A series LCR circuit with `L= 0.12H, C=480 nF,` and `R=23 Omega` is connected to a `230V` variable frequency supply.
(a) What is the source frequency for which current amplitude is maximum? Find this maximum value.
(b) What is the source frequency for which average power absorbed by the circuit is maximum? Obtain the value of maximum power.
(c ) For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency?
(d) What is the Q-factor of the circuit?

Explanation

Here, `L = 0.12 H, C = 480 nF = 480 xx 10^(-9) F, R = 23 Omega `volt, `E_(0) = sqrt2 E_(v) = sqrt2 xx 230` valt.
`I_(0) = (E_(0))/(sqrtR^(2) + (omega L - 1//omega C)^(2))`
`I_(0)` would be maximum, when `omega_(r) = omega = (1)/(sqrt(LC)) = (1)/(sqrt(0.12 xx 480 xx 10^(-9))) = 4166.7 rads^(-1)`
`I_(0) = (E_(0))/(R ) = (sqrt2 xx 230)/(23) = 14.14` amp.
(b) Average power absorbed by the circuit is maximum when `I = I_(0)` at `omega = omega_(r)`
`P_(av) = (1)/(2) I_(0)^(2) R = (1)/(2) xx (14.14)^(2) xx 23 = 2299.3` watt
(c ) Power transferred to the circuit is half the power at resonance frequency, when
`Delta omega = (R )/(2 L) = (23)/(2 xx 0.12) = 95.83 rad s^(-2)`
`:.` Frequencies at which power transferred is half `= omega_(r) +- Delta omega = 4166.7 +- 95.83`
`=4262.53` and `4090.87 rad s^(-1)`
Current amplitude at these frequencies is `(I_(0))/(sqrt2) = (14.14)/(1.4141) = 10` amp.
(d) `Q = (omega_(r) L)/(R ) = (4166.7 xx 0.12)/(23) = 21.74`