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Why both Sines and Cosines with the Fourier series?

Phase Shifting.

Phase Shifting.

So far with the Fourier Series we have made some interesting waveforms but what happens in real life? Things are never as clean in real life as on paper. (Except maybe my smudged up papers.)   If you think of a perfect square wave with an infinite number of odd harmonics, what would happen if it was sent into a real circuit?   More than likely some of the frequencies would get attenuated.  Even if it had perfect bandpass, it would have some time delay getting through the circuit.   That constant time delay would cause higher frequencies to shift more in relation to the period of one cycle.

All of this means that all of those higher frequency sine waves are no longer in phase with each other.   So, how do we deal with that?

A Phasor Plot of a Sine wave, a Cosine wave and the Sum of the two.

A Phasor Plot of a Sine wave, a Cosine wave and the Sum of the two.

We have already talked about one way of doing it when we talked about using imaginary numbers and phasers when we dealt with simple filters in RC and RL circuits.  Imaginary numbers can be used with Fourier Analysis, but that requires more math and more theory.  We will just do it by talking about both the sine component and the cosine component.

As seen in the two pictures the sum of these two waves also yields a sinusoidal wave, but the strength of the two determines how much the sum wave is shifted,.

Depending upon your outlook things are getting very interesting or very complex.  It actually is not,.  Since the sine and cosine are 90 degrees out of phase with each other, we are basically back to our old friend the right triangle and a orthogonal relationship.  I explain all this thoroughly in my video I posted last night.
As promised I have several related posts that I will link here.

Each of those pages will have other links to previous posts.   It seems like we always keep coming back to right triangles.

The next post will be heading toward real life effects… I think you will enjoy it and maybe it will make the math more interesting.

Gary

The video associated with this post is embedded below.

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