An AC generator supplies an rms voltage of 115 V at 60.0 Hz. It is connected in
ID: 1260978 • Letter: A
Question
An AC generator supplies an rms voltage of 115 V at 60.0 Hz. It is connected in series with a 0.700 H inductor, a 3.20 ?F capacitor and a 311 ? resistor.
What is the impedance of the circuit?
What is the rms current through the resistor?
What is the average power dissipated in the circuit?
What is the peak current through the resistor?
What is the peak voltage across the inductor?
What is the peak voltage across the capacitor?
The generator frequency is now changed so that the circuit is in resonance. What is that new (resonance) frequency?
Explanation / Answer
a) the impedance of each element is:
R = 291 ohms
Xl = 2 * pi * freq * inductance = j157.0796 ohms
Xc = -1/ 2 * pi * freq * capacitance = -j884.1941 ohms
Since resistances add in series the total resistance is 291 - j727.1145 which is a phasor of 783.1836 ohms at -68.1881 degrees.
b) the current thru the resistor is the 120V RMS voltage divided by 291 ohms = 0.4124 amps or 412.4 milliamps
c) the average power dissipated in the circuit is P = I^2 * Rtotal = (0.4124)^2 * 783.1836 = 133.1990 watts
d) the peak current thru the resistor is the peak voltage thru the resistor, where Vrms / 0.7071 = Vpeak, so
120 / 0.7071 = 169.7056 volts = Vpeak. The peak current is 169.7056 volts / 291 ohms = 0.5832 amps or 583.2 milliamps. The peak voltage across the resistor is the peak voltage divided by the reactance
169.7056 volts * (291 / 291 - j727.1145) = 169.7056 volts * (0.1381 +j0.3450) volts
= 169.7056 volts * (0.3761 ohms at 68.1881 degrees) = 63.0560 volts
e) the peak voltage across the inductor is the peak voltage divided by the reactance
169.7056 volts * (j157.0796 ohms / 291 - j727.1145) = 169.7056 volts * (-0.1862 +j0.0745) volts
= 169.7056 volts * (0.2006 ohms at 158.1881 degrees) = 34.0371 volts
f) the peak voltage across the capacitor is the peak voltage divided by the reactance
169.7056 volts * (-j884.1941 ohms / 291 - j727.1145) = 169.7056 volts * (1.0481 -j0.4195) volts
= 169.7056 volts * (1.1290 ohms at -21.8119 degrees) = 177.8769 volts
g) the resonance is 1/ 2* pi* sqrt (L*C) = 118.6271 Hertz
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