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A planetary gear train can provide different speed ratios and rotation directions by changing which of its three main members acts as the input, output, or fixed member.

For a simple planetary gear set, the three main members are the sun gear, ring gear, and planet carrier. The planet gears connect these members and rotate on their own axes while also revolving with the carrier.

The key to calculating a planetary gear train is to look at the relative speeds of the sun gear, ring gear, and carrier. The same gear set can work as a reducer, speed increaser, reversing mechanism, or differential depending on how these members are connected.

Basic Speed Relationship

For a simple planetary gear train, let:

• $z_s$ = number of teeth on the sun gear
• $z_r$ = number of teeth on the ring gear
• $n_s$ = speed of the sun gear
• $n_r$ = speed of the ring gear
• $n_c$ = speed of the planet carrier

The fundamental speed relationship is:$$\frac{n_s-n_c}{n_r-n_c} = -\frac{z_r}{z_s}$$

This equation is often called the Willis equation.

It can also be written as:$$n_s+\frac{z_r}{z_s}n_r = \left(1+\frac{z_r}{z_s}\right)n_c$$

This relationship allows the speed of one member to be calculated when the other two speeds and the relevant tooth counts are known.

For a standard simple planetary arrangement, the tooth counts also satisfy the geometric relationship:$$z_r=z_s+2z_p$$

where $z_p$ is the number of teeth on a planet gear. This relationship comes from the concentric arrangement of the sun gear, planet gears, and ring gear. KHK lists this as one of the basic tooth-number conditions for planetary gear mechanisms.

An important point is that the planet gear tooth count does not directly determine the reduction ratio in a simple planetary gearbox. The ratio mainly depends on the tooth counts of the sun gear and ring gear, while the planet gear size must still satisfy the required geometry.

How the Transmission Ratio Changes

The transmission ratio should be defined before performing any calculation.

In this article:$$i=\frac{n_{in}}{n_{out}}$$

A positive value indicates that the input and output rotate in the same direction. A negative value indicates opposite rotation.

For a simple planetary gear set, the main operating combinations are as follows.

Fixed Member	Input	Output	Transmission Ratio	Direction
Ring gear	Sun gear	Carrier	$i=1+\frac{z_r}{z_s}$	Same
Ring gear	Carrier	Sun gear	$i=\frac{1}{1+z_r/z_s}$	Same
Sun gear	Ring gear	Carrier	$i=1+\frac{z_s}{z_r}$	Same
Sun gear	Carrier	Ring gear	$i=\frac{1}{1+z_s/z_r}$	Same
Carrier	Sun gear	Ring gear	$i=-\frac{z_r}{z_s}$	Opposite
Carrier	Ring gear	Sun gear	$i=-\frac{z_s}{z_r}$	Opposite

The first arrangement is the one most commonly associated with a basic planetary reducer. The ring gear remains stationary, the sun gear receives the input, and the carrier provides the output. In this configuration, the output speed decreases while the output torque increases.

The other arrangements use the same mechanical components but produce different relationships between speed, torque, and direction.

Planetary Gear Ratio Calculation

Consider a simple planetary gear set with:$$z_s=20$$$$z_r=60$$

The ring gear is fixed, the sun gear provides the input, and the carrier provides the output.

Using:$$i=1+\frac{z_r}{z_s}$$

we get:$$i=1+\frac{60}{20}=4$$

The planetary gear set therefore provides a 4:1 reduction ratio.

If the sun gear rotates at:$$n_s=1500\ rpm$$

the carrier speed becomes:$$n_c=\frac{n_s}{i}$$$$n_c=\frac{1500}{4}=375\ rpm$$

So the gear set changes the input from 1500 rpm to 375 rpm.

This example also shows why planetary gearboxes are useful in compact reduction applications. A fixed ring gear keeps the system coaxial while the planet gears transfer the load between the sun gear and carrier.

