How do you calculate the gear ratio of a planetary gear system?

For a planetary system with sun gear (S), planet gears (P), and ring gear (R), using the fundamental equation: Ns/Nr = 1 + Zr/Zs. With ring fixed and sun as input, carrier output ratio = 1 + Zr/Zs. With carrier fixed and sun input, ring output ratio = -Zr/Zs (reverse). Planetary systems achieve high ratios (3:1 to 10:1 per stage) in compact space, used in automatic transmissions, wind turbines, and heavy machinery.