How does the X-Y mode of an oscilloscope work?

Question

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.

Answer 1

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.