-bucket relationship I = C times (dV/dt). With a fixed maximum current pouring into a fixed bucket, there is a hard ceiling on how fast the internal node, and therefore the output, can ramp. A classic 741 manages only about 0.5 V/µs; a snappy modern part might do tens or even hundreds of V/µs.
The give-away symptom: feed a big, fast sine and the output cannot keep up, so it leaves the curve and runs in straight ramps — the sine arrives as a triangle. Put numbers on it. A 0.1 V peak sine at 100 kHz needs a top slope of 2 times pi times 100,000 times 0.1, about 0.063 V/µs — trivial. The same 100 kHz but at 5 V peak needs 2 times pi times 100,000 times 5, about 3.1 V/µs — far beyond a 741's 0.5, so it slews into a triangle. Identical frequency, different amplitude, different fate: that is the fingerprint that tells slewing apart from plain bandwidth roll-off, which would simply shrink every amplitude by the same factor.
Full-power bandwidth and the swing ceiling
Now fuse the two ideas. For a sine of peak amplitude A at frequency f, the steepest part of the curve has slope 2 times pi times f times A. The output can trace it only while that slope stays at or below the slew rate. Solve for the highest frequency at which you can still get the full swing without slewing, and you get the full-power bandwidth: f_max = slew rate / (2 times pi times A).
Run the 741. With slew rate 0.5 V/µs = 0.5 times 10^6 V/s and a full ten-volt peak (A = 10), f_max = 0.5 times 10^6 / (2 times pi times 10), about 8 kHz. Read that twice: even though the small-signal bandwidth at gain 10 was 100 kHz, you can only get the full 10 V amplitude up to roughly 8 kHz. Above that you must either accept a smaller swing or reach for a faster chip — swap in a part with 20 V/µs and f_max climbs to about 318 kHz at the same 10 V. Small-signal speed and full-amplitude speed are different numbers, and the slow one usually decides.
There is a second ceiling on amplitude that has nothing to do with speed: the output voltage swing. The output can never pass the supply rails, and a classic op-amp cannot even reach them — it stops one to two volts short of each. A rail-to-rail output stage gets within tens of millivolts of each rail (best at light load), which is why those parts shine on low single supplies. Ask for more swing than the part allows and the output clips flat-topped — a large-signal limit that is separate from slewing but often met on the same day, because pushing amplitude pokes both at once.
Reading it off the part — and honest caveats
So before you trust a circuit at speed, run a quick two-front check — one for the small-signal ceiling, one for the large-signal ceiling.
- Look up two numbers: the GBW (or unity-gain bandwidth) and the slew rate. Both sit on the front page of the part's specs.
- Small-signal check: compute closed-loop bandwidth = GBW / your gain. Is your highest signal frequency comfortably below it — say at most a third of it, to keep the response flat? If not, lower the gain or pick a higher-GBW part.
- Large-signal check: compute the required slope 2 times pi times f times A at your biggest output amplitude and highest frequency together. Is it below the slew rate? If not, you are slew-limited — shrink the swing, drop the frequency, or get a faster part.
- Swing check: make sure your peak output fits inside the output swing for your supply and load; otherwise it clips. Both checks must pass — the worse one wins.
Every number above lives on the datasheet, and the next guide is devoted to reading one fluently. Be honest about the simplifications, though. The neat "gain times bandwidth is constant" only holds for an internally-compensated part with one dominant pole; for a real chip it is roughly true near unity gain, and not true at all for decompensated or uncompensated types. Slew rate can differ slewing up versus down. Distortion creeps in well before the output fully triangulates — the curve sours as you approach the slew limit, not only when you cross it. And settling time, how long the output takes to land within a hair of its target, is yet another spec you may need.
One last thread, picked up by the next guide. That same deliberate internal pole that fixes the GBW also costs phase as frequency climbs. Chase more bandwidth by weakening the compensation and you erode phase margin — and a feedback loop with too little phase margin breaks into oscillation. So bandwidth, slew rate, and stability are three faces of one internal compromise. Get comfortable with these AC limits here; the next guide is where feedback bites back.