Planetary Gear Ratio Calculation and Formula

A planetary gear ratio is calculated from the tooth counts of the sun, ring, and planet gears, together with which member is held, driven, and used as the output. Enter the sun and ring tooth counts and choose the fixed and input members to calculate the signed speed ratio and the output-to-input speed relationship.

Formula

For a simple planetary set, the Willis equation relates the speeds of its three central members:

(ωs − ωc) / (ωr − ωc) = −Nr / Ns

Here, ωs is the sun speed, ωr is the ring speed, and ωc is the carrier speed. Speeds can be in revolutions per minute (rpm), radians per second, or another consistent angular-speed unit. Ns is the number of sun teeth and Nr the number of internal ring teeth; tooth counts are dimensionless.

The calculator assumes one member is fixed and the remaining two are the input and output. Its signed ratio is ωout / ωin. A negative result means output and input turn in opposite directions; its reciprocal is the commonly quoted reduction ratio, input speed divided by output speed.

• Carrier fixed: ωr/ωs = −Ns/Nr when the sun is input and ring is output; swapping input and output gives −Nr/Ns.
• Sun fixed: ωc/ωr = Nr/(Ns + Nr) when the ring is input and carrier is output.
• Ring fixed: ωc/ωs = Ns/(Ns + Nr) when the sun is input and carrier is output.

More generally, for a fixed member and a specified input, the output-to-input ratio is found by setting the fixed member’s speed to zero in the Willis equation and solving for the remaining speed. The selected members must be different; fixing the input would leave no driven motion in this simple calculation.

Worked Example

Use the calculator defaults: a 30-tooth sun, a 70-tooth ring, carrier fixed, and sun input. The ring is the output. Substituting the fixed carrier speed, ωc = 0, into the Willis equation gives:

1. (ωs − 0) / (ωr − 0) = −70 / 30.
2. Rearrange for output divided by input: ωr / ωs = −30 / 70.
3. Calculate the signed ratio: ωout/ωin = −0.428571…, which rounds to −0.429 to three decimal places.

Thus, for every 1 rpm of sun speed, the ring turns at about 0.429 rpm in the opposite direction. The equivalent reduction ratio in input-to-output form is 70/30 = 2.333:1. If the sun turns at 1,000 rpm, the ideal ring speed is approximately −429 rpm.

How a Simple Planetary Set Works

A basic planetary, or epicyclic, gear set has a central sun gear, an internally toothed ring gear around it, and one or more planet gears meshing with both. The planets are mounted on a carrier. The carrier can rotate, so planetary gears combine gear rotation about their own axes with motion around the sun.

The Willis equation accounts for that moving carrier. This is why a planetary ratio cannot be determined from tooth counts alone: the same gears can produce different ratios, directions, and outputs depending on which member is held and which member is driven. For a standard simple set, compatible tooth geometry also requires the ring-to-sun tooth difference to accommodate the planet gear arrangement; real designs must satisfy the geometry and assembly constraints, not just a speed equation.

Using the Ratio and Choosing Members

Use the signed result when direction matters, such as checking whether a shaft reverses. Use the positive reduction ratio when comparing speed reduction in the familiar “input to output” format. For example, an input-to-output reduction of 3:1 means the output turns at one-third the input speed in magnitude, assuming the ideal kinematic relationship.

To choose a configuration, identify the desired function first: a fixed carrier, sun, or ring; which part receives power; and which part drives the load. Then use tooth counts to calculate its ratio. If a target ratio is not met, changing the fixed member may be simpler than changing gears, but it also changes output direction and which shaft must be supported or driven.

These calculations describe speed, not torque capacity or efficiency. In an ideal lossless gear train, a speed reduction corresponds to a torque increase by the inverse speed ratio; actual torque and efficiency depend on tooth design, materials, lubrication, bearings, loading, and losses. Do not treat the ratio as a rating for a particular gearbox.

Common Mistakes and Limits

• Confusing ratio conventions: “gear ratio” may mean output/input or input/output. State the convention and direction whenever reporting a result.
• Ignoring the fixed member: tooth counts do not select a unique ratio unless the held, input, and output members are specified.
• Dropping the sign: a negative signed speed ratio indicates opposite rotation under the chosen positive direction convention. The magnitude alone does not show direction.
• Mixing tooth counts and dimensions: use actual integer teeth, not pitch diameters or gear module values in the tooth-count formula. Keep speed units consistent when calculating rpm.
• Overlooking real construction: the equation assumes an ideal, simple planetary arrangement. Compound planetary systems, additional stages, clutches, flexible parts, and losses require further analysis.

Round only after calculating the ratio; premature rounding can affect later speed estimates. If two members are coupled to external gearing or a motor, include those stages separately. The calculator does not determine whether a design is strong enough, quiet, lubricated correctly, or manufacturable.

Frequently Asked Questions

What is the formula for a planetary gear ratio?
The general speed relation for a simple set is (ωs − ωc)/(ωr − ωc) = −Nr/Ns. Set the fixed member’s speed to zero, then solve for the chosen output speed divided by input speed.

How do you calculate planetary gear ratio with the ring gear fixed?
For sun input and carrier output, the ratio is ωc/ωs = Ns/(Ns + Nr). For example, with 30 sun teeth and 70 ring teeth, the carrier turns at 30/100, or 0.30, of sun speed in the same direction.

What happens when the carrier is fixed?
The sun and ring rotate in opposite directions. With the sun as input and ring as output, ωr/ωs = −Ns/Nr. Reversing the input and output gives the reciprocal signed ratio.

Does a planetary gear ratio depend on the number of planet gears?
For an ideal simple planetary set, the basic speed ratio depends on sun and ring tooth counts and on the member roles, not on how many planets are installed. Planet count affects load sharing, layout, and design constraints.

Is a 3:1 planetary gear ratio a speed increase or reduction?
It depends on the convention. If 3:1 means input speed to output speed, it is a threefold reduction. Always identify the input, output, fixed member, and whether the ratio is input/output or output/input.

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in Ns | Sun gear teeth | - | 30 | 1
in Nr | Ring gear teeth | - | 70 | 1
sel fixed | Fixed member | Carrier=1, Sun=2, Ring=3
sel input | Input member | Sun=1, Ring=2, Carrier=3