Question

A television channel is assigned the frequency range from 54 MHz to 60 MHz. A series RLCtuning circuit in a TV receiver resonates in the middle of this frequency range. The circuit uses a 14 pF capacitor.

Part B:

In order to function properly, the current throughout the frequency range must be at least 50% of the current at the resonance frequency. What is the minimum possible value of the circuit's resistance?

Express your answer to two significant figures and include the appropriate units.

Answer 1

Given that

A television channel is assigned the frequency range from 54 MHz to 60 MHz.

A series RLCtuning circuit in a TV receiver resonates in the middle of this frequency range (f) =54+60/2 =57*10⁶Hz

The circuit uses a capacitor(C) =14 pF =14*10^-12F

a)

The angular frequency is given by w =2pif =2*3.14*57*10⁶Hz =357.96*10⁶rad/s

The resonating freqency is given by

w =Sqrt(1/LC)====>L =1/w²C =1/(357.96*10⁶rad/s)²(14*10^-12F) =5.574*10-7H=0.5574uH

b)

Given that

In order to function properly, the current throughout the frequency range must be at least 50% of the current at the resonance frequency.

Then I2/I1 =Z1/Z2 =(1/2)

Z2 =2Z1

At the resonance freqency , impedance =R

Z₂ =2*Z₁

Then Z₂² =4*R²

R²+(XL-XC)² =4R²

(XL-XC)² =3R²

We know that

XL =wL =(357.96*10⁶rad/s)(0.5574*10^-6) =199.526ohm

XC =1/wC =1/((357.96*10⁶rad/s)(14*10^-12) =199.543ohm

(199.526-199.543)² =3R2

2.89*10^-4 =3R²

The minimum possible value of the circuit is R =9.814*10^-3ohm =9.814mH