3.1 Amplitude modulation

1.1 Terminology used in electronic communication system
1.2 Bandwidth of signals
1.3 Bandwidth of transmission medium
1.4 Earth's atmosphere

2.1 Propagation of electromagnetic waves in the atmosphere

3.1 Amplitude modulation
3.2 Frequency modulation
3.3 Necessity of modulation

In amplitude modulation, the amplitude of carrier wave is varied in accordance with the amplitude of modulating signal.

Let $c(t) = {A_c}\sin {\omega _c}t$ represent carrier wave

$m(t) = {A_m}\sin {\omega _m}t$ represent modulating signal

The amplitude modulated signal is represented as, $$\begin{equation} \begin{aligned} {c_m}\left( t \right) = \left( {{A_c} + {A_m}\sin {\omega _m}t} \right)\sin {\omega _c}t \\ {c_m}\left( t \right) = {A_c}\left( {1 + \frac{{{A_m}}}{{{A_c}}}\sin {\omega _m}t} \right)\sin {\omega _c}t \\ {c_m}\left( t \right) = {A_c}\sin {\omega _c}t + \mu {A_c}\sin {\omega _m}t\sin {\omega _c}t \\ {c_m}\left( t \right) = {A_c}\sin {\omega _c}t + \frac{{\mu {A_c}}}{2}\cos \left( {{\omega _c} - {\omega _m}} \right)t - \frac{{\mu {A_c}}}{2}\cos \left( {{\omega _c} + {\omega _m}} \right)t \\ \\\end{aligned} \end{equation} $$

where,

${\omega _c} = 2\pi {\upsilon _c}$: Angular frequency of carrier wave

${\omega _m} = 2\pi {\upsilon _m}$: Angular frequency of modulating signal

$A_m$: Amplitude of the modulating signal

$A_c$: Amplitude of the carrier wave

$\mu = \frac{{{A_m}}}{{{A_c}}}$: Modulation index.

Note: In practice, modulation index $\mu $ is kept $ \leqslant $ to avoid distortion.

The amplitude modulated signal contains three frequencies.

${\upsilon _c}$: Carrier frequency

${\upsilon _{SB}} = \left( {{\upsilon _c} \pm {\upsilon _m}} \right)$: Side band frequencies

Frequency of lower side band, $${\upsilon _{S{B_L}}} = {\upsilon _c} - {\upsilon _m}$$

Frequency of upper side band, $${\upsilon _{S{B_U}}} = {\upsilon _c} + {\upsilon _m}$$

The frequency spectrum of the amplitude modulated signal is shown in the figure.

Bandwidth of amplitude modulated $(AM)$ signal $ = {\upsilon _{S{B_U}}} - {\upsilon _{S{B_L}}} = 2{\upsilon _m}$

Average power per cycle in the carrier wave is, $${P_c} = \frac{{A_c^2}}{{2R}}$$

where $R$ is the resistance

Total power per cycle in the modulated wave, $${P_t} = {P_c}\left( {1 + \frac{{{\mu ^2}}}{2}} \right)$$

If $I_t$ is $rms$ value of total modulated current and $I_c$ is the $rms$ value of unmodulated carrier current then,

$$\frac{{{I_t}}}{{{I_c}}} = \sqrt {1 + \frac{{{\mu ^2}}}{2}} $$