Output Speed Calculation

Once the transmission ratio is known, output speed is straightforward:$$n_{out}=\frac{n_{in}}{i}$$

For a reduction ratio of 4:1:$$n_{out}=\frac{1500}{4}=375\ rpm$$

For a speed-increasing arrangement, the ratio is less than 1. Such as, when the ring gear remains fixed and the carrier drives the sun gear:$$i=\frac{1}{1+z_r/z_s}$$

With $z_s=20$ and $z_r=60$:$$i=\frac{1}{4}=0.25$$

If the carrier rotates at 375 rpm, the sun gear rotates at:$$n_s=\frac{375}{0.25}=1500\ rpm$$

The same gear set has therefore changed from a 4:1 reducer to a 4:1 speed increaser simply by changing the input and output members.

Planetary Gear Torque Calculation

Speed and torque change in opposite directions when a planetary gear train reduces speed.

The basic power relationship is:$$P=T\omega$$

where:

• $P$ = mechanical power
• $T$ = torque
• $\omega$ = angular speed

For an ideal gear train with no losses:$$T_{out}=T_{in}i$$

Real planetary gearboxes have mechanical losses, so actual output torque is lower:$$T_{out}\approx T_{in}i\eta$$

where $\eta$ is the transmission efficiency.

Using the previous 4:1 reduction example, suppose the input torque is:$$T_{in}=10\ N\cdot m$$

and the transmission efficiency is 95%:$$T_{out}=10\times4\times0.95$$$$T_{out}=38\ N\cdot m$$

Under ideal conditions, the output torque would be 40 N·m. The difference represents the power lost through friction, bearing losses, gear meshing, and other mechanical effects.

Planetary gear systems can achieve high torque capacity because multiple planet gears share the transmitted load. The actual load-sharing level depends on the number of planets, manufacturing accuracy, carrier stiffness, assembly conditions, and the design of the gear set.

What Happens When Two Members Are Locked Together?

A planetary gear set can also provide a 1:1 drive.

When any two of the three main members are locked together, the third member must rotate at the same speed because the relative motion between the gears disappears.

Therefore:$$i=1$$

The sun gear, planet gears, ring gear, and carrier rotate together as one unit.

This type of arrangement is useful when a transmission needs a direct-drive condition in addition to its reduction or speed-increasing modes. A practical planetary drive can use clutches or other locking mechanisms to change between different operating conditions.

Why Planet Gear Teeth Still Matter

Although the planet gear tooth count does not directly appear in the basic reduction ratio for a simple planetary arrangement, it remains important to the design.

The tooth numbers determine the physical size and position of the gears. For a standard simple planetary arrangement:$$z_r=z_s+2z_p$$

The number and spacing of planet gears also affect assembly. When several planets share the load, their positions must allow correct meshing with both the sun and ring gears.

For example, KHK specifies tooth-number and planet-spacing conditions when configuring stock gears into a planetary mechanism.

Planet gear dimensions also affect tooth strength, bearing size, clearance, and the available space inside the ring gear. Therefore, gear ratio is only one part of the planetary gear design.

FAQ

What is the basic formula for a planetary gear train?

For a simple planetary gear train, the basic speed relationship is:$$\frac{n_s-n_c}{n_r-n_c} = -\frac{z_r}{z_s}$$

It relates the speeds of the sun gear, ring gear, and carrier to the tooth counts of the sun and ring gears.

How do you calculate the ratio of a planetary gear?

First identify the fixed member, input member, and output member. Then substitute the corresponding speeds into the planetary gear speed equation.

For the common arrangement with the ring gear fixed, sun gear input, and carrier output:$$i=1+\frac{z_r}{z_s}$$

Does the number of planet gear teeth affect the gear ratio?

Not directly in a simple planetary gear set. The basic ratio depends on the sun gear and ring gear tooth counts. However, the planet gear tooth count still determines the geometry and must satisfy the required meshing and concentricity conditions.

How do you calculate planetary gear output torque?

Start with the input torque and transmission ratio:$$T_{out}\approx T_{in}i\eta$$

where $\eta$ represents transmission efficiency. For an ideal calculation, set $\eta=1$.

Can the same planetary gear set provide both reduction and speed increase?

Yes. Changing which member acts as the input and which member remains fixed changes the transmission ratio. The same planetary gear set can therefore work as a reducer, speed increaser, or reversing mechanism.

What happens when two planetary gear members are locked together?

The planetary gear set operates at a 1:1 ratio because all three main members rotate at the same speed. The gear set effectively moves as a single unit.