Give the expression for the impedance of an LCR circuit and obtain the expression for the resonant frequency. What is the power factor of the resonant circuit?

Explanation

Impedance of a series LCR circuit is given by
`Z=sqrt(R^(2)+(X_(L)-X_(C))^(2))=sqrt(R^(2)+(omegaL-(1)/(omegaC))^(2))`
Where, `omega=2pif`, f being frequency of ac.
i.e., `omega_(0)^(2)LC=1`
`therefore omega_(0)=(1)/(sqrt(LC))or2pif_(0)=(1)/(sqrt(LC))`
`therefore f_(0)=(1)/(2pisqrt(LC))`, where `f_(0)` is known as the resonant frequency.
Power factor = cos R/Z and Z = R at resonant frequency.
Power factor = cos R/R = 1.
Voltage is in phase with current.