Maximum modulation factor of a two tone signal

[opas31]

Homework Statement

Finding the maximum modulation factor for a two tone signal that can be employed without over modulation occurring

Relevant Equations

(Mt) = sqrt(m1^2+m2^2)

I'm stuck on this because v is a 2 tone signal, so it's not as simple as Am/Ac. The teacher said I will need to differentiate it and equate it to zero, which I thought made sense. Differentiating v gives me: [- mwsin(wt) - mwsin(2w*t)], so there's still unknown variables. I don't know how I'm supposed to equate that to zero and find m when there are still variables that will not cancel.

I have seen another (looks easier) method where the Total Modulation Factor (Mt) = sqrt(m1^2+m2^2), where m1 = Am1/Ac and m2 = Am2/Ac. It does make sense, but I've spent a long time looking for where this, or something similar, is written down and haven't found anything so far.

Could anyone provide some help or point me towards some guidance please?

Thanks

 

[alan123hk]

As far as I know, the maximum modulation factor of Traditional Amplitude Modulation should be 1. When the modulation factor >1, the transmitted wave will be distorted, so that the signal wave cannot be exactly reproduced. This is independent of whether the modulating signal is is single tone or multi-tone. So I was wondering if you are sure you are looking for the maximum modulation factor and not the maximum amplitude of the modulating signal.

 

[opas31]

I don't think I worded the post very well. I am looking for the maximum amplitude of the modulating signal (v).

 

[alan123hk]

I agree that the following question is inappropriate, what we are really asking for is the maximum amplitude of the modulating signal.

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So all we have to do is differentiate this equation, which you've already done.
$$ \frac {dv}{dt} = -V\omega~sin(\omega t)-V\omega~sin(2\omega t)$$
The next step is to find what $\omega t$ is when it is equal zero, that is to say we need to solve the following equation.
$$ sin(\omega t)+sin(2\omega t)=0$$
Note that this equation has three solutions.

 

[alan123hk]

We are actually looking for what $~\omega t~$ is when the amplitude of the modulating signal v is maximum, then the value of $~V~$ can also be determined accordingly.

$$sin(\omega t)+sin(2\omega t)=0~~~\rightarrow~~~sin(\omega t)+2sin(\omega t)cos(\omega t)=0~~~~ \rightarrow~~~~sin(\omega t)\left( 1+2cos(\omega t) \right)=0$$
$$ sin(\omega t)=0~~ \text {when} ~~ \omega t =0~~ \text{or}~~ \pi ~~~~~~\text{and}~~~~~~1+2cos(\omega t)=0~~\text{when}~~\omega t = 2.094$$
Because the maximum amplitude of the modulating signal must be equal to the amplitude of the carrier which has been set to 1V, then the maximum value of $~V~$ seem to have the following possibilities.

$$\mathbf{v}= -1=Vcos(\omega t)+\frac{V}{2}cos(2\omega t)~~~~~\rightarrow ~~~~ V=\frac {-2} {2cos(\omega t)+cos(2\omega t)} = ~-\frac {2}{3}, ~ 2~~~\text{or}~~1.33 $$
But the value of $2$ is actually invalid, so in the end only $~-\frac{2}{3}~$ and $~1.33~$ are really applicable.

 

[opas31]

Thanks, that's really helpful. Also matches with the plot I've made.