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

Question

A television channel is assigned the frequency range from 54 MHz to 60 MHz A series RLC tuning circuit in a TV receiver resonates in the middle of this frequency range. The circuit uses a 17 pF capacitor. What is the value of the inductor? Express your answer to two significant figures and include the appropriate units. 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.

Explanation / Answer

Voltage applied be V
Current at resonant frequency = V/R

Current must be atleast 50%.
Current will be least at corner frequency
At corner frequancy,
Current = V/sqrt (R^2 + (2*pi*f*L - 1/(2*pi*f*C ) )^2)
= V/sqrt (R^2 + (2*pi*54*10^6*0.46*10^-6 - 1/(2*pi*54*10^-6*17*10^-12 ) )^2)
= V/sqrt (R^2 + (156.07 - 173.37 )^2)
= V/sqrt (R^2 + 299.93)

use:
0.5*Current at resonant frequency = other current
0.5* V/R = V/sqrt (R^2 + 299.93)
0.5/R = 1/sqrt (R^2 + 299.93)
0.5*sqrt (R^2 + 299.93) = R
R^2 + 299.93= 4R^2
R = 10 ohm
Answer: 10 ohm

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