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Trouble deriving series Q

  1. Apr 17, 2013 #1
    I have a derivation from a book that says

    Q = 2π Es / Ed

    Where Es is the energy stored in the resonant components. Dividing both by the period at resonance gives...

    Q = ωo Es / Pd

    This is where I'm stuck. The book says Es = 1/2 LI^2 at the instant that all of the energy is being stored in the inductor. Then it goes on to say the power dissipated in the series resistance, Pd is equal to 1/2 I^2 R. Why is it the average power?


    Of course these both simplify down to ωoL/R.

    And, how do you go from 1/2 LI^2 as the energy of the inductor to ωLI^2 as the power? The power is 4πf times the energy?

    I think I posted this in the wrong section, sorry.
     
    Last edited: Apr 17, 2013
  2. jcsd
  3. Apr 17, 2013 #2

    AlephZero

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    The power dissipated in the resistor at any instant in time is ##I^2R##.

    The circuit is oscillating so ##I^2 = I_{\text{max}}^2 \sin^2 \omega t##.

    When you take the average power by integrating over one cycle of the oscillation, you get ##I_{\text{max}}^2 R/2##.

    For the second question, average power = energy / time. The energy in the inductor changes between ##0## and ##LI_{\text{max}}^2/2## every half cycle, or in time ##\pi / \omega## seconds.
     
  4. Apr 17, 2013 #3
    Thank you very much, that helps a lot. Too bad my teacher could not explain this to me.
     
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