A series alternating current circuit comprises a resistor with resistance 30\ \Omega, an i… — JAMB UTME

A series alternating current circuit comprises a resistor with resistance $30\ \Omega$ , an inductor with inductive reactance $80\ \Omega$ , and a capacitor with capacitive reactance $40\ \Omega$ . What is the total impedance of the circuit?

$50\ \Omega$ Answer
B
$150\ \Omega$
C
$70\ \Omega$
D
$40\ \Omega$

Answer

The impedance of the circuit is

50\ \Omega

The impedance

Z

of a series RLC circuit is calculated using the formula

Z = \sqrt{R^2 + (X_L - X_C)^2}

. Substituting the given values

R = 30\ \Omega

X_L = 80\ \Omega

, and

X_C = 40\ \Omega

yields

Z = \sqrt{30^2 + (80 - 40)^2} = \sqrt{900 + 1600} = 50\ \Omega

Step-by-Step Solution

Calculate the net reactance (

X_{net}

)

X_{net} = X_L - X_C = 80\ \Omega - 40\ \Omega = 40\ \Omega

Inductive and capacitive reactances are

180^\circ

out of phase in a series AC circuit.

Apply the series impedance formula

Z = \sqrt{R^2 + X_{net}^2} = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50\ \Omega

Resistance and net reactance act at

90^\circ

phase to each other, requiring the Pythagorean relation.

Key Concept

Impedance of a Series RLC Circuit

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