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In a series LCR circuit, obtain an expre...

In a series LCR circuit, obtain an expression for the resonant frequency.

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Resonant frequency of LCR circuit : The current amplitude Io for a series LCR circuit is given by
`I_0=(E_0)/(sqrt(R^2+(omegaL-1/(omegaC))^2))`
Clearly, `I_0` becomes zero both for `omega rarr 0 and omega rarroo` . The value of `I_0` is maximum when
`omegaL-1/(omegaC)=0`
LCR series circuit is said to be in resonance when inductive reactance is equal to capacitive reactance i.e., `omegaL = 1/(omegaC)`
or `omega=1/(sqrt(LC))`
Then, impedance, `Z=sqrt(R^2+(omegaL-1/(omegaC))^2)`
Clearly, the impedance is minimum. The circuit is purely resistive. The current and voltage are in the same phase and the current in the circuit is maximum. This condition of the LCR circuit is called resonance condition. The frequency at which the current ampitude `I_0` attains a peak value is called natural or resonant frequency of the LCR circuit and is denoted by `f_r` .
Thus , `omega_r =2pif_r=1/(sqrt(LC))`
or `f_r =1 /(2pisqrt(LC))`
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