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I recently used an oscilloscope in X-Y mode to draw the phase ellipsis of two voltages. I then used the formula phi = arcsin(2y/B) where y is the value of the ellipsis at x = 0 and B is the total distance from the highest point of the ellipsis to the lowest.

Now I really want to know why this works, thank you.

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I guess You speak of an oscilloscope, don't You? –  Georg Oct 2 '11 at 10:54
    
Yes, sorry. I will update the question. –  Althalos Oct 2 '11 at 12:00

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The first thing to notice is where 2y comes from. You have two values at x=0. But for symmetry reasons, they differ only in sign. Hence, they're +y and -y, and their difference is 2y. Similarly, your Y signal has an amplitude A, and thus can have range from -A to +A. The difference is 2A, which you call B. Therefore, 2y/B is the ratio of the Y signal at x=0 and the max Y signal.

But why is this important? Let's look at some special cases. If the phase shift is zero, your ellipsis becomes degenerate. It's a line, and y=0. If the phase shift is pi/2, it's a circle and y=B/2. Clearly the shape matters, and y/B is a measure of that shape.

Not let's put in numbers:

x = A sin(f*t)
y = A sin(f*t+phi)

x=0 means sin(f*t) = 0 and therefore y = A sin(phi). Substituting B, you get x=0 => y = B/2 sin(phi) => sin(phi) = B/2y QED.

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I think the XY-mode of an oscilloscpe works by some switches, which connect the second channel amplifier to the horizontal plates in the Braun tube, instead of the time sweep circuit which is normally connected to those plates. –  Georg Oct 8 '11 at 17:07
    
@Georg: Modern scopes are a bit more complex, but that's indeed how it used to work. –  MSalters Oct 10 '11 at 7:47

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