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# Graphing the RC Time Constant (Error)

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#### crashsite

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Because my math skills are so poor, I sometimes try to figure things out or try to verify them using graphs or charts or pictures. Somelthing I've been wondering about for awhile and finally got around to trying out is to try to verify an idea I have about RC time constants (similar for RL). Way back in electronics class we were told that T=RC and that, after 5 time periods the capacitor or inductor was considered to be fully "charged". After the firrst time period, it was 62.3% charged. Well, you know all about that.

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?

wow - this is over complicating the problem.You're not going to get accurate results from a hand drawing. Do you have MS Excel? How about Open Office? You can use the spreadsheets to create accurate graphs really easy.

In cell A1, enter a value for a resistor (in Ohms)
In cell A2, enter a value for a capacitor (in Farads)
In cell A3, enter an initial voltage

In column A, starting at A4 enter a linearly increasing array of numbers (.001,.002,.003,...,n) that represent time
In cell B4 enter "=$A$3*(1-exp(-A4/($A$1*$A$2)))" then copy that cell downward.
Now create a line graph of the two columns and analyze.

Iteration

wow - this is over complicating the problem.You're not going to get accurate results from a hand drawing. Do you have MS Excel? How about Open Office? You can use the spreadsheets to create accurate graphs really easy.

....Now create a line graph of the two columns and analyze.

The purpose of the exercise wasn't to make an accurate graph or to come up with a way to iteratively solve for the points for the line. There are a number of ways to do that. But, all that does is show that I can solve for the points and draw the line.

My intent was to try to prove to myself that my understanding of the exponential nature and scaling of the line is right. That the curve came out with an error that was substantially greater than I had expected (knowing the resolution I was working at) is the part that concerns me.

I know it kind of looks like mechanical scans of inked lines on graph paper but, it's actually screen shots taken directly from the CAD program so, it at least accurately represents what was calculated by the program to draw the Bezier curves into the computer's display memory.

still makes no sense...

garbage in yields garbage out

if you want to understand the curve then you need to start with an accurate curve and then analyze that

Hello,

The obvious answer is that the program is not drawing the exponential curve
correctly. It's as simple as that.
When drawn correctly the first time constant will appear at y=1-1/e
which is close to 0.632 as you noted.
The solution is to find a program that CAN draw the curve accurately, then
do everything over again. Better yet, set the drawing programs grid to be
10x10 and that way you can see everything line up (or not).

Last edited:
Okay, I feel better about it, now...

The obvious answer is that the program is not drawing the exponential curve
correctly. It's as simple as that.

Okay, thanks. That helps. I went back to the CAD program and redrew the Bezier curve, using the same points but, at a different scale on the screen and it did draw differently.

Now, I'm more comfortable that the RC curve it works as I had thought but, can see that I'd better watch out for trusting that those curves will accurately and repeatedly give the "right" view of things in a CAD program.

Thanks also for the correction. It's 63.2%, not 62.3%.

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