Question

A series RCL circuit has a resonant frequency of 1200 Hz. When operating at a frequency other than 1200 Hz, the circuit has a capacitive reactance of 7.00 and an inductive reactance of 41.0. What are the values of (a) L and (b) C? Note: The ac current and voltage are rms values and power is an average value unless indicated otherwise.

Answer 1

As we know that the impedence of series RLC circuit is given by -

Z= sqrt[R^2 + (Lw - 1/Cw)^2]

Now at resonance -

w=wo, (wo=2*pi*fo & fo = 1200 Hz (resonant freq.))

Z has to be the least (Z=R) or current the MAXIMUM

Lwo = (1/Cwo) or wo^2 = 1/LC or

LC = 1 / 4*pi^2*fo^2 ---------------------------------------------(i)

When this circuit operates on any other frequency (v), then

Inductive reactance= L.(2*pi*v) = 41--------------------------------(ii)
Capacitive reactance= 1 / (C*2*pi*v) = 7----------------------------(iii)

Now, multiply (ii) and (iii), we get -

L / C = 287---------------------------------(iv)

Multiply (i) and (iv)

L^2 = [1 / 4*pi^2*fo^2] * 287

=> L = sqrt(287)/2*pi*fo
L = 16.9 / (2*3.14*1200) = 0.002243 = 2243 x 10^-6 H = 2243 micro H

And -

C = L /150 = 2243 / 150 = 14.95 micro farad.

(a) L = 2243 H

(b) C = 14.95 F