Question

A circuit is constructed with an AC generator, a resistor, capacitor and inductor as shown. The generator voltage varies in time as ? =V_a - V_b = ?_msin?t, where ?_m = 120 V and ? = 221 radians/second. The inductance L = 352 mH. The values for the capacitance C and the resistance R are unkown. What is known is that the current in the circuit leads the voltage across the generator by ? = 58 degrees and the average power delivered to the circuit by the generator is P_avg = 68 W.

1) What is I_max, the amplitude of the current oscillations in the circuit?

2) What is R, the value of the resistance of the circuit?

3) What is C, the value of the capacitance of the circuit?

4) The value of ? is now changed, keeping all other circuit parameters constant, until resonance is reached. How was ? changed?

"? was decreased" or

"? was increased"

5)

What is the average power delivered to the circuit when it is in resonance?

Answer 1

2)

current leads voltage

so Xc>Xl

now

tan (58) = (Xc-Xl)/R

=> Xc -Xl = 1.6*R........... (1)

we know average power

P = (V_rms)^2 * R / |Z|^2 = 7200* R / {R^2 + (Xc-Xl)^2} = 7200*R/ { R^2+2.56R^2}

solving we get

2022.47191/R = 68

R = 29.74223 ohms

now

Xl= wL = 221*352*10^-3 = 77.792 ohms

now from equation 1 we get

Xc = 1/wc = 1.6R+ Xl = 125.3796

=> C = 1/(221*125.3796) = 36.0895 uF (micro farad)

1) I_max = V/|Z| = 120 / sqrt{R^2 + (Xc-Xl)^2} = 120/ (1.8868*29.74223) = 2.13836 A

4) resonance

w= 1/sqrt(LC) = 280.5680

w is increased

5) in resonance avg power delivered = 1/2*V^2/R = 242.0800 watts