Question
The carrier frequency of a transmitter is provided by a tank circuit of a coil of inductance $49 \mu H$ and a capacitance of 2.5 nF. It is modulated by an audio signal of 12 kHz. The frequency range occupied by the side bands is :
The correct answer is
442 kHz – 466 kHz
To solve this problem, we need to determine the carrier frequency of the transmitter and then use this to find the frequency range occupied by the sidebands when modulated by an audio signal.
- First, calculate the carrier frequency of the transmitter using the given inductance and capacitance values. The resonance frequency \((f_c)\) of a tank circuit is given by the formula: \(f_c = \frac{1}{2\pi\sqrt{LC}}\) where: - \(L = 49 \, \mu H = 49 \times 10^{-6} \, H\) - \(C = 2.5 \, nF = 2.5 \times 10^{-9} \, F\)
- Substitute these values into the formula: \[ f_c = \frac{1}{2\pi\sqrt{49 \times 10^{-6} \times 2.5 \times 10^{-9}}} \] Simplifying under the square root: \[ f_c = \frac{1}{2\pi\sqrt{1.225 \times 10^{-13}}} \] \[ f_c = \frac{1}{2\pi \times 3.5 \times 10^{-7}} \] \[ f_c \approx \frac{1}{2\pi \times 3.5 \times 10^{-7}} \approx 450.66 \, kHz \]
- The carrier frequency f_c is approximately 450.66 kHz.
- Now, determine the range of frequencies occupied by the sidebands when modulated by an audio signal of 12 kHz. In amplitude modulation, the sidebands extend by the audio frequency around the carrier frequency. Therefore, the sideband frequencies are (f_c - f_m) and (f_c + f_m), where f_m = 12 \, kHz.
- Thus, the range of frequencies occupied by the sidebands is:
- Lower sideband frequency: f_{LSB} = f_c - f_m = 450.66 \, kHz - 12 \, kHz = 438.66 \, kHz
- Upper sideband frequency: f_{USB} = f_c + f_m = 450.66 \, kHz + 12 \, kHz = 462.66 \, kHz
- Therefore, the frequency range occupied by the sidebands is approximately 442 kHz – 466 kHz.
- Checking the options, the correct answer is: 442 kHz – 466 kHz
This frequency range corresponds to the expected sideband distribution for the given carrier and modulation frequencies in amplitude modulation.
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