A circuit is constructed with an AC generator, a resistor, capacitor and inducto

Question

A circuit is constructed with an AC generator, a resistor, capacitor and inductor as shown. The generator voltage varies in time as ? =Va - Vb = ?msin?t, where ?m = 120 V and ? = 270 radians/second. The inductance L = 268 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 ? = 41 degrees and the average power delivered to the circuit by the generator is Pavg= 56 W.

1) What is Imax, 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?

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

Explanation / Answer

a)

Average power

Pav=(1/2)*Vm*Im*cos(o)

56=(1/2)*120*Im*cos41

Im=1.2367 A

b)

impedane

Z=Vmax/Imax =120/1.2367

Z=97.03 ohms

Power factor

PF=cos(o)=R/Z

R=97.03*cos41

R=73.23 ohms

c)

Inductive reactance

XL=WL=270*0.268=72.36 ohms

since

Z=sqrt[R^2+(XL-Xc)^2]

97.03^2=sqrt[73.23^2+(72.36-Xc)^2]

Xc=8.7 ohms

since

Xc=1/WC

=>C=1/W*Xc =1/270*8.7

C=4.257*10^-4 F or 425.7 uF

d)

at resonance

XL=Xc

so impedance

Z=R

Rms Voltage

Vrms=Vmax/sqrt(2) =120/sqrt(2)

Vrms=84.85 Volts

average power

P=Vrms^2/R =84.85^2/73.23

P=98.32 Watts

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