A series RLC circuit with a resistance of 40Ω, a capacitor of 6.7mF, and an inductor of 0.20H is connected to a 127V/60Hz voltage source.

Question

Select the alternative that presents the value of the circuit impedance.

(A) Z = 40Ω - j75.0Ω .

(B) Z = 40Ω + j2.51Ω .

(C) Z = 40Ω + j0.251Ω .

(D) Z = 40Ω + j75.0Ω .

(E) Z = 40Ω + j25.1Ω .

Answer 1

Given:

R = 40 Ω

C = 0.0067 F

L = 0.20 H

V = 127 V

f = 60 Hz

Find:

Z = ? Ω + j ? Ω

Assume:

All components are ideal with 100% efficiency

Solution:

In the equations following, the angular frequency, ω, will be needed and not the linear frequency, f.

ω = 2 * π * f = 2π * 60 Hz ∴ ω ≈ 376.99 rad / s

Resistors are in phase with the series RLC circuit impedance

∴ Z_R = R + j 0 ∴ Z_R = 40 Ω

Capacitors are out of phase with the series RLC circuit impedance by π/2 and are behind the impedance phase.

∴ Z_C = 0 - j ( 1 / ωC ) = - j ( 1 / ( 376.99 rad / s * 0.0067 F ) ) ∴ Z_C ≈ - j 0.396 Ω

Inductors are out of phase with the series RLC circuit impedance by π/2 are are ahead the impedance phase.

∴ Z_L = 0 + j ωL = j ( 376.99 rad / s * 0.20 H ) ∴ Z_L ≈ + j 75.398 Ω

The series RLC circuit impedance total is the linear sum of all the components' impedances.

Z = ZR + ZC + ZL = [ 40 + ( - j 0.396 ) + j 75.398 ] Ω ∴ Z ≈ 40 Ω + j 75.00 Ω

Thus, the correct answer choice is D.

I hope this helps! Message me in the comments if you have questions!