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Electrical Engineering - Control Systems - 2cnd Order System

  1. Feb 25, 2014 #1
    1. The problem statement, all variables and given/known data

    Given a electrical circuit as one below

    SPnOUPs.png

    One can find the transfer function

    9PPdYxi.png

    For second order systems this can be written as

    OWMjA6f.png

    I know that the damping coefficient can be found using the following formula

    xBvs1cn.png

    In the over damped case ζ>1 can easily occur depending on the values of the components above.

    My question is how would you plot a bode plot when ζ>1? I know that when 0<=ζ<=1 Θ can be found using, Θ=arccos(ζ). In the over damped case you would get the arccosine of a value greater than one. In which case Θ is imaginary so how would you go about plotting it such a angle? I know how to solve such a equation such as arccosine(2), but am unsure of the physical conclusions from such a equation. I'm just intrigued by learning physical representations of the trigonometric functions that fall out side of their real domains. I would like to learn more on this subject.

    Thanks for any help.
     
  2. jcsd
  3. Feb 26, 2014 #2

    NascentOxygen

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    Staff: Mentor

    You replace ##s## by ##jω## and this gives an expression with a complex number in the denominator. Determine the magnitude of the expression, and its angle.
     
  4. Feb 27, 2014 #3

    rude man

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    Homework Helper
    Gold Member

    When the system is overdamp[ed the roots of the characteristic equation both lie on the real negative axis. So that's actually much easier to Bode-plot than for an underdamped system.

    What are the Bode plots of ab/(s+a)(s+b)?
     
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