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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.

Gennaro Arguzzi
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  • 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: http://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
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    @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 07:39
  • @dcorking Sorry, but I couldn't find an example for "divided into" meaning opposite of "divided by", to me they are the same thing. 12 divided into 4 is 3. 12 divided by 4 is also 3, not 1/3 and not 48. Maybe it's a language trick that I've missed. – Gürkan Çetin Sep 16 '17 at 19:11
  • As agentp said, it is clumsy wording, that I am surprised to see in a textbook. It is a common informal turn of phrase, at least in North America and Europe. – dcorking Sep 17 '17 at 08:24
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    @GurkanCetin see http://www.macmillandictionary.com/dictionary/british/divide-into please – Gennaro Arguzzi Sep 17 '17 at 19:52
  • Thanks a lot @dcorking and GennaroArguzzi , I had never noticed the phrase divided into in this use. – Gürkan Çetin Sep 17 '17 at 20:15

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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."

Mika Sundland
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agentp
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