Question

In an amplitude modulator circuit, the carrier wave is given by, \[C\left( t \right) = 4\sin \left( {2000\pi t} \right)\] while modulating signal is given by, \[m\left( t \right) = 2\sin \left( {200\pi t} \right)\]. The values of modulation index and lower side band frequency are:
A. 0.5 and 0.9kHz
B. 0.5 and 10kHz
C. 0.3 and 9kHz
D. 0.4 and 10kHz

Answer 1

Hint:An important step in communication is the modulation, which involves superimposing of an audio frequency wave on a high frequency carrier wave. Both the carrier wave and the modulating signal can be written in terms of sine waves. Equating the original equations of carrier wave and the modulating signal with the ones given in the problem can give the values for frequencies and amplitudes which are required to obtain the modulation index and lower side band frequency.

Formula Used:
Modulation index (\[A\] ) is given by:
\[A = \dfrac{M}{C}\]
where, \[C\] is the carrier amplitude and \[M\] is the modulating amplitude.
Lower side band frequency (\[LSB\]) is given by:
\[LSB = {\omega _c} - {\omega _m}\]
where, \[{\omega _c}\] is the carrier frequency, \[{\omega _m}\] is the modulating frequency.

Complete step by step answer:
Modulation is the process of variation of some characteristics of the carrier wave in accordance with the signal. Carrier wave is the radio wave which carries the message. It is a high frequency wave. On the other hand, signal is the message to be communicated. It is a low frequency wave. The carrier wave and the modulating signal are written in terms of sine wave as;
\[C\left( t \right) = C\sin \left( {2\pi {\omega _c}t} \right)\] \[ \to (1)\]
and
\[m\left( t \right) = M\sin \left( {2\pi {\omega _m}t} \right)\] \[ \to (2)\]
Here,\[C\left( t \right)\] is the carrier signal, \[m\left( t \right)\] is the modulating signal, \[C\] is the carrier amplitude, \[M\] is the modulating amplitude, \[{\omega _c}\] is the carrier frequency, \[{\omega _m}\] is the modulating frequency and \[t\] is the time period.

In the problem, the carrier wave is given by,
\[C\left( t \right) = 4\sin \left( {2000\pi t} \right) \\
\Rightarrow C\left( t \right) = 4\sin \left( {2 \times 1000\pi t} \right)\] \[ \to (3)\]
And the modulating signal is given by,
\[m\left( t \right) = 2\sin \left( {200\pi t} \right) \\
\Rightarrow m\left( t \right) = 2\sin \left( {2 \times 100\pi t} \right)\] \[ \to (4)\]
Equating equations (3) and (4) with equations (1) and (2). Thus,
\[C = 4\], \[M = 2\], \[{\omega _c} = 1000Hz\] and \[{\omega _m} = 100Hz\]
Modulation index (\[A\] ) is given by \[A = \dfrac{M}{C} = \dfrac{2}{4} = 0.5\]
Lower side band frequency (\[LSB\]) is given by
\[LSB = {\omega _c} - {\omega _m} \\
\Rightarrow LSB = 1000Hz - 100Hz \\
\Rightarrow LSB = 900Hz \\
\therefore LSB = 0.9kHz\]

Hence, option A is the correct answer.

Note: Carrier wave is the high frequency wave and signal is the low frequency wave.This is because carrier wave carries the message and the signal is the message itself. The carrier wave is actually given by \[C\left( t \right) = C\sin \left( {2\pi {\omega _c}t + \phi } \right)\] and the modulating signal is given as \[m\left( t \right) = M\sin \left( {2\pi {\omega _m}t + \phi } \right)\] where \[\phi \] is the phase of the signal at the start of the reference time. But \[\phi \] can be omitted and approximated to zero. Thus one can obtain the equations for the carrier wave and the modulating signal in the form of equations (1) and (2).