The In(put)s and Out(put)s of Planetary Gears

From hybrid cars to space-bound robotics, one of engineering’s most versatile mechanisms hides in plain sight: the planetary gear. By letting gears orbit like miniature solar systems, this compact design packs serious torque, smooth motion, and multiple speed options into places where traditional gear trains simply won’t fit. In this post, we’ll explore how they work, how to design them, and why they’ve powered technology from 19th-century watchmaking to modern wind turbines.

Planetary gears, also known as epicyclic gear trains, are a unique type of gear system widely used for their compactness and versatility. In a planetary gearset, multiple gears—called planets—rotate around a central gear known as the sun. These planets are held in place by a carrier and mesh with both the sun gear and an outer ring gear (sometimes called the annulus). This ingenious arrangement allows planetary gear systems to deliver high torque in a small space and enables multiple gear ratios or directions of rotation.

Specific examples highlight how essential planetary gears are:

• In the Toyota Prius hybrid, planetary gears form the heart of the Hybrid Synergy Drive’s power split device. This device seamlessly blends power from the gasoline engine and the electric motor, allowing the car to switch between driving modes efficiently.

• Automatic transmissions in many cars use planetary gearsets to shift between gears smoothly and compactly.

• In wind turbines, planetary gearboxes step up rotational speed from the low-speed rotor to the high-speed generator.

• Aerospace actuators and robotic arms employ planetary gears for precise control in limited space.

• Consumer devices like wristwatches utilize miniature planetary gear trains to synchronize complex movements.

Planetary gears are treasured in engineering because they can handle higher loads, offer multiple speeds and torque outputs, and make efficient use of available space—all while providing smooth and reliable operation.

Planetary gears can be designed and manufactured with standard gear-making techniques. Like any other gears, the pitch (tooth spacing, typically expressed as diametral pitch in imperial units or module in metric), must be the same for all teeth on every gear for them to mesh properly. For more information on how pitch is used, please see our blog post here:

How do we know how many teeth to use? The approach is simple:

Decide on how many teeth you would like for your sun and planet gears, and the ring gear teeth will be solved for as follows:

Where:

R= The number of teeth on the ring gear
P= The number of teeth on each Planet Gear
S= The number of teeth on the Sun Gear

Similarly, if needing to solve for the teeth on the sun gear with determined Ring and Planet teeth, the following may be used:

For solving for planetary teeth, with known ring and sun teeth, the following equation can be used:

In a planetary gear train, gear ratio depends on which component (sun, ring or carrier) is fixed, which is the input, and which is the output. Below is a chart of formulas for each of these situations:

The formula that describes the relationship between planetary gears, now called Willis’s equation, dates back to Robert Willis, a 19th-century British engineer and professor at Cambridge. In 1841, Willis published Principles of Mechanism, a landmark text on the geometry and motion of machine components. One of his key contributions was a general kinematic formula for epicyclic (planetary) gear trains, allowing the relative speeds of any two members to be calculated when the third is held or moving at any speed.

This equation unified earlier ad-hoc methods for analyzing complex gear trains, making it possible to treat fixed, rotating, and moving carriers in one expression. While modern derivations often use vector or relative velocity notation, the underlying relationship is essentially unchanged from Willis’s original work nearly two centuries ago.

The Willis equation has several forms, the most classic and well-known form is as follows:

Where:

ωs,ωr,ωc are angular velocities of the sun, ring, and carrier

S and R are the sun and ring tooth counts

The minus sign denotes opposite rotation between the sun and ring

It is a general kinematic relationship that applies to any planetary gear train. The equation can also be written to discuss numbers of teeth. Let us rearrange the Willis Equation to be written in terms of tooth numbers:

Cross Multiplying:

Expanding:

Collecting Carrier Terms On One Side:

Substitute ωi=2πTi and cancel 2π:

Reorder:

Where

 R=The number of teeth on the ring gear

S=The number of teeth on the Sun Gear

Tc=Number of turns (or velocity) on the Planet Carrier

Tr=Number of turns (or velocity) of the ring gear

Ts=Number of turns (or velocity) of the sun gear

The Willis equation can be used to derive the formulas for gear ratios in a planetary system.

When we hold a gear constant, it turns to zero. For example, if we hold the planet carrier stationary, the above equation will have zero values for planet gears:

Becomes:

Simplifying:

We will arrange this formula by putting the sun gear variables (S and TS) on one side and Ring gear on the other (R and Tr).

We can find the speed ratio by dividing both sides by -S*Ts

Simplifying:

To make things simple, we can move the tooth numbers to the other side of the equation to express the speed to numbers of teeth.

If we wish to know the gear ratio of our planetary system with a stationary planetary gear carrier, we simply divide the number of teeth of the sun gear by the number of teeth of the ring gear. The negative sign indicates opposite direction of rotation; the magnitude gives the speed ratio. For simplicity, the formula above is not shown with the negative sign.

Perhaps we would like to hold the ring gear constant and examine the gear ratio between the sun gear and planet carrier. For this, we will use our original formula:

We will zero out the ring gear velocity:

Now to find the gear ratio, we will move all terms of R and S on one side of the equation:

So if we wish to find the ratio between the sun and planet gear, simply divide the number of teeth of the ring gear by the sun gear and add 1.

