A circuit is constructed with an AC generator a resistor cap

A circuit is constructed with an AC generator, a resistor, capacitor and inductor as shown. The generator voltage varies in time as =V_a - V_b = _msint, where _m = 120 V and = 633 radians/second. The values for the remaining circuit components are: R = 76 , L = 126.6 mH, and C = 14F.

1)

What is Z, the impedance of the circuit?

2)

What is I_max, the magnitude of the maximum value of the current in the circuit?

A

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.

degrees

4)

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

seconds

5)

What is V_C = V_d - V_a, the voltage across the capacitor, at time t = 0? Note that V_C is a signed number.

V

Solution

Z = sqrt [R² + (X_L -X_C)²]

X_L = L w = 126.6 * 10^-3 * 633 = 80.14 ohm

X_C = 1 / C w = 1 / [14 * 10^-6 * 633] = 112.84 ohm

Z = sqrt [76² + (80.14 -112.84)²]

= 82.74 ohm

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2) I_max = E_max / Z = 120 / 82.74

= 1.45 A

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3) cos phi = R/Z

phi = cos^-1(R/Z) = cos^-1(76/82.74)

= 23.22 deg

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4) t = distance / speed

distance = pi * 66.78 / 180 = 1.166

t = 1.166 / 633 = 0.00184 s