Slew rate

The slew rate SR is the maximum rate at which an op-amp output voltage can change, in volts per microsecond:

$\text{SR}={\left|\frac{d{v}_O}{dt}\right|}_{\max }$

For a 741 it is about $0.5\text{V/}\mu \text{s}$; fast modern parts reach hundreds of V/µs. It is a large-signal limitation caused by a finite internal current charging the op-amp’s internal compensation capacitor — that current cannot exceed a fixed value, so $dv/dt=i/C$ is capped no matter how big the demanded change is.

Distinct from output saturation

Op-amp output saturation is an amplitude limit (how high $v_O$ can go); slew rate is a rate limit (how fast it can get there). Even when the steady-state output sits comfortably within the rails, the op-amp may be unable to move fast enough to trace a fast-changing waveform. Feed a small step and the output does not jump — it ramps at the slew rate. Feed a large fast sine and the output cannot keep up on the steep parts, so the sinusoid degenerates into a triangle wave (constant-slope ramps between turning points). This is slew-rate limiting.

Output cannot change faster than SR; above $f_M=SR/(2\pi V_{O,max})$ large sines distort.

When it bites: the full-power bandwidth

A sine wave $v_O(t)=V_O\sin (\omega t)$ has derivative $\frac{d{v}_O}{dt}=\omega V_O\cos (\omega t)$, whose maximum magnitude is $\omega V_O$ (steepest at the zero crossings). The op-amp can reproduce this sine undistorted only while that peak slope stays below the slew rate: $\omega V_O\le \text{SR}$. The highest frequency at which a full-amplitude sine of peak $V_{O,\max }$ survives is therefore

$f_M=\frac{\text{SR}}{2\pi V_{O,\max }}$

the Full-power bandwidth. For a 741 driving a $\pm 10\text{V}$ swing, $f_M=\frac{0.5\text{V/}\mu \text{s}}{2\pi \cdot 10\text{V}}\approx 8\text{kHz}$. Above $\sim 8\text{kHz}$ large signals near the rails turn triangular.

The key subtlety: this is an amplitude-and-frequency limit. Small signals (well below the rails) have a small $\omega V_O$ even at high frequency, so they never hit the slew rate and stay linear all the way up to the small-signal gain-bandwidth limit $f_t$ (see Open-loop gain). So an op-amp has two independent bandwidths: small-signal $f_t$ and large-signal (full-power) $f_M$. A circuit may be limited by either, so check both.