= Rise time
{title2=$t_r$}
{wiki}
<Rise time> measures how long a response takes to traverse a specified fraction of its transition from an initial level to a final level. A common convention is the 10–90-percent <rise time>. Normalize an upward transition by
$$
z(t)=\frac{y(t)-y_0}{y_\infty-y_0},\qquad t_r=t_{0.9}-t_{0.1},
$$
where $t_p$ is the crossing time at normalized level $p$. The percentages and crossing rule must be stated, especially when a waveform overshoots, rings or crosses a threshold repeatedly.<Understanding Oscilloscope Bandwidth, Rise Time and Signal Fidelity by Tektronix>[Printed pp. 3–4, rise-time convention and step response]{ref}
For an original <first-order low-pass filter> with <time constant> $\tau$, its normalized response to a delayed ideal step is $z(t)=1-e^{-(t-t_d)/\tau}$ for $t\geq t_d$. Solving for the threshold gives
$$
t_p=t_d-\tau\ln(1-p),\qquad
t_r=\tau\ln9\approx2.197\tau.
$$
A pure delay changes both crossings by the same amount and therefore does not change their difference. For $\tau=1\,\mathrm{ms}$, the crossings occur about 0.105 and 2.303 milliseconds after the response begins, giving a 2.197-millisecond <rise time>.
The same one-pole model has a minus-3-decibel <bandwidth> $B=1/(2\pi\tau)$, so
$$
Bt_r=\frac{\ln9}{2\pi}\approx0.3497.
$$
The familiar $t_r\approx0.35/B$ relation therefore follows from this model and these thresholds. It is not a universal identity for every amplifier or <oscilloscope>. Different <frequency responses> can share a minus-3-decibel bandwidth while producing different transients.<Understanding Oscilloscope Bandwidth, Rise Time and Signal Fidelity by Tektronix>[Printed p. 3, one-pole assumption and counterexample discussion]{ref}
<Rise time> also differs from <settling time>. The example reaches 90 percent at $2.303\tau$ but does not reach the one-percent error band until $4.605\tau$. A waveform can rise quickly and then ring outside its required final tolerance.
To measure a device, apply a characterized step, record the response through the probe and <oscilloscope>, estimate the initial and final levels, and determine the stated threshold crossings. The source, device, probe and readout all contribute to the recorded waveform. Probe loading, limited sampling and noise can change the inferred crossings; a displayed fast edge is not automatically the device's undisturbed response.<Understanding Oscilloscope Bandwidth, Rise Time and Signal Fidelity by Tektronix>[Printed pp. 4–6, step generator and probe effects]{ref}
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