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Determining Constants in Second Order Circuit Analysis

  1. Nov 20, 2013 #1
    1. The problem statement, all variables and given/known data
    This was an electrical engineering lab, dealing with the steady state response of an RLC circuit (diagram attached). The main part of the lab consisted of experimentally determining the circuit's critical resistance value, and viewing the overdamped and underdamped responses. Now, I've hit the point where I'm determining the theoretical values for what we measured. I'm starting with the underdamped case, and found s1 and s2 using equation 1 (below). My final objective here is to determine i(t), which, I gather, requires that you solve for the constants A1 and A2, which appear in several of the equations below. Also, FYI I have already solved for ωo, ωd and [itex]\alpha[/itex].
    Thanks in advance.


    2. Relevant equations

    (1) s=-[itex]\frac{R}{2L}[/itex][itex]\pm\sqrt{(\frac{R}{2L})^{2}-\frac{1}{LC}}[/itex]

    (2) i(t)=A1e(s1t)+A2e(s2t)


    3. The attempt at a solution

    After looking around for a while, it seems as though the way to solve for these two is by creating a system of equations comprised of v(0) and the derivative of i(0), however every single source I've found has described it differently and my own lab simply says "solve for it" and nothing else on the subject.

    so far I'm fairly sure that one of the equations is V(0)=A11+A2=1 (we used a 1V signal). The other equation varies, and I've had trouble finding one that makes sense. The closest I think I've gotten was:

    [itex]\frac{i_{c}(0)}{C}[/itex] = S1A1+S2A2


    However nowhere is that equation mentioned in my lab and I don't see how it would make sense.

    Thanks again
     

    Attached Files:

  2. jcsd
  3. Nov 21, 2013 #2

    rude man

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    Homework Helper
    Gold Member

    This is the so-called homogeneous solution ( ih, solution to the reduced equation. It's called 'reduced' since we set the equation = 0).

    To comlete the picture you need to add the 'particular solution' due to the voltage input:

    Assume ip = Csin(wt) + Dcos(wt)
    where the subscript "p" on i refers to the 'particular solution'.

    Solve for C and D using the orig. equation & equating coefficients of like terms (2 equations, 2 unknowns C and D), then solve for A and B by using i = ih + ip and your initial conditions on i and di/dt (two more equations, 2 unknowns A and B). This is standard stuff.

    So your 'general solution' is i(t) = ih + ip
    = Aexp(s1t) + Bexp(s2t) + Csin(wt) + Dcos(wt).

    You realize s1 and s2 are complex so I hope you're familiar with the Euler identity for exp(jθ).
    i(t) will of course wind up being a real number.
     
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