Hello everyone. This is my first post on here. I figured that I would give it a shot. Just as a quick background: I'm studying electrical engineering and I'm working an internship for a machine manufacturing company now. One of the projects I have been assigned is to come up with a system to compensate periodic disturbances. To do this, I need to be able to calculate the phase delay so that the compensation signal I feed into the controller will have the proper amount of phase delay in it. Referring to my attached diagram, you can see there is a photo eye directly above the first "pulley". This first larger circle is a roll of toilet paper. As the machine takes paper away, or unwinds it, the bigger roll diameter gets smaller until it reaches the end of the roll (core diameter of 1.7 inches). Assuming that the photo eye is the zero point and working our way around the circle counter-clockwise, I need to calculate how much phase delay will be from when the photo eye sees the disturbance, to when the paper is unwinding off of the roll. As you can see in the diagram, when the unwind is rolling over the top this phase should be somewhere between (3*pi)/2 and 2*pi. When it is underwinding, it should be somewhere between pi and (3*pi)/2. I am really only trying to calculate one way as of now (overwinding) because I believe the math will be the same or very similar in nature when I figure out how to get this done. I have attempted drawing many triangles (via scratch paper sorry for no shown work) and have been stuck in a loop with having not enough information to be able to solve the triangles. Theta (red) is the angle that I need. Keep in mind that the x,y point where the paper is leaving the unwind roll will be changing as the radius changes. The big roll is centered at (0,0) and the 2" diameter roll where the paper lands is centered at (8.5,-0.5) by the nature of the physical machine (I can't change that). I'm not sure that I did a great job detailing what I want but I've been stuck on this for about a week at my job and need to get an answer quickly. If you have any questions please feel free to ask and I will do my best to field them. Thank you!