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RLC circuit problem

  • Thread starter Saitama
  • Start date
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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!
 

collinsmark

Homework Helper
Gold Member
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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:
3,810
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The way you've defined it, Z is the impedance, not the reactance.
Ah yes, thanks a lot collinsmark! :smile:
 

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