What is Amplitude Modulation? Complete Mathematical Derivation of AM Wave

In this topic, first we will understand the process of amplitude modulation and then write the mathematical derivation of AM wave. I suggest you to take a pen and paper while reading this article and write down each step of the derivation while reading.

Definition: Amplitude modulation is defined as the process, in which the amplitude of carrier signal is changed in accordance with the amplitude of modulating signal. To produce AM wave, a carrier signal of frequency f_c is mixed with another low frequency modulating signal of frequency f_m in a circuit known as amplitude modulator.

Suppose carrier signal is given by –

e_c = V_c.sin(ω_c).t  … (1)

And the modulating signal is given by –

e_m = V_m.sin(ω_m).t  … (2)

Now when they are mixed in amplitude modulator circuit, AM wave is produced. The peak amplitude of this AM wave is given by –

A = V_c + e_m = V_c + V_m.sin(ω_m).t  … (3)

Now let ‘m’ be the ratio of peak amplitude of modulating signal to peak amplitude of carrier signal. The value of ‘m’ is known as modulation index –

m = V_m / V_c        i.e.       V_m = m.V_c

Putting this value of V_m in equation (3), we get –

A = V_c + m.V_c.sin(ω_m).t  = V_c (1 + m.sin(ω_m).t ) … (4)

Now the equation of AM wave is given by –

e_mod = A.sin(ω_c).t  … (5)

where, e_mod = instantaneous amplitude of amplitude modulated (AM) wave

Putting the value of equation (4), in equation (5), we get –

e_mod = (V_c (1 + m.sin(ω_m).t )).sin(ω_c).t  … (6)

By solving equation (6), we get –

e_mod = V_c.sin(ω_c).t + m.V_c ((sin(ω_c).t ).(sin(ω_m).t )) … (7)

Applying trigonometric rule, finally we get –

(sin(x).sin(y)) = ½ sin(x – y) – ½ sin(x + y). So applying this rule to equation (7), finally we get –

Bandwidth of AM Wave

The above given final equation of AM wave consists of three components –

Original i.e. unmodulated carrier frequency: ω_c = 2π.f_c

The Lower Side Band (LSB): ω_c – ω_m = 2π.( f_c – f_m )

And the Upper Side Band (USB): ω_c + ω_m = 2π.( f_c + f_m )

The difference of USB and LSB is called as the bandwidth –

BW = (2π.( f_c + f_m )) – (2π.( f_c – f_m )) = 2f_m

Important Definitions

These are the two important definitions in order to understand the next topic of importance of sidebands in AM process –

Modulation index (m)

It is defined as the ratio of peak amplitude of modulating signal to the peak amplitude of carrier signal.

m = V_m / V_c

Percentage modulation (M)

When modulation index ‘m’ is multiplied with 100, that is expressed in percentage, it is called as percentage modulation M.

M = m × 100 = ( V_m / V_c ) × 100