Amplitude Modulation of a wave

[Prashasti]

Homework Statement

I am utterly confused. When I was reading my textbook , I found something unacceptable.


While deriving an expression for a modulated wave,
It's been given that
"A sinusoidal carrier wave can be represented as c (t) = A_c sin
(ωt + Φ)
where c (t) is the signal strength of the carrier wave.
Let m (t) = A_m sinω_mt represent the message or the modulating signal.
The modulated signal c_m (t) can be written as
c_m= (A_c+A_msin ω_mt) sin ω_ct

I wonder how's it possible! Shouldn't it be c_m (t) = A_csin ω_ct + A_m sinω_mt ?

But then I made an adhoc assumption - which was not satisfactory - but I thought it could be justified from a more rigorous application of mathematics. So, I continued reading

On the next page, I found something in contrast to my "assumption".
In the topic "Production of amplitude modulated wave" -
According to my textbook "Here the modulating signal A_m
sinω_mt is added to the carrier signal A_csinωt to produce the signal x (t). This signal x (t) = A_m sinω_mt + A_c sin ω_ct is passed through a square law device."

Now this equation for x (t) is different from the one which was used (in the textbook) earlier.
What even is happening?

 

[ehild]

If the sum of two signals of different frequencies ω₁ and ω₂ is the input of a linear network, the output would be again the sum of two signals with the same frequencies.
Using a nonlinear device, one, for example, that produces the square of the input, it will "mix" the frequencies. Expand (Asin(ω₁t) + Bsin(ω₂t ))², what do you get? ( there will be components with frequencies of 2ω₁, 2ω₂, ω₁+ω₂, ω₁-ω₂).
Semiconductor diodes, transistors have nonlinear characteristics, and can be used for "mixing frequencies"

 

[Prashasti]

If the sum of two signals of different frequencies ω₁ and ω₂ is the input of a linear network, the output would be again the sum of two signals with the same frequencies.
Using a nonlinear device, one, for example, that produces the square of the input, it will "mix" the frequencies. Expand (Asin(ω₁t) + Bsin(ω₂t ))², what do you get? ( there will be components with frequencies of 2ω₁, 2ω₂, ω₁+ω₂, ω₁-ω₂).
Semiconductor diodes, transistors have nonlinear characteristics, and can be used for "mixing frequencies"

Ok.
So, the problem lies in the arrangement of all that in my textbook..

 

[Prashasti]

They mentioned the result before deducing the expression for the same.
I got it!
Thanks!

 

[HalfLostAndSearching]

I can understand your confusion and frustration. It is important to carefully review and analyze the equations and assumptions presented in textbooks and scientific literature. It is possible that there may be typos or errors in the equations, or that different authors may use slightly different equations or assumptions.

In the case of amplitude modulation, both equations you mentioned are correct and can be used to represent the modulated signal. The first equation, cm (t) = (Ac + Am sin ωmt) sin ωct, is known as the "product modulator" equation and is commonly used in the analysis of amplitude modulation. The second equation, x (t) = Am sin ωmt + Ac sin ωct, is known as the "sum modulator" equation and is commonly used in the practical implementation of amplitude modulation.

It is important to note that both equations are derived from different assumptions and models, but they ultimately represent the same modulated signal. It is up to the reader to carefully examine and understand the context in which each equation is used.

In the production of amplitude modulated waves, a square law device is used to produce the sum of the carrier and modulating signals, resulting in the second equation x (t). This is a practical approach to implementing amplitude modulation and may not necessarily match the theoretical analysis using the first equation cm (t).

I would recommend consulting with your teacher or a fellow scientist to further clarify any confusion and ensure a thorough understanding of the topic. It is also important to critically analyze and question the information presented in textbooks and scientific literature. As scientists, it is our responsibility to strive for accuracy and clarity in our work.