Guide
Gear ratio formulas for planetary, cycloidal, harmonic and RV reducers
The ratio formula for each precision reducer type, tooth-count tables for standard ratios, and how to get output speed and torque, with worked examples.
Key takeaways
- Planetary: i = 1 + Zring ÷ Zsun. Two stages multiply.
- Cycloidal: i = lobes = pins − 1.
- Harmonic: i = Zflex ÷ (Zcircular − Zflex), usually Zflex ÷ 2.
- RV: i = 1 + (Zspur ÷ Zsun) × Zpins.
Each formula below assumes the usual arrangement for automation gearboxes: the housing is fixed, the motor drives the input and the output is the slow member. Change tooth counts live in the ratio explorer on each of our segment pages.
Planetary gearbox
Sun gear input, ring gear fixed, planet carrier output: i = 1 + Zring ÷ Zsun. The planets do not appear in the ratio, but their teeth are fixed by the geometry: Zring = Zsun + 2 × Zplanet. For three equally spaced planets, (Zsun + Zring) must divide by 3.
| Ratio | Sun | Planet | Ring | Check |
|---|---|---|---|---|
| 3:1 | 24 | 12 | 48 | 1 + 48/24 = 3 |
| 4:1 | 18 | 18 | 54 | 1 + 54/18 = 4 |
| 5:1 | 12 | 18 | 48 | 1 + 48/12 = 5 |
| 7:1 | 12 | 30 | 72 | 1 + 72/12 = 7 |
| 10:1 | 9 | 36 | 81 | 1 + 81/9 = 10 |
Above about 10:1 the sun becomes too small and the planets too large to fit, so higher ratios use two stages: 15 = 3 × 5, 20 = 4 × 5, 25 = 5 × 5, 35 = 7 × 5, 50 = 10 × 5, 70 = 7 × 10, 100 = 10 × 10. The output turns the same direction as the input.
Cycloidal reducer
An eccentric input drives a disc with one lobe fewer than the ring has pins: i = Zlobes ÷ (Zpins − Zlobes) = Zlobes. Output is taken through pins in holes in the disc, and turns opposite to the input.
| Ratio | 11 | 17 | 23 | 29 | 35 | 43 | 59 | 71 | 87 | 119 |
|---|---|---|---|---|---|---|---|---|---|---|
| Pins | 12 | 18 | 24 | 30 | 36 | 44 | 60 | 72 | 88 | 120 |
Harmonic drive (strain wave gear)
Wave generator input, circular spline fixed, flexspline output: i = Zflex ÷ (Zcircular − Zflex). The difference is two teeth, matching the two engagement zones of the elliptical wave generator. The output turns opposite to the input.
| Ratio | 30 | 50 | 80 | 100 | 120 | 160 |
|---|---|---|---|---|---|---|
| Flexspline | 60 | 100 | 160 | 200 | 240 | 320 |
| Circular spline | 62 | 102 | 162 | 202 | 242 | 322 |
RV reducer
A sun gear drives spur gears on the crankshafts, and the cranks drive the cycloid discs inside a pin ring, with the carrier as output: i = 1 + (Zspur ÷ Zsun) × Zpins. The output turns the same direction as the input.
| Ratio | Sun | Spur | Check |
|---|---|---|---|
| 57:1 | 15 | 21 | 1 + 1.4 × 40 = 57 |
| 81:1 | 16 | 32 | 1 + 2.0 × 40 = 81 |
| 105:1 | 15 | 39 | 1 + 2.6 × 40 = 105 |
| 121:1 | 12 | 36 | 1 + 3.0 × 40 = 121 |
| 153:1 | 10 | 38 | 1 + 3.8 × 40 = 153 |
From ratio to output speed and torque
Output speed = input speed ÷ i. Output torque = input torque × i × η. Example: a 3,000 rpm, 1.27 N·m servo on a 50:1 two-stage planetary gearbox (η ≈ 0.94) gives 60 rpm and 1.27 × 50 × 0.94 = 59.7 N·m. Always check the result against the gearbox's rated and peak torque, not only the motor.
References
- ANSI/AGMA 6123, Design manual for enclosed epicyclic gear drives.
- C. W. Musser, Strain wave gearing, US Patent 2,906,143 (1959).
Frequently asked questions
What is the formula for a planetary gear ratio?
With the ring gear fixed, sun input and carrier output: i = 1 + Z_ring ÷ Z_sun. A 12-tooth sun in a 48-tooth ring gives 1 + 48 ÷ 12 = 5:1.
How is the harmonic drive ratio calculated?
With the circular spline fixed and the flexspline as output: i = Z_flex ÷ (Z_circular − Z_flex). The difference is normally 2 teeth, so a 200-tooth flexspline gives 100:1.
How do I calculate output torque from the ratio?
Output torque = input torque × ratio × efficiency. For example 1.27 N·m × 50 × 0.94 = 59.7 N·m.