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Homework Help: Step response of a first order system

  1. Feb 9, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the unit step response of the transfer function...

    a) [tex]G(s)\,=\,\frac{4}{s\,+\,4}[/tex]

    b) [tex]G(s)\,=\,\frac{2}{0.2s\,+\,1}[/tex]



    2. Relevant equations

    General first order step response equation...

    [tex]C(s)\,=\,R(s)\,G(s)\,=\,\frac{a}{s(s\,+\,a)}[/tex], where [tex]R(s)\,=\,\frac{1}{s}[/tex]

    then do an inverse Laplace transform...

    [tex]c(t)\,=\,1\,-\,e^{-at}[/tex]



    3. The attempt at a solution

    Part a) is simple enough. I just plugged into formula above and got [tex]c(t)\,=\,1\,-\,e^{-4t}[/tex]

    However, part b) is where I am confused. To get the G(s) into the form needed (i.e. ~ [itex]\frac{a}{s\,+\,a}[/itex]), I divided both the numerator and denominator by 0.2...

    [tex]G(s)\,=\,\frac{2}{0.2s\,+\,1}\,=\,\frac{10}{s\,+\,5}\,=\,2\left(\frac{5}{s\,+\,5}\right)[/tex]

    But now the form is not exactly as needed in the first order system equations. What do I do?

    I tried taking out a 2 from the numerator, and got an answer, just not sure if it's right though.

    Is this right for part b)...

    [tex]c(t)\,=\,2\,\left[1\,-\,e^{-5t}\right][/tex]
     
    Last edited: Feb 9, 2010
  2. jcsd
  3. Feb 9, 2010 #2
    That's right, remember that the Laplace transform is a linear operation. If f(t) has a Laplace transform of F(s), then a*f(t) has a Laplace transform of a*F(s). Assuming "a" is a scalar quantity. The same linearity is true for inverse Laplace transforms.
     
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