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RLC circuit problem

  1. Mar 28, 2014 #1
    1. The problem statement, all variables and given/known data
    A series LCR circuit with ##L=0.125/\pi## H, ##C=500/\pi## nF and ##R=23\,\Omega## is connected to a 230 V variable frequency supply. For what reactance of circuit, the power transferred to the circuit is half the power at resonance?


    2. Relevant equations



    3. The attempt at a solution
    At resonance,
    $$f=\frac{1}{2\pi\sqrt{LC}}=2000\,Hz$$
    Hence, the power transferred at resonance is given by ##P=V^2_{rms}/R=2300\,\,W##.

    When the power transferred is half, let the reactance be Z, hence,
    $$P'=\frac{V^2_{rms}}{Z}\cos\phi=\frac{V^2_{rms}}{Z}\frac{R}{Z}$$
    As per the question:
    $$\frac{V^2_{rms}}{Z}\frac{R}{Z}=\frac{1}{2}\times 2300$$
    $$\Rightarrow \frac{230\times 230 \times 23}{Z^2}=\frac{1}{2}\times 2300$$
    $$\Rightarrow Z=23\sqrt{2} \,\,\Omega$$
    But this is incorrect. The correct answer is ##23\,\,\Omega##. :confused:

    Any help is appreciated. Thanks!
     
  2. jcsd
  3. Mar 28, 2014 #2

    collinsmark

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    Homework Helper
    Gold Member

    The way you've defined it, Z is the impedance, not the reactance.

    The reactance, X is such that Z = R + jX, and it's the X that you need to solve for.

    Note that the magnitude squared of Z is, Z2 = R2 + X2. That will come in useful. :smile:
     
    Last edited: Mar 28, 2014
  4. Mar 30, 2014 #3
    Ah yes, thanks a lot collinsmark! :smile:
     
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