It's also related to the natural logarthm,

*e*which seems to be a value selected to make the other stuff work out (so that the reactive component, in the unit of F, charges in 5 time constants). Penciled into a book around here, somewhere, I even have a formula for calculating the state of charge at selected times and, I've even worked through it and it works. I don't understand it but, it works.

The curve seems to be exponential so, I got into my TurboCAD and made an exponential curve. I started with an 8 X 8 grid since that's a pretty simple way to do it and then worked out my points and connected them with a Bezier curve. It's not perfect due to the rounding off of the chart resolution (and, I kind of wonder about the Bezier curve...but, it seems to work a lot like the old, french curve, from days past) but, it's pretty close and I got the expected, nice smooth curve.

Then, I superimposed that curve onto a 5 X 10 grid. 5 to represent the 5 time constants (Y axis) and 10 to give an easily readable decimal value (X axis). I expected the line to pass through the first vertical at about 62% but it doesn't. In fact, it does so at about 66%, which is pretty far off.

I redrew the curve to pass through at least the 62.5% and 88% levels for the 1st and 2nd time constant lines (green line) and it's pretty easy to see that there's a fair error in the curve somewhere. But, I'm not sure why.

So, now I'm wondering if my whole notion of how this works might be askew. Is there an obvious answer to all this that I'm just missing?