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First Order System's Time Constant

  1. Jul 19, 2011 #1
    I have a question on a the units of a first order system's time constant.
    If i have a first order system the basic transfer function will be:
    where K is the Gain, and tau is the system's time constant.
    tau's units, according to what i've learned, are [sec].
    but aren't the s plane's units in [rad/sec] (s=jw+sigma)?
    That means that tau should be given in [sec/rad] to match the "1"-'s units in the transfer function.
    I know that rad can be considered "unitless" but when dealing with actual numbers it matters if the system's time constant is 1 [sec] or 1[sec/rad]= 2*pi [sec].

    My question is specifically about the units of tau in the transfer function,
    not when it is used in the decay rate of e (e^(-t/tau)), there it has to be sec.

    I'll appreciate a clarification.

  2. jcsd
  3. Jul 19, 2011 #2


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    welcome to pf!

    hello yanaibarr! welcome to pf! :wink:

    tau is always in seconds …

    the difference between radians and (eg) degrees is absorbed into the k :smile:
  4. Jul 20, 2011 #3
    Re: welcome to pf!

    No, one may use any unit for tau. For exponential decay, Ae^(-t/tau), the exponent (-t/tau) should be unit-less.
    Last edited: Jul 20, 2011
  5. Jul 23, 2011 #4
    Re: welcome to pf!

  6. Jul 23, 2011 #5


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    sorry, i don't know, i haven't come across the s-plane :redface:
  7. Jul 24, 2011 #6
    Re: welcome to pf!

  8. Jul 24, 2011 #7
    The s-plane is what u get after using the Laplace Transform.
  9. Jul 24, 2011 #8
    Re: welcome to pf!

  10. Oct 23, 2011 #9
    I've stumbled at the same problem. All learning materials seem to expose the concept but none gives example with exact units.

    So, if I want a frequency break at 1 Hz, should I write 1/(s+1) or 1/(s+2Pi)? Second seems more plausible. However, when Laplace-transfromed, it gives e-2pi t meaning that time constant is T = 1/2pi. Yet, I'm customed that periods are measured in seconds rather than seconds per radian. I mean that 2pi is not usually a part of period. But, wikipedia article on time constant does not clarify what are the units.

    Last edited by a moderator: Apr 26, 2017
  11. Oct 24, 2011 #10
    Last edited by a moderator: Apr 26, 2017
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