Question

A series RLC circuit has a 242 Hz resonant frequency, and is driven by an AC voltage source at 315 Hz. The circuit has a 75.0 $\Omega$ resistor and a 66.0 mH inductor. The maximum current is 1.20 A.

a) Find the capacitance in the circuit.

b) Find the maximum voltage of the source.

c) Find the average power of the circuit

d) Find the phase angle $\phi$ .

Answer 1

a)

resonant frequency, fo = 242 Hz

AC frequency, f = 315 Hz

resistance, R = 75 ohm

inductance, L = 66 mH

max current , Io = 1.2 A

a)

We know , resonant frequency, fo = 1/sqrt(LC)

So, C = 1/(fo^2*L) = 1/(242^2*66*10^-3)

= 2.59*10^-4 F = 259 uF <-----------answer

b )

maximum voltage = maximum current*R = Io*R = 1.2*75 = 90 V

c)

Z = sqrt(R2+(2*pi*f*L-1/(2*pi*f*C))^2) = sqrt(75^2+(1*3.14*315*0.066-1/(2*3.14*315*2.59*10^-4))^2)

= 98.2 ohm

So, power factor = cos $\phi$ = R/Z = 75/98.2 = 0.764

Imax = 1.2 A

Vmax = 90 V

average power, Pavg = Imax*Vmax*cos $\phi$ / 2 = 1.2*90*0.764/2 = 41.3 W <-------answer

d)

phase angle = acos(0.764) = 40.2 degrees <-------------answer

Answer 2