# sine wave amplitude calculation

• Apr 27th 2012, 02:35 PM
suzuki
sine wave amplitude calculation
Hi,

I'm having some troubles determining the amplitude/magnitude of the following equation.

$\displaystyle A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3)$

Since each part is at a different frequency, i cannot sum the magnitudes of each part.

I have also thought about using variations of the double/triple angle formulae and some basic trigonometric identities, so that I can write the equation under a single frequency, but by doing so, I am introducing some higher order terms, which seem to negate the ability to sum the amplitudes together.

For example, the term with $\displaystyle 5\omega$ would look like
$\displaystyle 16\cos^5(\omega t)-20\cos^3(\omega t)+5\cos(\omega t)$

I suppose another method would be to plot the first equation and then record the amplitude, but i would like a more generalised approach to solve this problem. All input is welcome and appreciated.
• Apr 27th 2012, 04:37 PM
ignite
Re: sine wave amplitude calculation
I will show how to do it using an example and then you can generalize it.
$\displaystyle f(t)=cos(2t)+2sin(3t)$
$\displaystyle f(t)=\sqrt{1^2+2^2}(\frac{1}{\sqrt{1^2+2^2}}cos(2t )+\frac{2}{\sqrt{1^2+2^2}}sin(3t))$
$\displaystyle sin(u)=\frac{1}{\sqrt{1^2+2^2}};cos(u)=\frac{2}{ \sqrt{1^2+2^2}}$
$\displaystyle \Rightarrow f(t)=\sqrt{5}(sin(u)cos(2t)+cos(u)sin(3t))=\sqrt{5 }sin(u+2t)$

So amplitude in this case would be $\displaystyle \sqrt{5}$.
• Apr 29th 2012, 08:18 PM
suzuki
Re: sine wave amplitude calculation
Hi,

Thanks for your answer. I actually have a question about this. When you found that it is $\displaystyle \sqrt(5)$, is this a peak value or the average value? I plotted your example in matlab, and found that the peak amplitude value is 3. What is this method that you are using? Could you please further explain your method?

• Apr 29th 2012, 09:34 PM
ignite
Re: sine wave amplitude calculation
$\displaystyle \sqrt{5}$ is the maximum value.Please check the graph at Wolfram|Alpha
• Apr 29th 2012, 10:02 PM
suzuki
Re: sine wave amplitude calculation
Hi,
Sorry I must have made a mistake there.

I'm having some trouble visualizing how to do it for three terms. im assuming the magnitude can be found by $\displaystyle \sqrt(A^2+B^2+C^2)$. But when you have to break it down into cosine and sine terms, it seems to be quite difficult. If A = 1, B = 2 and C = 3, i can see that maybe you can break the C term into A+B, but for the general case, this is not so simple. Please advise.

Thanks again.
• Apr 30th 2012, 02:16 PM
suzuki
Re: sine wave amplitude calculation
Seems like another problem i run into is that the simplification does not necessarily work if all my terms are cosines. Is there any further help that can provided for this? I am really stuck on how to solve this...
• Apr 30th 2012, 02:22 PM
ignite
Re: sine wave amplitude calculation
I am extremely sorry.The method I described above works only when the frequencies are same,but amplitudes are different.
In general case,it is not possible to write it as a simple sine or cosine formula.
• Apr 30th 2012, 02:29 PM
ignite
Re: sine wave amplitude calculation
Quote:

Originally Posted by ignite
I will show how to do it using an example and then you can generalize it.
$\displaystyle f(t)=cos(2t)+2sin(3t)$
$\displaystyle f(t)=\sqrt{1^2+2^2}(\frac{1}{\sqrt{1^2+2^2}}cos(2t )+\frac{2}{\sqrt{1^2+2^2}}sin(3t))$
$\displaystyle sin(u)=\frac{1}{\sqrt{1^2+2^2}};cos(u)=\frac{2}{ \sqrt{1^2+2^2}}$
$\displaystyle \Rightarrow f(t)=\sqrt{5}(sin(u)cos(2t)+cos(u)sin(3t))=\sqrt{5 }sin(u+2t)$

So amplitude in this case would be $\displaystyle \sqrt{5}$.