Where did the rule of thumb that the rise time of an exponential curve is 2.2 times the time constant of that rise come from?
You've probably used the rule of thumb that the rise time of an exponential curve is 2.2 times the time constant of that rise, but maybe you never looked at where that rule came from. Here's the answer.
The timing versus voltage or current or whatever relationship for an exponentially rising waveshape is illustrated as follows.
Figure 1 Exponential curve timing
Using the above, we find the time values T10 and T90, which we define as being the times when the rising curve has come to the 10 percent and the 90 percent points of the excursion toward the "final value."
Figure 2 Timing of the 10 percent and 90 percent points
Since the rise time is defined as the time interval between the 10 percent and 90 percent points shown above, the difference between those two values is found to be the time constant times 2.197224 ...
Oh what the heck! Call it 2.2 times Tau and ascribe that to an engineering approximation.
John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).