In the circuit shown in the figure, the resistors are each 1.0 , the capacitors

Question

In the circuit shown in the figure, the resistors are each 1.0 , the capacitors are each 220 µF, and the inductors are each 30 mH. (a) What are the equivalent values of resistance, capacitance, and inductance? (b) What is the approximate resonant frequency of the circuit? (c) What is the impedance of the circuit at this frequency? (d) What is the phase difference between the driving emf and the current in the circuit? (e) What is the half-width /d of the resonance? (f) What is the Q-factor of the circuit?

Explanation / Answer

Given that Resistors, R1 = R2 = 1 Capacitors, C1 = C2 = 220 F = 220 * 10^-6 F Inductors, L1 = L2 = 30 mH = 0.03 H a) The resultant resistance, R = R1R2/(R1+R2) = 0.5 The resultant capacitance, C = C1 + C2 = 440 * 10^-6 F The resultant inductance, L = L1 + L2 = 0.06 H b) Resonant frequency, f = 1 / (2LC)                                    = 30.97 Hz c) XL = L = 2fL = 11.7 XC = 1/C = 11.68 Impedence : Z = [R2 + (XL - XC)2]                          = 0.5 d) tan = wL - 1/ wC    Phase difference, = tan^-1 (wL - 1/ wC)                                    = 1.4 deg e) WE know that the formula for the Q factor as        Q = Tan = 0.02 Now we have formula for the half width as       /d = (1/Q) = 50 f) The Q factor is given by           Q = 0.02

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