# Time constant - formula

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.

• Are you sure he says the sentence that follows the semi-colon? The unit of time constant is seconds (or Time), but the rate of change divided to total change is 1/second. For a general explanation this page is useful: controlguru.com/process-gain-is-the-how-fast-variable-2 – Gürkan Çetin Sep 15 '17 at 19:27
• Hi @GurkanCetin, I reported the words which I read on the book. – Gennaro Arguzzi Sep 15 '17 at 19:33
• Ok, then it's counter intuitive for me. I would be OK if total change was divided to change rate, resulting in a time value. – Gürkan Çetin Sep 15 '17 at 19:43
• @Gurkan Book says 'divided into' so rate of change is the denominator. Therefore the sentence is dimensionally correct. 'Divided into' is the reverse of 'divided by'. – dcorking Sep 16 '17 at 7:39
• @GurkanCetin see macmillandictionary.com/dictionary/british/divide-into please – Gennaro Arguzzi Sep 17 '17 at 19:52

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."