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RLC Circuits

  1. Mar 11, 2014 #1
    1. The problem statement, all variables and given/known data

    In the circuit in the following figure, the resistor is adjusted for critical damping. The initial capacitor voltage is 15 V, and the initial inductor current is 6 mA.

    Figure_P08.44.jpg

    Find the numerical value of R.

    Find the numerical values of i immediately after the switch is closed.

    Find the numerical values of di/dt immediately after the switch is closed.

    Find v_C(t) for t≥0.

    2. Relevant equations

    x_C(t) = K_1 e^(s_1*t)+K_2 * t * e^(s_1*t)
    (General solution to 2nd order differential equation)
    damping ratio = [itex]\zeta[/itex] = [itex]\alpha[/itex]/[itex]\omega[/itex]_0
    [itex]\alpha[/itex] = R/(2L), [itex]\omega[/itex]_0 = 1/[itex]\sqrt{LC}[/itex]
    s_1,2 = -[itex]\alpha[/itex] +/- ([itex]\alpha[/itex]2-[itex]\omega[/itex]_02)^(1/2)
    KVL

    3. The attempt at a solution

    Find the numerical value of R.

    Critical damping, so zeta = 1. Using the equation for damping ratio, I solved for R to be 1250 Ω (correct answer).

    Find the numerical values of i immediately after the switch is closed.

    Once the switch is closed, I found i to be 6.00 mA due to the continuity principle i_C(0-) = i_C(0+) (correct answer).

    Find the numerical values of di/dt immediately after the switch is closed.

    Here is where I'm having trouble. I started by using the general solution and evaluating at t=0 to find K_1 = 6 mA.

    s_1 = 0, s_2 = -10000

    So I now have:

    i(t) = 0.006 + K_2 * t * e^(-10000*t)

    EDIT: Just realized it's s_1 in the exponent of both terms, however I'm still not getting the correct answer. Is it possible I have the values for s_1 and s_2 mixed up? Assuming I have them right, the equation should be:

    i(t) = 0.006 + K_2*t

    Obviously I need to solve for K_2 here to be able to differentiate the equation and find di(0)/dt, but I'm not sure how I would go about doing that. I can't plug in t=0 because that would remove K_2 from the equation, and I don't know any other boundary conditions that I could make use of.

    Find v_C(t) for t≥0. **where t is in milliseconds**

    Currently working on this using a similar approach to above.

    EDIT: Not getting this one either. I started with v_C(0) = 15V, which was given in the problem. I used that boundary condition in the homogeneous equation to solve for K1 = 15.

    V_C(t) = 15*e^(0t) + K_2*t*e^(0t)

    Now to solve for K_2, I needed dv_C(0)/dt which I calculated using i_C(0) = C * dv_C(0)/dt and got 18750. Differentiating the homogeneous equation and evaluating at t=0, I got:

    0*K_1+K_2=18750

    Therefore K_2 should = 18750. I'm not sure if I need to convert the units since t is in milliseconds, but the equation I got was:

    15+18750*t, which is obviously incorrect.
     
    Last edited: Mar 11, 2014
  2. jcsd
  3. Mar 11, 2014 #2

    NascentOxygen

    User Avatar

    Staff: Mentor

    Hi zealeth. So the circuit you've shown is not representative of the circuit in use here, because what you show does not indicate how the inductor can have an initial non-zero current?

    Do you happen to know the correct answer for ##di/dt## at ##t=0^+## ?
     
  4. Mar 11, 2014 #3
    The circuit I've shown is what was given with the problem statement. And no, I do not know what di/dt is at t=0+.
     
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