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Units of constants in transfer functions?

  1. Jul 13, 2016 #1
    Hi All

    Probably a very basic question.

    What are the units of the constants in transfer functions?

    It we take a look at the transfer function of a second order system we then have:

    H(s) = ω02/(s2+2ζω0s+ω02)

    ω0 is the natural resonance frequency and has a unit of rad/sec. ζ is the damping and has no unit. Hence everything ends up being without an unit if s also has the unit rad/sec. So for so good.

    If we now look at the transfer function for a first order response:

    H(s) = a/(s+a)

    where a = 1/τ, where τ is the time constant. The unit of τ is sec. Assuming again that the unit of s is rad/sec, a/(s+a) is not without a unit.

    When looking at more complex transfer functions the units become more confusion to me.

    I am most likely missing a small detail. However, I will be very grateful if someone can tell me what I am missing.

    Thanks.
     
  2. jcsd
  3. Jul 13, 2016 #2
    Remember the definition of the transfer function
    $$ H(s) = \frac{Y(s)}{X(s)} $$
    It will always have units
    $$ \frac{\text{units of }Y(s)}{\text{units of }X(s)} $$
    If ##X(s)## and ##Y(s)## (or ##x(t)## and ##y(t)## ) have the same units, the transfer function is unit-less.
    I am not sure exactly where your confusion lies.
    $$Y(s) \propto \int y(t) e^{-st}\,dt$$
    ##Y(s) ## should then have units $$\text{units of }Y(s) = \text{units of }y(t) * \text{ time}. $$
     
  4. Jul 13, 2016 #3
    Hi MisterX

    Thanks for your reply.

    I do understand that the input and output of a system often have units. However, from your definition of the transfer function above you can rewrite like this: H(s)=K⋅G(s) Where K is the gain and has some units that are specifik to the system of interest.

    We are then back to the situation where the transfer function, without the gain, of a first order system is: G(s) = a/(s+a) And we have a = 1/τ, where τ is the time constant. The unit of τ is sec. Assuming that the unit of s is rad/sec, we have (1/sec)/(rad/sec+1/sec) which does't make any sense to me.
     
  5. Jul 13, 2016 #4
    Yes your confusion comes from the fact that radians aren't a unit. Consider the relation between arc length and radius
    $$S = \theta R $$
    Both arc length ##S## and radius ##R## are lengths, measured in meters for example, and so ##\theta## in radians is unitless. Radians/second is the same unit as ##1/second##
     
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