Question

A television channel is assigned the frequency range from 54 MHz to 60MHz. A series RLC tuning circuit in a TV receiver resonates in the middle of this frequency range. The circuit uses a 19 pF capacitor.

A. What is the value of the inductor?
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?

Answer 1

To find the value of the inductor, calculate the resonant frequency of the circuit and use the formula for resonant frequency. The value of the inductor is 13.5 μH. To find the minimum possible value of the circuit's resistance, multiply the current at resonance frequency by 0.5.

Step-by-step explanation:

To find the value of the inductor in the RLC tuning circuit, we need to calculate the resonant frequency of the circuit and then use the formula for resonant frequency:fr = 1 / (2π√(LC))

Since the resonant frequency is in the middle of the frequency range, which is 54 to 60 MHz, the resonant frequency would be (54 + 60) / 2 = 57 MHz = 57*10^6 Hz.

Using the formula mentioned earlier, we can calculate the value of the inductor:L = (1 / (4π²f²C))

By substituting the values, we get L = 1.35 x 10^-5 H or 13.5 μH (microhenries).

To find the minimum possible value of the circuit's resistance, we need to multiply the current at the resonance frequency (Ir) by 0.5. The current at the resonance frequency can be found using the formula:Ir = Vr / XL

where Vr is the voltage at the resonance frequency and XL is the reactance of the inductor, which can be calculated using the formula: XL = 2πfL

By substituting the values, we can calculate the reactance and current at the resonance frequency, and subsequently determine the minimum value of the circuit's resistance.