Hello please I need soluions for this quetion with clear steps. thanks A circuit

Question

Hello please I need soluions for this quetion with clear steps. thanks

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 ? = 835 radians/second. The values for the remaining circuit components are: R = 90 ?, L = 41 mH, and C = 14?F

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

2)What is Imax, the magnitude of the maximum value of the current in the circuit?

3)What is ?, the phase angle between the generator voltage and the current in this circuit. The phase ? is defined to be positive if the generator voltage the current leads, and negative otherwise.

4)What is t1, the first time after t = 0 when the voltage across the inductor is zero?

5)What is VC = Vd - Va, the voltage across the capacitor, at time t = 0? Note that VC is a signed number

Explanation / Answer

a)

Inductive reactance

XL=WL=835*(41*10-3)=34.235 ohms

Capacitive reactacne

Xc=1/WC =1/835*(14*10-6)=85.543 ohms

Impedance

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

Z=sqrt[902+(34.235-85.543)2]

Z=103.6 ohms

b)

Imax=Emax/Z =120/103.6

Imax=1.158 A

c)

Phase angle

o=tan^-1(XL-Xc/R) =tan^-1(34.235-85.543

o=-29.7 degrees

d)

t1=90-29.7/835

t=0.07223 s or 72.23 ms

e)

Vc=I*Xc=1.158*85.543

Vc=-99 Volts

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