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In a series RCL circuit the dissipated power drops by a factor of 8.0 when the frequency of the generator is changed from the resonant frequency to a nonresonant frequency. The peak voltage is held constant while this change is made. Determine the power factor of the circuit at the nonresonant frequency. Note: The ac current and voltage are rms values and power is an average value unless indicated otherwise.

Respuesta :

Answer:

[tex]cos\phi = \frac{1}{2\sqrt2}[/tex]

Explanation:

Average power is defined as

[tex]P = I_{rms}V_{rms} cos\phi[/tex]

so here we know that

[tex]I_{rms} = \frac{V_{rms}}{z}[/tex]

[tex]P = \frac{V_{rms}^2}{z} cos\phi[/tex]

here we know that

[tex]cos\phi = \frac{R}{z}[/tex]

so we will have

[tex]P_{resonance} = 8 P_{nonresonance}[/tex]

[tex]\frac{V_{rms}^2}{z_1} \times \frac{R}{z_1} = 8 \frac{V_{rms}^2}{z_2} \times \frac{R}{z_2}[/tex]

so we have

[tex]z_1^2 = \frac{z_2^2}{8}[/tex]

[tex]z_1 = \frac{z_2}{2\sqrt2}[/tex]

at resonance power factor is given as

[tex]\frac{R}{z_1} = 1[/tex]

so when it is at non resonance condition then we have

[tex]cos\phi = \frac{R}{z_2}[/tex]

[tex]cos\phi = \frac{z_1}{2\sqrt2 z_1}[/tex]

[tex]cos\phi = \frac{1}{2\sqrt2}[/tex]

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