Consider an RLC series circuit (30 ohm, 250 mH, 20 microF) driven at 115 Vrms, 60 Hz.
a. What is the current in this circuit?
b. What frequency would bring this circuit into resonance?
c. Alternatively, what capacitance added in parallel to the existing capacitor would bring this circuit into resoance at 60 Hz?
d. What is the current at resonance?
The linear frequency is 60 Hz, so the angular frequency is 377 rad/sec.
The inductance is 250 mH, so the inductive reactance is 94.3 ohm.
The capacitance is 20 microF, so the capacitive reactance is 133 ohm.
Thus the net reactance is -38,3 ohm; the circuit is capacitive.
Following from the net reactance and the resistance of 30 ohm, the magnitude of the impedance is 48.7 ohm.
a. The rms current in the circuit is 115 V / 48.7 ohm = 2.36 A.
b. The circuit resonates at an angular frequency which is the reciprocal of the square root of the product of the inductance and the capacitance. This value is 447 rad/sec; the corresponding linear frequency is 71 Hz.
c. A capacitance such that the capacitive reactance equals the inductive reactance would bring the circuit into resonance. A total capacitance of 28 microF would have the desired reactance of 94 ohm. Thus an additional capacitance of 8 microF ould be required in parallel.
d. In resonance, the impedance is just the resistance (since XL = XC). Thus the rms current at resonance is 115 V / 30 ohm = 3.8 A.