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Question
A 25 μF capacitor, a 0.10 H inductor, and a 25Ω resistor are connected in series with an AC source whose emf is given by e = 310 sin 314 t (volt). What is the frequency, reactance, impedance, current, and phase angle of the circuit?
Solution
Data: C = 25 µF = 25 x 10-6 F, L = 0.10 H, R = 25Ω, e = 310 sin (314 t)[volt]
Comparing e = 310 sin (314 t) with
e = e0 sin (2πft), we get,
the frequency of the alternating emf as
f = `314/(2pi) = 314/(2(3.14))` = 50 Hz
Reactance = `|omega"L" - 1/(omega"C")| = |2pi"fL" - 1/(2pi"fC")|`
`= |2(3.14)(50)(0.10) - 1/(2(3.14)(50)(25 xx 10^-6))|`
`= |31.4 - (100 xx 10^2)/((3.14)(25))|` = |31.4 - 127.4|
= 96 Ω
`"Z"^2 = "R"^2 + (omega"L" - 1/(omega"C"))^2 = (25)^2 + (96)^2 = 9841 Omega^2`
∴ Impedance, Z = `sqrt9841`Ω = 99.2 Ω
Peak current, `"i"_0 = "e"_0/"Z" = 310/99.2`A
∴ `"i"_"rms" = "i"_0/sqrt2 = 310/((1.414)(99.2))`A = 2.21 A
cos Φ = `"R"/"Z" = 25/99.2 = 0.2520`
∴ Phase angle,
Φ = cos-1(0.2520) = 75.40° = 1.316 rad
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