Source-free 2nd order lin. circuit

  1. RLC second order linear network question:
    So, we are given this equation which is the same for Vc(t) and iL(t) expressed as x(t):

    2nd deriv of x(t) + R/L(1st deriv of x(t)) + 1/(LC)(x(t)) = 0;

    And in one of the problems it asks to find both equation for the Vc(t) and iL(t) for t < 0, and now I am confused, it seems to me that they are the same, since the solution is the same for both of them:

    aS^2 + bS + c = 0;
    because the coefficients are the same from differential equation, so there are the same roots for Vc(t) and iL(t), and roots are w/t imaginary part, just reals.
    Am I wrong?

    Thanks a lot.
     
    Last edited: Mar 26, 2005
  2. jcsd
  3. Andrew Mason

    Andrew Mason 6,856
    Science Advisor
    Homework Helper

    Chegg
    I am not sure I understand what you are asking exactly, but perhaps this will help.

    An LC circuit with resistance is simply a damped harmonic oscillator. The differential equation:

    [tex]L\ddot x(t) + R\dot x(t) + \frac{1}{C}x(t) = 0[/tex]

    has the general solution:

    [tex]x = A_0e^{-\gamma t}sin(\omega t+\phi)[/tex]

    where [itex]\omega^2 = \omega_0^2 - \gamma^2[/itex] and
    [itex]\omega_0^2 = 1/LC[/itex] and
    [itex]\gamma = R/2L[/itex]

    The relationship between V and I in an RLC circuit is:

    [tex]V = IZ[/tex] where is the impedance: [itex]Z^2 = R^2 + (\omega L - 1/\omega C)^2[/itex]

    AM
     
    Last edited: Mar 28, 2005
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