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 ? = 493 radians/second. The inductance L = 242 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 ? = 36 degrees and the average power delivered to the circuit by the generator is Pavg = 122 W.

1)

What is Imax, the amplitude of the current oscillations in the circuit?

A

2)

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

?

3)

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

?F

4)

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

? was decreased

? was increased

5)

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

W

Explanation / Answer

1.

Average Power delivered

Pav =(1/2)*Emax*Imax*cos(o)

122=(1/2)*120*Imax*cos36

Imax=2.51 A

2.

Impedacne

Z=Emax/Imax =120/2.51

Z=47.75 ohms

Power Factor

Cos(o)=R/Z

=>R=Z*cos(o) =47.75*cos36

R=38.63 ohms

3.

Inductive reactacne

XL=WL =493*(242*10^-3)

XL=119.31 ohms

since Impedacne

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

47.75=sqrt[38.63^2+(119.31-Xc)^2]

787.8=(119.31-Xc)^2

119.31-Xc=28.07

Xc=91.24 ohms

Since Capacitive reactance

Xc=1/WC

=>C=1/WXc =1/493*91.24

C=2.223*10^-5 F or 22.23 uF

4.

W is decreased

5.

at resonance

XL=Xc

so impedance

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

Z=R=38.63 ohms

Rms Voltage

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

Vrms=84.85 Volts

Average power delivered

Pav=Vrms^2/R =84.85^2/38.63

Pav=186.4 Watts

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