Question

An RLC circuit has a capacitance of 0.29 μF .

Part A: What inductance will produce a resonance frequency of 92 MHz ?

Part B: It is desired that the impedance at resonance be one-fifth the impedance at 20 kHz . What value of R should be used to obtain this result?

Answer 1

Part A.

resonance frequency is produced when

Inductive reactance = Inductive capcitance

XL = Xc

w*L = 1/(w*C)

w = 2*pi*f

2*pi*f*L = 1/(2*pi*f*C)

L = 1/((2*pi*f)^2*C)

Using given values:

L = 1/((2*pi*92*10^6)^2*0.29*10^-6)

L = 1.032*10^-11 H

L = 10.32*10^-12 H = 10.32 pH

Part B.

At resonance impedance is given by:

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

Since XL = Xc, So

Z1 = R, at resonance

Now at 20 kHz, impedance will be

Z2 = sqrt (R^2 + (XL - Xc)^2)

Given that

Z1 = Z2*(1/5)

R = sqrt (R^2 + (XL - Xc)^2)/5

25R^2 = R^2 + (XL - Xc)^2

24R^2 = (XL - Xc)^2

R = |(XL - Xc)|/sqrt 24

R = |(w*L - 1/(w*C))|/sqrt 24

R = |(2*pi*20000*10.32*10^-12 - 1/(2*pi*20000*0.29*10^-6))|/sqrt 24

R = 5.6 Ohm

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