1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Laplace and pole zero diagram.

  1. Jan 7, 2013 #1
    The following diff. equation describes the functionality of a system with respect to time. However, it is not known how the system will behave when stimulated. Apply a forward Laplace transform to determine damping ratio and pole zeros. Plot a pole zero diagram and comment on stability.

    d2y/dt2 + 6dy/dt + 10y = 63dy/dt + 63x


    Putting like terms on either side of the equation:

    d2y/dt2 + 6dy/dt - 63dy/dt + 10y = 63x

    = d2y/dt2 - 57dy/dt + 10y = 63x

    Laplace transform:

    s2Y(s) - 57sY(s) + 10Y(s) = 63 .... (63 because unit impulse used to stimulate?)

    Simplify for Y(s):

    Y(s).(s2 - 57s + 10) = 63

    Solve for Y(s):

    Y(s) = 63/(s2 - 57s + 10)

    Using quadratic formula to find poles:

    s= (57 +/- √572 - 4 x 1 x 10)/(2 x 1)

    = (57 +/- 56.648)/2 = 28.5 +/- 56.648 (I was expecting a complex number!)

    I think that I am nearly there, but I suspect that I have gone wrong.
     
    Last edited: Jan 7, 2013
  2. jcsd
  3. Jan 7, 2013 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    Factor out Y(s)F(s) = X(s) from the above transformed equation with X(s) = 1 (unit impulse input).

    Then Y(s)/X(s) = 1/F(s) and let H(s) = 1/F(s) so that now Y(s) = X(s)H(s). H(s) is now the system transfer function. Given any X(s) you can now compute Y(s) and consequently y(t).

    Note that the magnitude of the input impulse has nothing to do with pole/zero location, nor damping ratio. You only use it if you want to compute Y(s) and y(t).

    Note also that with an impulse input, all the poles & zeros of the output Y(s) are due to the system H(s) only.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Laplace and pole zero diagram.
  1. Zero and pole (Replies: 5)

  2. Pole zero plot (Replies: 3)

Loading...