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The planetary gear set mostly consists of Four main parts as shown in Figure Below.

• Sun gear
• Planet gear
• Ring gear
• Carrier

Each of these components can be the input, the output, or can be held stationary. Depending upon input, output, and stationery element the following gear ratios can be calculated.

S.No.	Input	Output	Stationary	Ratio
A	Sun (S)	Planet Carrier (C)	Ring (R)	1 + R/S
B	Planet Carrier (C)	Ring (R)	Sun (S)	1 / (1 + S/R)
C	Sun (S)	Ring (R)	Planet Carrier (C)	-R/S

Holding any two components together will deliver 1:1 gear reduction. The first gear ratio listed above is a reduction where the output speed is slower than the input speed. The second is an overdrive, where the output speed is faster than the input speed. The last is also reduction but with opposite output direction (hence represented by negative sign).

As most of the industrial applications demand speed reduction, our concern is with only the first type of reduction where the ring gear is held stationary. Input is given through the sun gear and output is the planet carrier.

Let’s look in detail how the  Gear ratio is derived for this kind of reduction

1. From the basic law of planetary gear set we have,

   

r_r= r_s + dp,

r_r=r_s+ 2r_p

2r_p= r_r-r_s        – (Eqn. 1)

Where, r_r= Radius of Ring Gear,

r_s = Radius of Sun Gear,

d_p = Diameter of Pinion Gear

r_p = Radius of Planetary Gear

Now, we know Linear velocity ‘V’= Angular velocity x Radius = N x r  (as ω = 2πN)

Therefore,  For an object moving in circle its Linear velocity is V= N x r

2. In planetary gear set, as the input is connected to sun gear and output is given to carrier

Gear ratio ‘i’= N_s/N_c

Where, Ns= Speed of Sun Gear, Nc= Speed of Carrier

3. Linear Velocity of the carrier is angular velocity x distance from the centre

V_c= N_c x (r_s + r_p)      – (Eqn. 2)

4. The instantaneous linear velocity of the point of contact between sun and planet is

V_s= N_s x r_s        – (Eqn. 3)

Also, instantaneous velocity equals to twice the velocity of carrier

V_s= 2V_c= 2N_c x (r_s + r_p)        – (Eqn. 4)

Equating Eqn. 3 and Eqn.4 we get,

N_s x r_s = 2 x N_c x (r_s + r_p)

i = Ns/Nc = 2 x (r_s + r_p) / r_s

= 2r_s + 2r_p / r_s

From Equation 1 we have,

2r_p= r_r – r_s,

=(2 r_s + r_r – r_s)/ r_s

= (r_s + r_r) / r_s

=1 + r_r / r_s

= 1 + Zr/Zs

Here, Zr is No. Of teeth on Ring Gear, Zs is No. Of teeth on sun gear. Hence Gear ratio for planetary gear set is, Ratio of Number of teeth of ring gear to number of teeth of Sun gear plus One. In this way the gear ratio for planetary gear set is derived…!!