In the final case, the sun gear will be stationary and we will calculate the ratio between the planetary gear carrier and ring gear:

Sun gear velocity moves to zero:

Diametral pitch plays a crucial role in the loads the system can handle. Planetary gears, such as those used in winches, hoists and heavy-duty drives, often require sizing based on load capacity. Below is a basic formula as a first pass sizing rule that can help determine the minimum diametral pitch based on loads in your system, noting that the output is an approximation and doesn’t account for shock loading, gear manufacturing defects or misaligned gears. Users should increase or decrease their diametral pitch as needed.

Where:

Inputs:

• S = allowable bending stress (psi) — often ~1/3 of material’s ultimate tensile strength for safety

• F = face width (inches)

• Y = Lewis form factor (unitless, varies with pressure angle and tooth count)

• N = number of teeth on the gear

• T = torque (in-lbf)

Output:

• DPmin = smallest safe diametral pitch to handle the load without exceeding bending stress

Lewis form factor is meant for use on spur gears, and so it is an approximation when applied to planetary gears. As a quick reference, approximate Lewis form factors for 20° tooth pressure angles are:

A vital property of your gearset is whether or not the planetary gears are in phase and evenly spaced. Consider the following picture:

You may notice that the black reference lines are all 120 degrees apart and evenly spaced, however the planet gears are not quite evenly spaced, and the teeth of each gear are in different amounts of engagement with each other. If the gears have the same angle between their centers, they are evenly spaced, and if the teeth have the same engagement everywhere, they are in phase. The example above is neither in phase nor evenly spaced.

In the following image, you can also see that planetary spacing is even, the planetary gears are all 120° apart. If you look closely, the teeth are in different phases to each other. This is an example of an evenly spaced out of phase geartrain.

Finally, in the last image, notice that all the planetary gears are 120 degrees apart as before, but now the teeth are all engaged in the same way with their surrounding gears. This is an example of an in phase, evenly spaced planetary set.

Whether the planets are evenly or unevenly spaced and whether their gear teeth are in phase or out of phase—are controlled almost entirely by your choice of tooth counts on the sun and ring gears. Get those numbers right, and you can control the look, feel, and function of your planetary set.

Here are the three main cases you’ll run into when designing planetary gears:

This is the “classic” look: planets placed at equal angles around the carrier, all with their gear teeth lined up in the same orientation relative to the sun.

How to design for it:

1. Add the sun gear’s teeth to the ring gear’s teeth. This total must be exactly divisible by the number of planets you want. This ensures equal angular spacing without tooth interference.

2. The ring gear’s tooth count must also be divisible by the number of planets. This second rule locks the planets into the same tooth phase.

Result: Symmetry, balance, and satisfying alignment. Great for visually appealing mechanisms or when you want the planets to engage the sun in exactly the same way.

At first glance, this looks similar to the in-phase layout: planets equally spaced around the carrier. But look closely at the teeth, and you’ll see that each planet is offset relative to the others.

How to design for it:

1. Again, the total of sun + ring teeth must be divisible by the number of planets to allow even spacing.

2. The ring gear’s tooth count must not be divisible by the number of planets.

Result: Even spacing, but the planets are out of phase. This can be useful for reducing the chance that all planets engage the sun gear teeth at the same point, which can help spread out wear patterns or alter noise and vibration characteristics. The exact phasing pattern depends on the remainder when the ring tooth count is divided by the number of planets.

Sometimes your ratio requirements or packaging constraints mean the evenly spaced condition just isn’t possible. In that case, you can still build a working planetary gearset, but the planets will need to be at irregular angles around the carrier.

How to design for it:

• Choose sun and ring gear tooth counts where the total (sun + ring) is not divisible by the number of planets.

• Planets are placed using spacing “ticks” based on that total tooth count, which means the angles between planets will differ.

• In certain special cases, you can use an “X” layout—pairs of planets directly opposite each other—if the tooth total divided by the number of planets leaves a half-tooth remainder.

Result: This layout is mechanically functional but lacks the visual symmetry of evenly spaced sets. Phasing will always be mixed as gears rotate—some planets will be closer to in phase, others further out—and load sharing may be less uniform.

Here is a summation table of the pros and cons of each configuration:

Planetary gears are more than just a clever mechanical trick—they’re a design solution that has stood the test of time, from 19th-century theory to 21st-century technology. From little baby teeth to large high strength gears, their unique geometry delivers high torque in a compact space, enables multiple speed ratios, and offers flexible configurations to meet a wide range of engineering needs. Whether you’re optimizing a hybrid drivetrain, building a precision robotic joint, or designing a gearbox for extreme loads at high input speeds, understanding tooth counts, phasing, and spacing rules gives you the power to control not just how your gearset works, but how smoothly, quietly, and efficiently it runs whether you are using 32 teeth or 320 teeth. Master the fundamentals, and planetary gears become less of a mystery and more of a tool you can bend to your will.