Question

I'm checking my answers since it's REALLY important I get this right. Please show how you attained the answer for each. Honestly, part C, the diagram, is what I have the most issue with.

An AC generator with emf amplitude E=220V and operating frequency 440Hz causes oscillations in a series RLC circuit having R=220 ohms, L=150 milliHenrys, and C=24.0 microFarads.

a)Find Inductive reactance XL, capacitive reactance XC, and impedence Z

b)Find the current amplitude I, power factor, average and peak power delivered to the circuit.

c)Draw(with proper orientation) a phasor diagram including phasors for current, and voltage across the resistor, inductor, and capacitor.

d) Find resonant frequency

e) Find average power at this resonant frequency

Answer 1

A)we know that Xl is given by:

Xl = 2 pi f L

Xl = 2 x 3.14 x 440 x 150 x 10^-3 = 414.48 Ohm

Xc = 1/ 2 pi f C

Xc = 1/2 x 3.14 x 440 x 24 x 10^-6 = 15.08 Ohm

We know that,

Z = sqrt (R^2 + (Xl - Xc)^2)

Z = sqrt (220^2 + (414.48 - 15.08)^2) = 455.98 Ohm

Z = 456 Ohm

Hence, Xl = 414.48 Ohm, Xc = 15.08 Ohm , Z = 456 Ohm

b)The current in the circuit will be:

I = V/Z = 220/456 = 0.48 A

phi = tan-1 (Xl-Xc/R) = tan-1 ((414.48 - 15)/220) = 61.15 deg

Power factor =pf = cos(phi) = 0.49

Peak Power = I^2 Z = 0.48^2 x 456 = 105.1 Watts

Hence, I = 0.48 A ; pf = 0.49 ; P = 105.1 W

c)Resonance occurs when

Xl = Xc

f = 1/2 pi sqrt (LC)

f = 1/2 x 3.14 x sqrt (0.150 x 24 x 10^-6 ) = 83.92 Hz

Hence, f = 83.92 Hz

e)I = V/R = 220/220 = 1 A

P = I^2 R = 1 x 220 = 220 W

Hence, P = 220 W