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प्रश्न
A series LCR circuit with L = 0.12 H, C = 480 nF, R = 23 Ω is connected to a 230 V variable frequency supply.
(a) What is the source frequency for which current amplitude is maximum. Obtain this maximum value.
(b) What is the source frequency for which average power absorbed by the circuit is maximum. Obtain the value of this maximum power.
(c) For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these frequencies?
(d) What is the Q-factor of the given circuit?
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उत्तर
Inductance, L = 0.12 H
Capacitance, C = 480 nF = 480 × 10−9 F
Resistance, R = 23 Ω
Supply voltage, V = 230 V
Peak voltage is given as:
V0 = `sqrt2 xx 230` = 325.22 V
(a) Current flowing in the circuit is given by the relation,
I0 = `"V"_0/(sqrt("R"^2 + (ω"L" - 1/(ω"C"))^2`
Where,
I0 = maximum at resonance
At resonance, we have
`ω_"R""L" - 1/(ω_"R""C")` = 0
Where,
ωR = Resonance angular frequency
∴ ωR = `1/sqrt("LC")`
= `1/sqrt(0.12 xx 480 xx 10^-9)`
= 4166.67 rad/s
∴ Resonant frequency, vR = `ω_"R"/(2π) = 4166.67/(2 xx 3.14)` = 663.48 Hz
And, maximum current `("I"_0)_"Max" = "V"_0/"R" = 325.22/23` = 14.14 A
(b) Maximum average power absorbed by the circuit is given as:
`("P"_"av")_"Max" = 1/2 ("I"_0)_"Max"^2 "R"`
= `1/2 xx (14.14)^2 xx 23`
= 2299.3 W
Hence, resonant frequency (vR) is 663.48 Hz.
(c) The power transferred to the circuit is half the power at resonant frequency.
Frequencies at which power transferred is half, = ωR ± Δω
= `2π ("v"_"R" ± Δ"v")`
Where,
Δω = `"R"/(2"L")`
= `23/(2 xx 0.12)`
= 95.83 rad/s
Hence, change in frequency, Δv = `1/(2π)Δω = 95.83/(2π)` = 15.26 Hz
∴ vR + Δv = 663.48 + 15.26 = 678.74 Hz
And, vR − Δv = 663.48 − 15.26 = 648.22 Hz
Hence, at 648.22 Hz and 678.74 Hz frequencies, the power transferred is half.
At these frequencies, current amplitude can be given as:
I' = `1/sqrt2 xx ("I"_0)_"Max"`
= `14.14/sqrt2`
= 10 A
(d) Q-factor of the given circuit can be obtained using the relation, Q = `(ω_"R""L")/"R"`
= `(4166.67 xx 0.12)/23`
= 21.74
Hence, the Q-factor of the given circuit is 21.74.
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