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.

  • $\begingroup$ 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 $\endgroup$ – Gürkan Çetin Sep 15 '17 at 19:27
  • $\begingroup$ Hi @GurkanCetin, I reported the words which I read on the book. $\endgroup$ – Gennaro Arguzzi Sep 15 '17 at 19:33
  • $\begingroup$ 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. $\endgroup$ – Gürkan Çetin Sep 15 '17 at 19:43
  • 1
    $\begingroup$ @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'. $\endgroup$ – dcorking Sep 16 '17 at 7:39
  • 1
    $\begingroup$ @GurkanCetin see macmillandictionary.com/dictionary/british/divide-into please $\endgroup$ – 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:

$$\frac{d parameter}{dt} = \frac{e^{-t/\tau}}{\tau}$$


$$\frac{e^{-t/\tau}}{\frac{e^{-t/\tau}}{\tau}} = \tau$$

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."


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.