Rise Time

Rise time and fall time measure how fast a signal transitions from one value to another.

The rise/fall time is defined as the time it needs to go from 10% to 90% of the transition (or vice versa).

Since every curve behaves the same way, we always have $V(t)$ at 90% when $\tau =0.105$, and $V(t)$ at 10% when $\tau =2.303$ (or vice versa in the discharging case). So, we always have:

$$t_{\text{rise}}=t_{\text{fall}}=2.2\tau$$

Using this information, we can determine or design the maximum frequency of digital systems.

For other percentages, we can find the time required would by:

1. Finding the percentage in terms of $-\frac{t}{\tau }$ by solving:

$$\begin{array}{r}{e}^{-\frac{t_i}{\tau }}=0.8\\ e^{-\frac{t_f}{\tau }}=0.2\end{array}$$

2. Once you’ve found $-\frac{t}{\tau }$, you can find $t$ since you already know $\tau$.