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RLC Zero State/Zero Input Response

  1. Jul 10, 2013 #1
    1. The problem statement, all variables and given/known data
    2013-07-10175923_zps68571203.jpg

    In the above diagram,

    iin(t) = -0.5u(-t) + 2u(t) A
    R = 2 Ω
    L = 1 H
    C = 8 mF

    Find the zero-input, zero-state, and complete responses of vC(t) and iL(t) for t > 0.

    2. Relevant equations

    σ = R/2L for series RLC circuits
    ωn = 1/(sqrt(LC)) for RLC circuits
    ωd = sqrt(ωn2 - σ2)
    x(t) = [Acos(ωdt) + Bsin(ωdt)]e-σt for underdamped source-free RLC circuits

    3. The attempt at a solution

    2013-07-10175846_zpsab495816.jpg

    This is what I have so far, and I think it's correct, but I am totally confused as to what I need to do to get the zero-input response. I'm not really sure if I understand it conceptually.
     
  2. jcsd
  3. Jul 10, 2013 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    I do't know why you started with w = 1/sqrt(LC), sigma, etc.

    How do you know it's an underdamped circuit?

    You might instead:

    1. determine VC and IL at t = 0 by inspection.

    2. apply a step input of current = 2A U(t) and compute VC and IL at and after t = 0+. You will have to solve the differential equation with the initial conditions VC(0) and IL(0) set up by Iin = - 0.5A U(-t).

    I also don't know what they mean by "zero-input". Not to mention "zero-state". I would just solve for VC(t) and IL(t), t > 0.


    If you've had the Laplace transform that's the easy way to do that.
     
  4. Jul 10, 2013 #3
    Determining w and sigma is helpful in that if w is greater than sigma, the circuit is underdamped.

    This is an introductory circuits course. Most people in the class have not taken differential equations, so they're not heavily used in the course.

    I would certainly like to just solve for v_C(t) and i_L(t), but the question is asking me to find the zero input and zero state responses.
     
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