AM Equation: UNIZOR.COM - Physics4Teens - Waves - Radio
Channel: Zor Shekhtman on UNIZOR Education
Date: 2021-12-18

UNIZOR.COM - Physics through a Looking Glass of Mathematician
Notes to a video lecture on http://www.unizor.com

AM Equation

Let's approach amplitude modulation (AM) from a mathematical standpoint.

High frequency oscillations of the electric current in the LC circuit with an inductor of inductance L and a capacitor of capacitance C can be described by a function
I(t) = Ac·cos(ωc·t)
where
I(t) is the circuit's own oscillation of the current,
Ac is the amplitude of these oscillations,
ωc = 1/√(L·C) is an angular frequency of these oscillations.

Simple air pressure oscillations resulting from a sound can be expressed by a similar function P(t) = As·cos(ωs·t)
where
P(t) is the oscillation of the air pressure around a source of sound,
As is the amplitude of these air pressure oscillations,
ωs is an angular frequency of these oscillations.

Note that our design requires ωc to be substantially greater than ωs.
Also note that real sound is a combination (superposition) of different overtones with different amplitudes and different angular frequencies and phases, like
P(t) = As1·cos(ωs1·(t+φs1)) +
+ As2·cos(ωs2·(t+φs2)) +
+ As3·cos(ωs3·(t+φs3)) + ...

Amplitude modulation alters the amplitude of the carrier's oscillations by changing it in synch with sound oscillations.
The simplest way to achieve it mathematically is to incorporate the sound waves into an amplitude of carrier's waves:
Im(t) = [Ac+P(t)]·cos(ωc·t)

Here is an example of this type of modulation of a carrier's signal.

Assume, the carrier's high frequency oscillations have an amplitude Ac=4 and angular frequency ωc=20, which can be described by an equation
I(t) = 4·cos(20t)
(see graph at UNIZOR.COM notes for this lecture)

The sound makes air pressure oscillations that combine two different tones, one with an amplitude As1=2 and angular frequency ωs1=1 and another with an amplitude As2=1 and angular frequency ωs2=2, which can be described by the following equation
P(t) = 2·cos(t) + cos(2t)
(see graph at UNIZOR.COM notes for this lecture)

Then the modulated signal can be described by an equation
Im(t) = [4+2·cos(t)+cos(2t)]·cos(20t)
that graphically looks like this:
(see graph at UNIZOR.COM notes for this lecture)