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Prove this! Total Voltage at Resonance

  1. Jul 24, 2011 #1
    Prove that at the resonance of a series RLC circuit, instantaneous velocity across the capacitor is equal to negative instantaneous velocity across the inductor, and thus, the total voltage across C and L is zero.

    Thanks!
     
  2. jcsd
  3. Jul 24, 2011 #2

    gneill

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    Please follow the posting template. You must show some work before help can be given (we can't just do your homework -- we can only assist you with your work).
     
  4. Jul 24, 2011 #3
    I know that impedance across the inductor is equal to the angular frequency times the inductance, and that the impedance across the conductor is equal to the reciprocal of the angular frequency times the capacitance.

    I have the equation: voltage across inductor (instantaneous) = amplitude of voltage across inductor times cos(angular freq x time + 90). There is an equivalent for voltage across conductor (instantaneous).

    I realize that instantaneous voltage across capacitor is equal to the negative instantaneous voltage across the inductor. Where I get lost is, how do I know what angular freq x time is to determine that the total voltage across C and L is zero?

    Thanks
     
  5. Jul 24, 2011 #4

    gneill

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    The condition given in the problem statement is "...at the resonance of a series RLC circuit,...". So you're interested in the case where the circuit is in resonance. That means that the frequency is the resonant frequency. What's the resonant frequency for a series RLC circuit?
     
  6. Jul 24, 2011 #5
    angular frequency/2(pi)
     
  7. Jul 24, 2011 #6
    or 1/(square root (LC) x 2pi)
     
  8. Jul 24, 2011 #7

    gneill

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    So, at the resonant frequency what are the impedances of the L and C components?
     
  9. Jul 24, 2011 #8
    Impedance of L = angular frequency times inductance = 2 x pi x f x L

    Impedance of C = 1/(angular frequency x capacitance) = 1/( 2 x pi x f x C)
     
    Last edited: Jul 24, 2011
  10. Jul 24, 2011 #9

    gneill

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    Plug in your angular frequency.
     
  11. Jul 24, 2011 #10
    Impedance of L = angular frequency times inductance = 2 x pi x f x L

    Impedance of C = 1/(angular frequency x capacitance) = 1/( 2 x pi x f x C)
     
  12. Jul 24, 2011 #11

    gneill

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    Yes, but you've just calculated the resonant frequency... so plug it in!
     
  13. Jul 24, 2011 #12
    :)

    Impedance of L = 2 x pi x (angular frequency/2pi) x L

    Impedance of C = 2 x pi / (2 x pi x angular frequency x C)
     
  14. Jul 24, 2011 #13

    gneill

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    Seriously, fromthepast? Going in circles isn't helping.

    You need to GET RID OF the variables f and/or "angular frequency" from your impedance expressions by plugging in the expression you found for the resonant frequency in terms of L and C. Simplify and compare the impedances.
     
  15. Jul 24, 2011 #14
    Doing that doesn't relate the instantaneous velocity of the inductor and the instantaneous velocity of the capacitor.

    vC = -vL
     
  16. Jul 24, 2011 #15

    gneill

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    What is the magnitude of the voltage across equal magnitude impedances in a series circuit? Remember, in a series circuit all the components have identical current at all times.

    It only remains to show that the voltages are phase shifted +90° and -90°, for a total of 180°.

    (The problem would be much simpler if you used the complex impedances).
     
  17. Jul 24, 2011 #16
    So how does saying that the phase angle is 180 connect vC = - vL to having total voltage across C and L equal zero?
     
  18. Jul 24, 2011 #17

    gneill

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    Because they are 180 degrees out of phase and equal. Draw a sine curve. Draw a sine curve that 180° shifted. Note that their values at any given time always sum to zero.
    sin(x + pi) = -sin(x).
     
  19. Jul 24, 2011 #18
    That still doesn't reason how when instantaneous voltage across a capacitor =
    negative instantaneous voltage across an inductor, total voltage across C and L is zero.

    ...
     
  20. Jul 24, 2011 #19

    gneill

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    They are in series, so the potential differences add.
     
  21. Jul 24, 2011 #20
    Yes, but what makes them add to zero?
     
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