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 2.5nF. It is modulated by an audio signal of 12kHz. The frequency range occupied by the side bands is:
A.) 18kHz – 30kHz
B.) 63kHz – 75kHz
C.) 442kHz – 466kHz
D.) 13482kHz – 13494kHz

Answer 1

Hint: An amplitude modulated signal has a carrier frequency and a modulating frequency. The frequencies of the side bands of the modulated signal are equal to the sum and difference of these two frequencies.

Complete step-by-step answer:
Modulation is a process used in communication in which the properties of a signal, like phase, amplitude or frequency are altered with respect to the nature of the signal which is to be transmitted from one place to another.

There are three types of modulations used in communication: Amplitude modulation (AM), Frequency modulation (FM), Phase modulation (PM)

An amplitude modulation (AM) wave contains three frequencies:

${f_C},{\text{ (}}{f_C} + {f_m})$ and $\left( {{f_C} - {f_m}} \right)$

where ${f_C}$ is the carrier frequency, ${f_m}$ is the modulating frequency and $\left( {{f_C} + {f_m}} \right)$ and $\left( {{f_C} - {f_m}} \right)$ are side-band frequencies. The side-bands signify the range over which a particular signal can be transmitted.

We are given the value of modulating frequency:

${f_m} = 12kHz$

The tank circuit consists of a coil which has following values of capacitance and inductance:

$\begin{gathered}
  C = 2.5nF \\
  L = 29\mu H \\
\end{gathered} $

The frequency of an LC tank circuit is given as:

$\omega = \dfrac{1}{{\sqrt {LC} }}$

Putting the values of C and L, we get the frequency to be:

$
  \omega = \dfrac{1}{{\sqrt {49 \times {{10}^{ - 6}} \times 2.5 \times {{10}^{ - 9}}} }} = 2857142.86Hz \\
   \Rightarrow {f_C} = \dfrac{\omega }{{2\pi }} = 454728.409Hz \simeq 454.728kHz \\
$

Now using this value of carrier frequency, we can calculate the side-band frequencies:

$
  \left( {{f_C} + {f_m}} \right) = 454.728 + 12 = 466.728kHz \\
  \left( {{f_C} - {f_m}} \right) = 454.728 - 12 = 442.728kHz \\
$

Hence, the correct answer should be option C if we ignore the digits after the decimal and take the integer value.

Note: The purpose of modulation is that it allows us to send a particular signal on frequency range. The role of the tank circuit in a communication channel is that it provides a carrier frequency. It is also used for tuning a circuit to a particular frequency.