**By George S. Bower and Bill Rollins**

GM already gave us some of the planetary gear set ratios in the referenced SAE article. They gave us the gear ratios with the ring gear locked for both planetary gear set 1 (PG1) and planetary gear set 2 (PG2). The ratios GM gave us are shown in figure 1.

**Figure 1: GM’s Supplied Gear Ratios for PG1 and PG2 with Ring Gear Locked**However, GM did not give us the gear ratios for the PG sets with the sun gear locked. It is necessary to have all the gear ratios in order to build a math model of the Gen 2 power train. Having all the gear ratios allows us to derive the speed relationships for the PG sets.

BillR (and George S.) came up with a rather ingenious way of deriving the other gear ratios. It is based on some simple math that is detailed in an article at woodgears.ca . The short version of the math follows.

We know the gear ratio of the PG sets with the ring gear locked. This gear ratio automatically implies the sun and ring gear tooth count as shown in the equation in figure 2.

**Figure 2: Equation for Number of Ring Gear Teeth as a function of Sun Gear Teeth.**Once we have the sun gear and ring gear tooth counts, the planet tooth count is easily calculated from the equation in figure 3.

**Figure 3: Equation for the Planet Gear Tooth Count as a Function of the Sun and Ring Gear Teeth**We don’t know the sun gear and ring gear tooth counts but we can assume a sun gear tooth count and then calculate what the ring gear and planet tooth counts would be based on the gear ratio that GM gave is. If we write a simple excel spread sheet we can just step thru a range of sun gear tooth counts and calculate the resulting ring gear and planet gear tooth counts as shown in figure 4.

**Figure 4: Excel Spread Sheet for Calculating the number of Ring Gear Teeth and Planet Teeth for an Assumed Sun Gear Tooth Count**We know the solution has to have an integer number of teeth. Once we see a configuration we know that an integral multiple of the solution would also apply. For example, the first solution shown on the spread sheet (13,27,7) would also apply to a gear set with 1,2,3,4,5 ….etc times the 13,27,7 solution.

We also have some GM supplied cutaways of the transmission that can guide us in our solution as shown in figure 5.

**Figure 5: Cutaway of Gen 2 Transmission Showing an Approximate PG2 Planet Tooth Count**We count approximately 7 teeth on PG2’s planet gear which would result in a tooth count of 28 which is very close to one of the solutions on the spread sheet 52,108,28 (multiple of 4). A multiple of 4 on PG1 solution gives a tooth count of 60,112,26.

The solutions of PG2 52,108,27 and PG1 60,112, 26 make sense if we look at a cross section of the Gen 2 gearbox shown below.

**Figure 6: Gen 2 Gearbox Cross Section Showing Relative Sizes of PG2 and PG1**We can see from figure 6 that PG2 and PG1 have approximately the same size ring gears. However PG1 seems to have a slightly larger diameter sun gear than PG2.

Both of these observations from the cross section seem to verify the solution that the authors have derived for the tooth counts on PG2 and PG 1 gear sets.

In addition, we examined this animation of the Gen 2 transmission.

At one point in the video, we can see the sun gear and ring gear of PG2. The tooth count helps verify the values that we obtained for PG2.

Once we have the solution to the tooth counts of both planetary gear sets we can derive all the gear ratios and the speed equations for the 2 gear sets as shown in figure 7.

**Figure 7: Final Gear Ratios and Speed Equations for the Gen 2 Planetary Gear Sets**Now that we have the gear ratios and the speed equations for the Gen 2 gearbox we can build a math model of the Gen 2 Power train. Results of the math model will be presented in an upcoming article.

The authors would appreciate any feedback from other Volt enthusiasts regarding these results.

Reference: SAE Paper 2015-01-1152: “The Next Generation Voltec Extended Range EV Propulsion System”, Conlon, Blohm et all General Motors dated 4/15/2015