a good question would be if I know I want a project an image of X size and I have a transparency of Z size, can i work backwards from there?
The formula for distance from focal point of lens to projection plane is (M+1)f=s’.
s’ = distance from focal point of lens to projection plane
M = magnification
f = focal length of lens
If the lens is 50 mm focal length then the distance between the focal point (somewhere in the middle of the lens) to where you want the projection (photolithography board) at a magnification of 0.5 (half size or 1:2) is 75 mm. Life size (M = 1 or 1:1) would require a distance of 100 mm.
The distance from the focal point of the lens to the object (transparency with dial) is s=s’/M.
s’ = same as above
M = same as above
s = distance from the focal point of the lens to the object
So continuing the example above, if you want half size (M = 0.5) then s = 37.5 mm. Life size would be 50 mm.
M = size of object projected divided by size of object in real life. You have a 54 mm transparency of a dial and want to reduce it to 27 mm then M = 27/54 = 0.5. It’s slightly more complicated if you have a 48 mm image of a dial and want it to be projected to 27 mm on your photolithography board. You have desired M = 27/48 = 0.56. Use that in the other equations.
The longer the lens focal length, the more working distance between the lens focal point and the projection plane, but the more bellows draw you need in an enlarger between the lens focal point and the transparency.
Note that finding the position of the focal point in the lens is not easy without manufacturer’s specs. Just approximate and make slight adjustments by moving everything a little each way to get exactly what you need. Then measure twice and write it all down!