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 ? = 647 radians/second. The values for the remaining circuit components are: R = 79 ?, C = 452 ?F. The value for L is unknown. What is known is that the voltage across the generator leads the current in the circuit by ? = 37 degrees.

1)What is Z, the impedance of the circuit?

2)What is L, the value of the inductor?

3)What is VL,max, the magnitude of the maximum voltage across the inductor?

4)What is VL = Vb - Vc, the voltage across the inductor, at time t = 0? Note that VL is a signed number.

Please help.

Explanation / Answer

a)

Power Factor

PF=cos(o)=R/Z

cos37=79/Z

Z=98.92 ohms

b)

Xc=1/WC =1/647*(452*10-6)

Xc=3.42 ohm

Impedance

Z=sqrt[R2+(XL-Xc)2]

98.92=sqrt[792+(XL-3.422]

XL-3.42=59.53 ohms

XL=62.95 ohms

since

XL=WL

=>L=XL/W=62.96/647

L=0.0973 H or 97.3 mH

c)

Imax=Emax/Z =120/98.92 =1.21 A

VL,max=Imax*XL=1.21*62.95 =76.36 volts

d)

VL=76.36 Volts

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