Shimano Nexus-7: how it works, how many teeth the gears have

Previously, I have tried to find “theoretically” the tooth counts of the Nexus-7 internal gear hub. Some nice numbers were obtained, but there were some suspects (you may think yourself what can be wrong with the numbers given in the link). Soon after, I disassembled the real hub and counted the teeth of all rings. Here are the real numbers. This is how the hub works:

  • The hub has two gear systems, each with own planet carrier. Call “first” the one which is closer to the sprocket.
  • The axle is immobile.
  • Each gear system has:
    • Two suns. Either can be immobilized by attaching to the axle with the pawls, activated by the shifter.
    • Two gears on the planet carrier, each meshing with own sun. The smaller gear on the planet carrier meshes with the ring.
    • One ring gear. The ring gear meshes with the smaller gear of the planets.
  • In lower gears 1 and 2, power input from the sprocket applies to the ring of the first gear system. This carrier lowers down the transmission ratio. Depending on which sun is locked to the axle by a pawl, two step-down ratios are obtained. The second gear system is not involved.
  • In higher gears 6 and 7, the first carrier is not involved. Power input from the sprocket applies to the planet carrier of the second gear system. Depending on which sun is locked to the axle by a pawl, two step-up ratios are obtained. The first gear system is not involved.
  • In middle gears 3, 4 and 5, both gear systems are involved, so that the first lowers the ration, the second raises it.

The table explains which exactly rings are engaged, and gives teeth numbers:

Gear Power
Sun Planet
the sun
the ring
Ring Exact
1 Ring1->Planets1 42 14 14 72 12/19 0.632
2 Ring1->Planets1 36 20 20/27 0.741
3 Gears 1 and 6 together 0.843
4 Gears 2 and 6 together 0.989
5 Gears 2 and 7 together 1.145
6 Planets2->Ring2 30 19 14 66 279/209 1.335
7 Planets2->Ring2 36 14 17/11 1.545

Note that in the step-up ring (Ring2), 30+19+14=63, while the ring has 66 teeth. I do not understand how so big difference (3 teeth) makes a working combination.

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