A 150\text{ V} (RMS) AC generator is connected across a series combination of a 12\ \Omega… — JAMB UTME

A $150\text{ V}$ (RMS) AC generator is connected across a series combination of a $12\ \Omega$ resistor, an inductor with an inductive reactance of $20\ \Omega$ , and a capacitor with a capacitive reactance of $11\ \Omega$ . What is the average electrical power consumed by this circuit?

A
$900\text{ W}$
$1200\text{ W}$ Answer
C
$1500\text{ W}$
D
$1875\text{ W}$

Answer

The average electrical power consumed by the circuit is

1200\text{ W}

To find the average real power dissipated in an AC series circuit, first determine the net reactance

X = X_L - X_C = 20\ \Omega - 11\ \Omega = 9\ \Omega

. Next, calculate total impedance using

Z = \sqrt{R^2 + X^2} = \sqrt{12^2 + 9^2} = 15\ \Omega

. The RMS current in the circuit is

I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{150\text{ V}}{15\ \Omega} = 10\text{ A}

. Since energy is dissipated only by resistance, the average real power is

P = I_{\text{rms}}^2 R = (10)^2 \times 12 = 1200\text{ W}

Step-by-Step Solution

Calculate the net reactance of the circuit.

X = X_L - X_C = 20\ \Omega - 11\ \Omega = 9\ \Omega

Inductive and capacitive reactances oppose each other in phase by

180^\circ

Calculate the total impedance of the series RLC circuit.

Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15\ \Omega

Resistance and net reactance add vectorially at a right angle.

Determine the RMS current flowing through the circuit.

I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{150\text{ V}}{15\ \Omega} = 10\text{ A}

Ohm's law for AC circuits states that current is the supply RMS voltage divided by total impedance.

Calculate the average real power dissipated by the circuit.

P_{\text{avg}} = I_{\text{rms}}^2 R = (10\text{ A})^2 \times 12\ \Omega = 1200\text{ W}

In an AC circuit, average power is only dissipated by resistive components, as ideal inductors and capacitors consume zero net real power over a complete cycle.

Key Concept

Power Dissipation in AC Series Circuits

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