How do I use the planetary gearbox calculator?

Enter the number of teeth on the sun, planet, and ring gears, the module (m), input torque, input speed, gearbox efficiency, and the number of planets. The tool will calculate the gear ratio, output speed, output torque, and approximate tangential forces on the sun and planet gears.

What is the gear ratio formula for a planetary gearbox?

The gear ratio is calculated using the formula: Gear Ratio = (Ring Gear Teeth + Sun Gear Teeth) / Sun Gear Teeth. For concentric geometry, the ring gear teeth should equal the sun gear teeth plus twice the planet gear teeth: Zr = Zs + 2 × Zp.

What are the three operating modes in a planetary gearbox?

The three operating modes are: 1) Sun drives carrier, ring fixed; 2) Sun drives ring, carrier fixed; 3) Carrier drives ring, sun fixed. Each mode has a different gear ratio and operational characteristics.

How is output torque calculated?

Output torque is approximated using the formula: Output Torque = Input Torque × Gear Ratio × Efficiency.

What are the typical applications of planetary gearboxes?

Planetary gearboxes are commonly used in applications requiring high torque and compact size, such as in automotive transmissions, robotics, conveyors, and wind turbines.

Planetary Gearbox Calculator

Unit system

Sun teeth, Zs

Ring teeth, Zr

Planet teeth, Zp

Planets Count

Module / Pitch

Input Torque

Input RPM

Efficiency %

Planetary gear calculator is a compact online tool that delivers immediate kinematic and load estimates for epicyclic gear trains. The utility computes gear ratios for three standard operating modes, returns output speed and torque, gives simple tangential force estimates on gear bodies and renders a clear geometry preview for rapid decision making. You can see a much clearer animation in the other calculator.

🛠 The calculator applies basic gear geometry and Willis relations to convert teeth counts and module into operational numbers. Input tooth counts for sun, planet and ring together with module, input torque and input speed. The tool computes the numerical gear ratio, output speed, output torque adjusted for efficiency and approximate tangential forces on the sun and each planet. Check out the interactive 3D-model here.

Table of Contents

Key input parameters

Parameter	Symbol	Notes
Sun gear teeth	Zs	Integer, typical minimum 6
Planet gear teeth	Zp	Same for every planet
Ring gear teeth	Zr	Ideal relation Zr = Zs + 2 × Zp for concentric geometry
Number of planets	N	Common choices 1 through 4, three gives symmetry
Module	m	Millimetres. Pitch diameter equals m times tooth count
Input torque	Tin	Newton metres at input shaft
Input speed	nin	Revolutions per minute at input shaft
Efficiency	eta	Decimal fraction, for example 0.96

Geometry and basic formulas

Pitch diameter equals module multiplied by tooth count

d = m × z

Pitch radius equals half the pitch diameter

r = d / 2 = m × z / 2

For ideal concentric geometry the ring tooth count equals sun plus two times planet

Zr = Zs + 2 × Zp

Kinematic core relation

The angular speeds of sun, carrier and ring satisfy the Willis equation

(ωs − ωc) / (ωr − ωc) = − Zr / Zs

From this relation the calculator derives closed form expressions for three operating modes

Planetary gearbox modes

Sun drives carrier, ring fixed

When the ring does not rotate the carrier to sun speed ratio is

ωs / ωc = 1 + Zr / Zs

Expressed in revolutions per minute the numeric ratio used by the tool is

i_sc = 1 + Zr / Zs

Sun drives ring, carrier fixed

With carrier fixed the ring speed is proportional to sun speed with opposite sign

ωr = − Zs / Zr × ωs

Magnitude of the gear ratio reported is

i_sr = Zr / Zs

Carrier drives ring, sun fixed

When the sun is held stationary the ring follows the carrier with a scale factor

ωr = ωc × (1 + Zs / Zr)

The calculator reports the useful inverse form when needed

i_cr = ωc / ωr = 1 / (1 − Zs / Zr)

Practical kinematic and power formulas

Output speed given input speed and numeric ratio equals input speed divided by ratio
Angular velocity in radians per second equals two pi times rpm divided by 60
Output torque approximated with efficiency equals Tin times numeric ratio times eta
Power equals torque multiplied by angular velocity so Pin = Tin × ωin and Pout ≈ eta × Pin

Worked examples

Sun drives carrier

Input setup: Zs = 17, Zp = 21, Zr = 59, N = 3, m = 2.5 millimetres, input speed 1200 rpm, input torque 40 N·m, efficiency 0.95

Check geometry: Zr equals Zs plus two times Zp, 59 equals 17 plus 42, geometry is compatible
Numeric ratio i equals 1 plus Zr divided by Zs, i = 1 + 59 / 17 = 4.4706

Computed results

Carrier speed equals 1200 divided by 4.4706 equals 268.48 rpm
Output torque equals 40 times 4.4706 times 0.95 equals 169.84 N·m
Pitch radii in millimetres, sun 21.25, planet 26.25, ring 73.75
Tangential force on sun equals input torque divided by sun radius in metres, 40 divided by 0.02125 equals 1882.35 newtons
Load per planet equals 1882.35 divided by 3 equals 627.45 newtons

Sun drives ring

Input setup: Zs = 12, Zp = 14, Zr = 40, input speed 2000 rpm, input torque 25 N·m, efficiency 0.94

Numeric ratio equals Zr divided by Zs, 40 divided by 12 equals 3.3333

Ring speed equals 2000 divided by 3.3333 equals 600 rpm
Output torque equals 25 times 3.3333 times 0.94 equals 78.33 N·m

Carrier drives ring

Input setup: Zs = 10, Zp = 15, Zr = 40, carrier speed 80 rpm, input torque 30 N·m, efficiency 0.96

Relative factor equals one minus Zs divided by Zr, one minus 10 divided by 40 equals 0.75

Ring speed equals 80 times 0.75 equals 60 rpm
Numeric inverse ratio equals 1 divided by 0.75 equals 1.3333
Output torque approximated equals 30 times 1.3333 times 0.96 equals 38.40 N·m

Geometry notes and practical cautions

Approximate centre distance between two wheels equals module times the sum of teeth divided by two

a ≈ m × (Z1 + Z2) / 2

Adding more planets reduces load per planet nearly in proportion to the planet count when load distribution is even

📝 Calculator outputs are engineering level estimates. Final design requires tooth profile, face width, material properties and contact analysis to verify strength and fatigue life. Efficiency defaults used reflect typical helical and spur gear behaviour. Worm gear efficiencies are significantly lower.

How to use the planetary gearbox calculator

Choose which element is driven and which element is fixed. That selection controls the calculation mode
Enter integer teeth counts Zs, Zp and Zr. If Zr differs from Zs plus two times Zp you will get a geometry warning
Enter module to obtain pitch diameters and radii in millimetres
Provide input torque and input speed. The calculator returns output speed, output torque and simple load estimates
Adjust efficiency and planet count to explore load distribution and system trade offs

Common pitfalls and tips

Verify the geometric relation Zr equals Zs plus two times Zp. If it fails the gearbox will need an offset or modified parts
A denominator close to zero in the carrier to ring formula produces very large ratios. Avoid teeth combinations that make Zs close to Zr
Keep units consistent. Module and diameters in millimetres, torque in newton metres and speeds in rpm yield correct force estimates

Use the planetary gearbox calculator as a fast, accurate way to screen concepts and compare variants early in the design process. For final component design undertake detailed tooth profile analysis and contact checks before production.