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.

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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.

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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.

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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?