[GeeTwo]

Originally Posted by MechEng83

The lower limit as Sun Gear diameter approaches Ring Gear diameter should be 2:1, as the ratio for a planetary gearbox as described is R/S+1.

The upper limit requires some geometry constructions of 3 equally spaced tangent inscribed circles. My literal back of the envelope derivation got messy, but the result came out to 1/(1-6/(3+2*sqrt(3)))+1 or 13.928203235:1

I'll try to post a coherent derivation when I get it worked out nicely on something more legible.

I should point out that these are the limits of the ratio, rather than achievable ratios under the stated conditions, as you would have gears or rollers moving in opposite directions in direct contact.

Correct on the limit of large sun gear is 2:1. On the lowest gear ratio, you're quite close but not quite right.