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If R^2= L/C, show the total power is independent of frequency

  1. Nov 27, 2016 #1
    1. The problem statement, all variables and given/known data
    You have a circuit supplied with AC voltage, it has 2 parallel branches, 1 with R and L, the other with R and C.

    a) If (frequency) w^2 = 1/LC, prove that the power in each branch is equal.
    b) If R^2 = L/C, show that the power is the circuit is independent of frequency.

    2. Relevant equations

    3. The attempt at a solution
    I am pretty sure I got a) wL=1/wC, so XL=XC, then I know currents are equal in each branch, and it's a parallel circuit so voltage is equal, therefore power is equal.

    But, I don't even know where to begin for b). I can see that R^2 = w/w * L * 1/C (and that is independent of w) but I really think there is more to it than that, some type of proof that R^2 = XL/XC??

    Please help!
     
  2. jcsd
  3. Nov 28, 2016 #2

    cnh1995

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    R2 is not XL/Xc. Find the correct relation among R, XL and Xc from R2=L/C. Find the equivalent impedance of the circuit and use the obtained relation in it.
     
  4. Nov 28, 2016 #3
    Ok, so I initially tried to work the result backwards. R^2 = L/C is the same as R^2 = L*1/C, this formula is independent of frequency, so I added frequency back in.
    w/w*L*1/C. Which gives wL*1/wC. and then wL = XL and 1/wC = XC. Therefore R^2 = XL/XC. The only way I could get ohms = ohms. I'm not sure how Henry/Farad = ohms^2.

    I understand that inductance of an inductor is determined by the details of its construction and is independent of the frequency of the circuit, same principle applies for the capacitor. But, finding the equivalent impedance of the circuit requires frequency.....still lost.....I don't know the correct relation among R, XL and Xc from R2=L/C.
     
  5. Nov 28, 2016 #4

    cnh1995

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    No. Your RHS is unitless while the LHS has the unit ohm2.
    Correct. So R2=??
    Also, are you familiar with the complex form of impedance?
     
  6. Nov 28, 2016 #5
    Response part 2 - I'm still drawing a blank for R^2. I don't think I know the complex form of impedance. I know that in a parallel circuit Z^-1 = 1/Z+1/Z+1/Z+..........but trying to do that in terms of R, C, and L got really messy and the result didn't give me any "ah-ha" moments.
     
  7. Dec 4, 2016 #6

    rude man

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    Gold Member

    So Z = jwL for the impedance of an inductor is not something you've seen before?
    If not you will have to deal with the trigonometric relations between applied voltage v and current i for inductors and capacitors:

    Let v = v0 sin(wt) across an inductor L or a capacitor C, then
    v = L di/dt so iL = v0/wL sin(wt - pi/2)
    and iC = C dv/dt = wC sin(wt + pi/2)

    Then show that the sum of the squares of the magnitudes of the branch currents |iL|2 + |iC|2 add up to a constant independent of w.
    Then wish you had covered complex electrical parameters! :smile:
     
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