# How to design planetary gears?

I am currently designing a planetary gear set, and I have designed the sun gear with a module of 2.25 and 35 teeth. Is there any formulae to calculate the internal gears size and amount of teeth?

Ideally I had like to have 3 small planet gears and 1 central gear.

For any standard planetary gearset, the following formula relating the numbers of teeth of the sun, annulus (internal) and planet gears applies:

$$N_s + 2N_p = N_a$$

Where $N$ is the number of teeth, and the subscripts correspond to sun, planet and annulus gears respectively. This is a purely geometric result, and comes from the fact that the pitch circles of the gears should be tangent to one another, and that the diameters of the pitch circles are directly proportional to the number of teeth.

Note how, with the number of teeth of the sun gear given to you, the number of teeth of the annulus gear can be varied by selecting the number of teeth of each planet. For example, given a sun gear of 35 teeth:

• choosing a planet with 20 teeth would yield an annulus with $35+2\times 20=75$ teeth.
• choosing a planet with 35 teeth would yield an annulus with $35+2\times 35=105$ teeth.

While there is a freedom to select the number of teeth on the planet and annulus, note that this selection will impact the value of the gear ratio. For example, having the sun/planet/annulus with 35/20/75 teeth will yield a 22:7 (i.e. 3.14:1) gear ratio from the sun gear to the planet carrier while fixing the annulus gear, and 35/35/105 teeth will yield a 4:1 gear ratio.

Finally, the diameter of each of the gears' pitch circles, including the annulus gear, is calculated as follows:

$$d=N\times m$$

where $m$ is the module of the gear.

The annulus gear's outermost diameter is arbitrary, but it should be kept greater than the root diameter, which for standard internal gears is as follows:

$$d_{root} = (N+2.5)\times m$$