Planetary Gear Ratio Calculator
The three ratios an epicyclic set can give depending on which member is held, and the planet tooth count.
Results
What this tool does
The same three gears give three completely different ratios depending on which one you hold still: ring fixed with the carrier out, sun fixed with the carrier out, or carrier fixed, which reverses direction. That is exactly how an automatic gearbox changes gear without disengaging anything — it simply changes which member the brake band holds.
Formula
fixed sprocket: i = 1 + Z sprocket ÷ Z sun
Variables
| Symbol | Meaning | Unit |
|---|---|---|
zs | Sun gear teeth | — |
zr | Ring gear teeth | — |
rp | Rotational speed | rpm |
I1 | Gear ratio | — |
I2 | Gear ratio | — |
I3 | Gear ratio | — |
ZP | Planet teeth | — |
NO | Rotational speed | rpm |
Worked example
- Sun gear teeth24
- Ring gear teeth72
- Rotational speed1450 rpm
- Gear ratio4.00000
- Gear ratio1.33333
- Gear ratio-3.00000
- Planet teeth24.0
- Rotational speed362.50 rpm
Limitations
- For work that must comply with a standard or be signed off, check the result against the applicable code and have it reviewed by a qualified engineer.
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
Frequently asked questions
Why must the planet teeth come out as a whole number?
Because the planet has to mesh with both the sun and the ring at the same time, and geometry forces the ring to equal the sun plus twice the planet. If the arithmetic gives a half tooth, that combination of sun and ring simply cannot be built with equal planet sizes. There is a second condition as well: for the planets to be equally spaced, the sum of sun and ring teeth must divide exactly by the number of planets.