A series circuit contains a 3.00-H inductor, a 3.00-F capacitor, and a 30.0- W r

Question

A series circuit contains a 3.00-H inductor, a 3.00-F capacitor, and a 30.0-W resistor connected to a 120-V (rms) source of variable frequency. Find the power delivered to the circuit when the frequency of the source is (a) the resonance frequency, (b) one-half the resonance frequency, (c) one-fourth the resonance frequency, (d) two times the resonance frequency, and (e) four times the resonance frequency. From your calculations, can you draw a conclusion about the frequency at which the maximum power is delivered to the circuit?

Explanation / Answer

a) At resonant frequency,

XL = Xc

and impedance Z = sqrt[ R^2 + (XL - Xc)^2 ]

and XL - Xc = 0

Z = R

R = 120^2 / 30 = 480 ohm

Pavg = Vrms^2 R / Z^2   = Vrms^2 / R = 120^2 / 480 = 30 W

b) resonant frequency, f = 1 / 2pisqrt(LC) = (1 / sqrt(3 x 3 x 10^-6)) / 2pi

f0 = 53.05 Hz

f = f0/2 = 26.53 Hz

XL = 2 pi f L = 500 ohm

Xc = 1 / (2 pi f C) = 2000 ohm

Z = sqrt[ 480^2 + (500 -2000)^2] = 1574.92

P = Vrms^2 R / Z^2 = 2.8 W

c)

f = f0/4 = 13.27 Hz

XL = 2 pi f L = 250 ohm

Xc = 1 / (2 pi f C) = 4000 ohm

Z = sqrt[ 480^2 + (250 - 4000)^2] = 3780.6 ohm

P = Vrms^2 R / Z^2 = 0.5 W

d)

f = 2f0 = 106.12 Hz

XL = 2 pi f L = 2000 ohm

Xc = 1 / (2 pi f C) = 500 ohm

Z = sqrt[ 480^2 + (500 -2000)^2] = 1574.92

P = Vrms^2 R / Z^2 = 2.8 W

e)

f = 4f0 = 212.32 Hz

XL = 2 pi f L = 4000 ohm

Xc = 1 / (2 pi f C) =250 ohm

Z = sqrt[ 480^2 + (250 - 4000)^2] = 3780.6 ohm

P = Vrms^2 R / Z^2 = 0.5 W

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