Question

An AC voltage of the form Δv = 95 sin 285t where Δv is in volts and t is in seconds, is applied to a series RLC circuit. If R = 52.0 Ω, C = 27.0 µF, and L = 0.240 H, find the following.

(a) the impedance of the circuit
Ω

(b) the rms current in the circuit
A

(c) the average power delivered to the circuit
W

Answer 1

(a) Impedance of the circuit:

R = 52.0 , C = 27.0 F, L = 0.240 H

Given expression of the AC voltage is -

Δv = 95 sin 285t

From the expression -

ω = 285

So -

Z = sqrt(R^2 + (ωL - 1/ωC)^2) = sqrt(52^2 + (285*0.24 - 1/285*27x10^-6)^2)

= sqrt(2704 + (68.4 - 1/0.007695)^2) = sqrt(2704 + (68.4 - 130.0)^2)

= sqrt(2704 + 3795) = 80.6 (Answer)

(b) RMS current in the circuit:

I rms = Vrms / Z = (Vmax/sqrt(2)) / Z = (95 / sqrt(2)) / 80.6 = 0.833 A (Answer)

(c) Average power delivered to the circuit:

P = vrms*irms*cos(φ) where φ = arctan((ωL - 1/ωC)/R) = arctan((285*0.24 - 1/285*27x10^-6)/52)

= arctan(( 68.4 - 130) / 52))

= arctan(-1.185) = -49.8 deg.

Therefore, average power P = 95/sqrt(2)*0.833*cos(-49.8) = 36.1 W (Answer)