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 ? = 747 radians/second. The values for the remaining circuit components are: R = 64 ?, L = 80.8 mH, and C = 14.3?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

1) Given that : e = Va - Vb = Emsinwt = 120sin(747t)
And, In series : R = 64 , L = 80.8 mH, and C = 14.3 F.
So,
XL = wL = 747*80.8*10^-3 = 60.4 ,
Xc = 1/wC = 1/747*14.3*10^-6 = 93.6 ,
Thus,
1.) Impedance, Z = 64 + j(60.4 - 93.6) = 64 - j33.2
AND
|Z| = SQRT[64^2+33.2^2] = 72.1

2) Imax = Em/|Z| = 120/72.1 = 1.66 A

3) z = 64 - j33.2
So, = arctan(-33.2/64) = -27.4 degrees Lagging

4) time = t = Distance/w = (pi*62.6/180)/747 = 0.001463 sec

5) Vc = Imax*Xc = 1.66*93.6 = 155.4 V

Vc = 155.4*sin(wt +- 62.6) (wt = 0)

Vc = 155.4*sin(+-62.6) = +-137.97

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