Planetary Gear Ratio & Speed Calculator
Enter tooth counts and one driving speed. The calculator uses the Willis equation for a simple planetary gear set (sun, ring, carrier).
What this planetary gear ratios calculator does
This calculator helps you quickly compute speed relationships in a simple planetary (epicyclic) gear train. You provide the sun gear tooth count, ring gear tooth count, choose which member is fixed, and set one input speed. The tool then solves for the remaining member speed, reports the gear ratio, and indicates whether the output rotates in the same direction or opposite direction.
Planetary gear basics: sun, planet, ring, and carrier
A planetary gearset typically includes:
- Sun gear: the center gear.
- Planet gears: gears that orbit the sun.
- Carrier: arm that holds the planet gears.
- Ring gear: internal gear surrounding the planets.
By changing which member is held, which member is driven, and which member is output, you can get large reduction, moderate reduction, or reverse direction behavior in a compact package.
Core equation used in the calculator
For a simple planetary set, speed relationships are governed by the Willis equation:
(ωr − ωc) / (ωs − ωc) = −Ns / Nr
Where:
- ωs, ωr, ωc are sun, ring, and carrier angular speeds
- Ns, Nr are sun and ring tooth counts
The calculator rearranges this equation automatically based on your fixed and input selections.
Common fixed-member configurations
1) Ring fixed, sun input, carrier output
This is a classic speed reduction setup. The carrier turns slower than the sun with increased torque potential (ignoring losses).
- Speed ratio magnitude: 1 + Nr/Ns
- Direction: usually same as input in this arrangement
2) Sun fixed, ring input, carrier output
Also produces reduction, but with different ratio characteristics. Useful where packaging or clutching strategy favors a fixed sun.
- Speed ratio magnitude: 1 + Ns/Nr (input/output interpretation depends on which member you define as input)
3) Carrier fixed (differential behavior removed)
With the carrier locked, the set behaves similarly to a simple gear pair relationship between sun and ring, with opposite rotation direction.
- Sun and ring rotate in opposite directions
- Magnitude relation tied to Nr/Ns
How to use this calculator correctly
- Use realistic tooth counts (ring teeth should be larger than sun teeth for a standard internal ring setup).
- Choose one fixed member and a different input member.
- Set input speed in RPM; positive/negative sign can represent direction convention.
- Read the output speed, signed ratio, and direction note.
Design and engineering notes
This calculator is ideal for concept-level sizing and kinematic checks. It does not model:
- Efficiency losses
- Bearing drag
- Tooth strength limits
- Deflection, noise, or thermal behavior
- Multi-stage coupled planetary systems
For final design, combine this with AGMA/ISO gear strength calculations, lubrication analysis, and transient dynamic simulation.
Quick worked example
Suppose Ns = 30, Nr = 90, ring fixed, sun input at 1200 RPM. You will get carrier output near 300 RPM, which is a 4:1 speed reduction (input/output). In ideal conditions, that implies about 4x torque multiplication before losses.
Frequently asked questions
Can this be used for automatic transmission gearsets?
Yes, for individual simple planetary relationships. Real transmissions often combine multiple sets and clutch states.
Why does the calculator show a negative ratio sometimes?
A negative signed ratio indicates output rotation opposite to input based on the sign convention.
What if output speed is near zero?
The ratio can become very large (approaching infinity), which indicates a near-stall output condition for the selected setup.