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Derive time-dependent current for circuit

  1. Sep 7, 2011 #1
    1. The problem statement, all variables and given/known data
    Derive an expression for the current I(t) as a function of time for the following circuit:

    ____ \____
    |.............. |
    &.............. -
    & ..............-
    | ...............|
    ___________

    This circuit may be a bit unclear - the && is an inductor L, the - - is a capacitor C, and the top part is a switch that is open for t<0 and closed at t=0. And the ... are filler.


    2. Relevant equations

    I have no idea I am struggling incredibly.


    3. The attempt at a solution

    Help help please
     
  2. jcsd
  3. Sep 10, 2011 #2
    What do you mean by filter???
     
  4. Sep 10, 2011 #3
    1. Use Kirchoff's Voltage Law (sum of voltages around the loop of the circuit):


    2. However, we need to determine the voltage across the inductor and capacitor.
    Just do research, the voltage of the inductor is:

    V_L = Ldi/dt

    V_C = (1/C) integral (idt)


    3. Using KVL: Ldi/dt + (1/C)integral(idt) = 0


    4. We need to turn this one to differential equations. So we differentiate term by term

    Ld^2i/dt^2 + (1/C)i = 0


    5. Use methods of differential equation to solve for i with respect to t.

    Lm^2 + (1/C) = 0

    m^2 + 1/(L*C) = 0; m = +- j/(L*C)

    I(t) = Acos(t/(LC)) + Bsin(t/(LC))


    6. The constants A and B can be derived using initial conditions.

    See here for more details:

    http://en.wikipedia.org/wiki/LC_circuit
     
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