MechClarityMechanical Engineering Calculators

Gears · Simple planetary kinematics

Planetary Gear Ratio Calculator

Calculate simple planetary gear ratio, output RPM, direction, planet teeth, and equal-spacing phasing for sun, ring, or carrier configurations.

Reference calculator #008

Define the simple planetary arrangement

Inputs stay in your browser. Values are normalized to canonical units before calculation.

Whole-number teeth on the central external sun gear.

Whole-number internal teeth on the surrounding ring gear.

Identical planets intended at equal angular spacing.

The selected member is held stationary at zero speed.

The input must differ from the fixed member; the third member becomes the output.

Enter a positive rotational speed; output direction is reported separately.

Calculated output

Results

planetary-gear-ratio/1.0.0
Carrier output speed300 rpm
Input / output speed ratio
4:1 reduction
Output member
Carrier
Output direction
Same as input
Calculated planet teeth
20 teeth
Fundamental ring / sun ratio
3:1
Equal-spacing phase check
Pass — quotient 20

Valid ideal simple-planetary kinematic result

Simple planetary gearset kinematic diagramA central sun, surrounding internal ring, equally spaced planets, and carrier arms with signed member speeds and fixed, input, and output roles.S 20TR 60T4 planets · 20 teeth eachMember speeds and rolesSun: 1200 rpmRing: 0 rpmCarrier: 300 rpmFixed: RingInput: SunOutput: Carrieri = 4:1 reduction
Pitch-circle schematic only. Planet tooth profiles, clearances, carrier pins, and rotation animation are not manufacturing geometry.
Scope and assumptions
  • The model is one simple coaxial planetary set with one sun, one internal ring, identical planets, and one rigid carrier.
  • Sun, planet, and ring gears share a compatible module, pressure angle, tooth system, and operating geometry.
  • One member is stationary, the selected input speed is a positive magnitude, and the third member is the output.
  • Reported ratio is the positive input-speed magnitude divided by output-speed magnitude; direction is reported separately.
  • Kinematics are ideal. Efficiency, torque, power split, load sharing, bearing loads, strength, backlash, deflection, and dynamics are not calculated.

Calculation engine: planetary-gear-ratio/1.0.0

Simple planetary speed equation

Zsωs + Zrωr = (Zs + Zr)ωc   ·   Zp = (Zr − Zs) / 2   ·   i = |ωin| / |ωout|

The calculator solves one ideal, simple, coaxial planetary gearset. Select one stationary member and one positive-speed input; the remaining member becomes the output. Signed member speeds determine direction, while the displayed ratio is always the positive input-speed magnitude divided by the output-speed magnitude.

Symbol Meaning Calculator convention
Zs Sun gear tooth count Positive whole number
Zr Internal ring gear tooth count Greater than Zs
Zp Calculated planet gear tooth count (Zr − Zs) / 2
ωs, ωr, ωc Signed sun, ring, and carrier speeds rpm
i Input/output speed ratio Positive magnitude

Six supported member configurations

Fixed member Input Output Input/output ratio Direction
Ring Sun Carrier 1 + Zr/Zs Same
Ring Carrier Sun Zs/(Zs + Zr) Same
Sun Ring Carrier 1 + Zs/Zr Same
Sun Carrier Ring Zr/(Zs + Zr) Same
Carrier Sun Ring Zr/Zs Opposite
Carrier Ring Sun Zs/Zr Opposite

Because this page defines ratio as input speed divided by output speed, a result above 1:1 is a reduction and a result below 1:1 is a speed increase. Some catalogs use different ratio conventions; compare member assignments and formula definitions before copying a value.

Worked ring-fixed example

For Zs = 20, Zr = 60, a fixed ring, and sun input at 1,200 rpm:

  1. Planet teeth: Zp = (60 − 20) / 2 = 20.
  2. Carrier speed: ωc = 20 × 1,200 / (20 + 60) = 300 rpm.
  3. Speed ratio: i = 1,200 / 300 = 4:1 reduction.
  4. The carrier rotates in the same direction as the sun.

If the carrier is fixed instead, the same 1,200 rpm sun input produces a ring speed of −400 rpm: a 3:1 reduction with opposite direction.

Planet spacing and assembly check

The basic equal-spacing phasing check requires (Zs + Zr) / number of planets to be a whole number. The default 20/60 set with four planets gives 80 / 4 = 20, so it passes this condition.

Passing does not prove that the planets fit physically. Adjacent-planet tip clearance, tooth thickness, profile shift, center-distance modification, carrier pin diameter, bearing space, manufacturing tolerances, and assembly method still require review. A non-whole quotient is reported as a warning rather than hiding the valid kinematic speed result.

Engineering scope and limitations

This V1 tool assumes compatible module, pressure angle, tooth system, and operating geometry for one sun, one internal ring, identical planets, and one rigid carrier. It excludes:

  • compound, stepped-planet, Ravigneaux, Wolfrom, differential, and multi-stage arrangements;
  • torque, power split, efficiency, friction, inertia, dynamic response, and load sharing;
  • tooth and rim strength, pitting, scuffing, bearings, shafts, carrier deflection, heat, and life;
  • backlash, profile shift, detailed interference, undercut, tolerances, accuracy grades, lubrication, and noise.

Use the result for transparent preliminary kinematics, then validate geometry, rating, and manufacturing details before approving a design.

Frequently asked questions

What equation does the planetary gear ratio calculator use?

It uses the simple planetary Willis relationship Zsωs + Zrωr = (Zs + Zr)ωc, where Z is tooth count and ω is signed angular speed for the sun, ring, and carrier.

What is the ratio when the ring gear is fixed?

With the ring fixed and the sun as input, the carrier reduction ratio is 1 + Zr/Zs. Reversing input and output gives the reciprocal speed ratio.

How are planet gear teeth calculated?

For a standard concentric simple planetary set with a common module and compatible tooth system, Zp = (Zr − Zs) / 2. The difference between ring and sun teeth must therefore be even.

When does the output rotate opposite to the input?

In this simple model, the sun and ring rotate in opposite directions when the carrier is fixed. Configurations with the sun or ring fixed produce the same input and output direction.

Does this calculator size or rate a planetary gearbox?

No. It does not calculate torque, efficiency, load sharing, tooth strength, bearing reactions, backlash, heat, lubrication, life, interference, or manufacturability.

References and review status

Reviewed . References support formula, terminology, and scope checks; they do not imply endorsement, certification, or standards conformity.