Question

In series.Lcr circuit.L =10mh c=10 micro f r=20 ohm is.Connected to ac source of v=200sinwt.Find the.Power at half.Power frequency

Answer 1

The frequencies which the power transferred to the circuit is half the power at the resonant frequency are 2162.27 rad/s and 4162.27 rad/s.

Explanation:

Given that,

Inductance = 10 mh

Capacitance = 10 μf

Resistance = 20 ohm

RMS Voltage = 200 volt

Suppose two angular frequencies for which the power transferred to the circuit is half the power at the resonant frequency

We need to calculate the resonant frequency

Using formula of resonant frequency

$\omega_{r}=\dfrac{1}{\sqrt{LC}}$

Put the value into the formula

$\omega_{r}=\dfrac{1}{\sqrt{10\times10^{-3}\times10\times10^{-6}}}$

$\omega_{r}=3162.27 rad/s$

We need to calculate the peak voltage

Using formula of peak voltage

$E_{0}=\sqrt{2}E_{v}$

$E_{0}=\sqrt{2}\times200$

We need to calculate the current

Using formula of current

$I_{0}=\dfrac{E_{0}}{R}$

Put the value into the formula

$I_{0}=\dfrac{\sqrt{2}\times200}{20}$

$I_{0}=14.14\ A$

We need to calculate the resonant frequency

Using formula of angular frequency

$\omega=\omega_{r}\pm\Delta \omega$

Where, $\Delta\omega=\dfrac{R}{2L}$

Put the value into the formula

$\Delta \omega=\dfrac{20}{2\times10\times10^{-3}}$

$\Delta\omega=1000\ rad/s$

We need to calculate the angular frequency at which power transferred is half at the resonant frequency

Using formula of angular frequency

$\omega=\omega_{r}\pm\Delta \omega$

Put the value into the formula

$\omega=3162.27\pm1000$

$\omega=2162.27 \rad/s$

$\omega=4162.27 rad/s$

Hence, The frequencies which the power transferred to the circuit is half the power at the resonant frequency are 2162.27 rad/s and 4162.27 rad/s.

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Topic : frequency

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