Full-power bandwidth

The full-power bandwidth $f_M$ is the highest frequency at which an op-amp can deliver an undistorted full-amplitude sine wave. It is set by the Slew rate, not by the small-signal gain roll-off:

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

where SR is the slew rate (maximum $|d{v}_O/dt|$) and $V_{O,\max }$ is the peak output amplitude you want to swing.

Where the formula comes from

A sine output $v_O(t)=V_{O,\max }\sin (\omega t)$ has maximum slope $\omega V_{O,\max }$ (at its zero crossings). The op-amp can trace the waveform faithfully only while that peak slope does not exceed the slew rate: $\omega V_{O,\max }\le \text{SR}$. The boundary frequency, with $\omega =2\pi f$, gives $f_M=\text{SR}/(2\pi V_{O,\max })$. Below $f_M$ the full-amplitude sine is clean; above it the op-amp cannot keep up on the steep parts and the sine degrades toward a triangle (slew-rate limiting).

Worked number. A 741 with $\text{SR}=0.5\text{V/}\mu \text{s}$ asked to swing $\pm 10\text{V}$:

$f_M=\frac{0.5\text{V/}\mu \text{s}}{2\pi \cdot 10\text{V}}=\frac{0.5\times 10^6\text{V/s}}{62.8\text{V}}\approx 8\text{kHz}$

So a 741 cannot produce a clean $\pm 10\text{V}$, $20\text{kHz}$ sine — even though $20\text{kHz}$ is far below its $1\text{MHz}$ small-signal gain-bandwidth.

Why it is a separate spec from $f_t$

The full-power bandwidth depends on amplitude (note $V_{O,\max }$ in the denominator): ask for a smaller swing and $f_M$ rises proportionally. The small-signal bandwidth from the finite Open-loop gain — roughly $f_t/|G|$ — does not depend on amplitude. They are two independent ceilings:

• Small signals (well below the rails) never hit the slew rate, so they stay linear up to the gain-bandwidth limit $f_t$.
• Large signals (near the rails) hit the slew rate first and distort at $f_M\ll f_t$.

A real amplifier can be limited by either, so always evaluate both: compute $f_t/|G|$ for the small-signal corner and $f_M=\text{SR}/(2\pi V_{O,\max })$ for the large-signal corner, and the lower one governs the signal you actually care about. Both are Real op-amp imperfections absent from the Ideal op-amp model.