Question

In AM wave, carrier power is given by:
A) ${P_c} = \dfrac{{2{E_c}^2}}{R}$
B) ${P_c} = \dfrac{{{E_c}^2}}{R}$
C) ${P_c} = \dfrac{{{E_c}^2}}{{2R}}$
D) ${P_c} = \dfrac{{{E_c}^2}}{{\sqrt {2R} }}$

Answer 1

Hint: The method of superimposing the low audio frequency baseband message signal on a high-frequency wave is called modulation. The low-frequency baseband signal is called a modulating signal. The high-frequency wave is called a carrier wave. And the resultant wave is called modulated waves.

Complete step by step solution:
In amplitude modulation, the amplitude of the carrier wave is varied in accordance with the amplitude of the audio frequency message signals. However, the frequency of the amplitude-modulated wave remains the same as that of the carrier wave (high-frequency wave), only the amplitude is changing.
The amplitude modulated wave is the sum of three high-frequency components. They are original carrier wave frequency, lower sideband frequency, and upper sideband frequency.
Hence, the power of the amplitude wave is equal to the sum of powers of the carrier, upper sideband, and lower sideband frequency components.
That is, \[{P_t} = {P_c} + {P_{USB}} + {P_{LSB}}\]
We know that the standard formula for the power of (1) sinusoidal signal is,
Power, $P = \dfrac{{{V_{rms}}^2}}{R}$ …………………..(1)
We have, ${V_{rms}} = \dfrac{{{V_0}}}{{\sqrt 2 }}$
$ \Rightarrow {V^2}_{rms} = \dfrac{{{V_0}^2}}{2}$
Substitute in equation (1) we get,
$P = \dfrac{{{V_0}^2}}{{2R}}$
Where ${V_{rms}}$ is the RMS value of the sine signal.
${V_0}$ is the peak value of the voltage that is the amplitude of the sinusoidal wave.
Then, the power of the carrier wave is given by,
${P_C} = \dfrac{{{V_0}^2}}{{2R}} = \dfrac{{{A_C}^2}}{{2R}}$
${A_C}$ is the amplitude of the carrier signal also represented as the energy of the carrier signal.
Then we get,
${P_C} = \dfrac{{{E_C}^2}}{{2R}}$

$\therefore $ Correct option is (c).

Note:
The purpose of communication is to transmit information or message signals. A message signal usually spreads over a range of frequencies, called the signal bandwidth. Because of this reason message signals are also called baseband signals. Information is converted into electrical form and suitable for transmission is called a signal. The word communication refers to the transfer of information or messages from one point to another point in an intelligible form.