The book "Instrument Engineers' Handbook vol. 1 - Liptak" said that for the first-order system forced by a step or an impulse, the time constant is the time required to complete 63.2% of the total rise or decay; at any instant during the process, the time constant is the quotient of the instantaneous rate of change divided into the change still to be completed.
How can I express the last sentence mathematically?
Thank you very much.
Does the book not give you the mathematics? The underlying expression is:
$$parameter = 1-e^{ -t/ \tau }$$
so you see at $t = \tau$
$$parameter = 1 - e^{-1} = 0.63$$
Now the second sentence says divide "change to be completed" which is $1 -parameter$ , or $e^{-t/\tau}$, by the first derivative:
$$\frac{d parameter}{dt} = \frac{e^{-t/\tau}}{\tau}$$
so
$$\frac{e^{-t/\tau}}{\frac{e^{-t/\tau}}{\tau}} = \tau$$
pretty simple..?
I will say the wording of that sentence is pretty awkward. I would reverse it as "the time constant is the quotient of the change still to be completed divided by the instantaneous rate of change."
