The torque translator

One equation, three members

Every planetary set you will ever encounter, in any gearbox, in any vehicle, obeys this. Learn it once.

A single planetary set has exactly three members that connect to the outside world: the sun, the ring, and the carrier (which holds the planet gears). The planets themselves never connect outward — they are idlers, meshing between sun and ring, doing nothing but transferring motion.

Where the equation actually comes from. Picture the carrier momentarily frozen — held still by hand. With the carrier fixed, the planets are just idler gears sitting in fixed locations, meshing sun-to-planet-to-ring exactly like an ordinary external/internal gear pair. In that frozen-carrier frame, the familiar gear relationship holds: ω_sun × teeth_sun = −ω_ring × teeth_ring (negative because an external gear meshing with an internal gear through an idler reverses direction once, then reverses back — net result, sun and ring counter-rotate relative to the carrier).

Now unfreeze the carrier and let it spin at its own speed ω_carrier. Every other member's speed has to be measured relative to the carrier, because that's the frame in which the simple gear relationship above is actually true. So you substitute (ω_sun − ω_carrier) for what was "ω_sun" and (ω_ring − ω_carrier) for what was "ω_ring", which gives:

Expand that and rearrange, and you land on the standard form used everywhere in transmission literature:

Three rotating members, one equation, two unknowns once you fix the third. Fix the speed of any one member (usually zero — grounded to the case), drive any second member as input, and the third is forced to be the output. There is no freedom left. That is the entire mechanism of every automatic transmission ever built.

Notice the reduction ratio in ring-fixed mode is always (1 + R) — never just R. That extra "+1" is the carrier's own geometric contribution: even with R close to zero (a tiny ring barely bigger than the sun), the carrier still can't exceed half the sun's speed, because the planets are physically orbiting as well as spinning. This is the detail that trips people up when they try to back-calculate tooth counts from a target ratio — the relationship is never a simple gear-pair ratio, it always carries that "+1" term from the orbiting motion.

Torque, not just speed. You called this a torque translator, and that's the more useful mental model than "gearbox" because torque is what actually moves the bus. In an ideal, lossless gear train, power in equals power out: P = T × ω is conserved. So if speed drops by a factor of 3 going through a 3:1 reduction, torque must rise by that same factor of 3 to keep power balanced. That's not a separate law — it falls straight out of the same Willis relationship, because the instant you fix the ratio of ω_in to ω_out, the ratio of T_out to T_in is just its reciprocal. A 3:1 speed reduction is, automatically, a 3:1 torque multiplication (minus a few percent for mesh friction). Real boxes lose roughly 2–4% per meshing planetary stage to friction and windage — worth knowing, but small enough that the Willis-equation math above is the dominant effect by a wide margin.

Three ways to wire one set

Same sun, same ring, same planets. Only the grounding point changes — watch the telemetry strip above as you switch modes.

Each mode reduces to its own clean formula straight from the Willis equation. With R = teeth(ring)/teeth(sun) and the default tooth counts above (sun 36, ring 72, so R = 2.0):

That gap between sun-fixed (1.50:1) and ring-fixed (3.00:1) on identical hardware is exactly why automatic transmission designers treat "which member do I ground" as the primary lever, and tooth count tuning as the secondary, fine-grained lever. Swapping the grounded member alone gets you into a completely different reduction regime.

Why three planets, and why not more. The diagram above defaults to three planet gears spaced 120° apart, which is the most common count in road-car transmissions. Two constraints govern this choice, and they fight each other:

• Load sharing. More planets split the torque across more simultaneous tooth contacts, so each individual tooth carries less load — this is why heavy-duty truck, bus, and tank final-drive planetaries commonly use 4, 5, or even 6 planets instead of 3. More planets, lower stress per tooth, for the same torque throughput.
• Assembly and timing. Planets aren't free to sit at arbitrary angles — for N evenly-spaced planets to all mesh correctly with both sun and ring simultaneously, the tooth counts must satisfy an integer constraint: (teeth_sun + teeth_ring) must be evenly divisible by N. Try to force a planet count that doesn't satisfy this and the gearset simply will not assemble — the last planet won't drop into mesh.

With sun = 36 and ring = 72 (sum = 108), valid planet counts are any N that divides 108 evenly while leaving room for the planets to physically clear each other: 3, 4, or 6 all work arithmetically; 5 does not (108/5 isn't an integer) without changing tooth counts slightly. This is the first place where your CPL-honed checklist instinct pays off directly — tooth count selection is a constraint-satisfaction problem before it's ever a ratio-optimization problem.

Bolt two sets together

The Simpson gearset — two planetary sets sharing a common sun shaft.

A single set gives you a handful of ratios from one grounding scheme. To get a useful spread — first gear deep enough to launch a loaded vehicle, top gear tall enough for cruise — production transmissions compound two or more planetary sets, sharing members between them so the second set's behaviour is shaped by the first.

The classic arrangement, found inside the Ford C4/C6, GM Turbo-Hydramatic, and Chrysler Torqueflite three-speeds, ties the sun gears of a front and rear set to one common shaft, and feeds the front carrier's motion into the rear ring:

Deriving the compound ratios by hand. The console above uses sun A = 36 / ring A = 72 (R_A = 2.0) and sun B = 32 / ring B = 76 (R_B = 2.375). Walk through 1st gear with those numbers, starting from the single-set ring-fixed formula you already know:

2nd gear uses the front set's ring-fixed relationship instead, with R_A = 2.0:

The step between 1st and 2nd here is 3.37/3.00 ≈ 1.12 — a deliberately small jump. Why the two sets use different tooth ratios at all (R_A = 2.0 vs R_B = 2.375) rather than identical sets: if both sets shared the exact same R, 1st and 2nd gear would collapse to suspiciously close ratios with no clean way to widen the spread, and you'd be leaning entirely on adding more sets just to get usable separation. Giving the rear set a slightly larger R than the front lets one pair of compounded sets alone produce a wider, more usable 1st-to-3rd spread without a third planetary stage — this is the actual design lever transmission engineers pull when laying out a Simpson box, not arbitrary part selection.

Adding overdrive

Same Simpson hardware. One more clutch element. One ratio above 1:1.

Three forward speeds plus reverse, as above, is the floor. To get a fourth — an overdrive, where output spins faster than input — you don't add new tooth geometry. You take the ring-fixed relationship from the single-set module and run the ports backward: drive the carrier instead of the sun, ground the ring, and read output off the sun side.

Deriving 4th gear by hand — the part most explanations skip. You already have the formula for ring-fixed mode: ω_carrier = ω_sun / (1 + R). Overdrive is the same equation solved for the opposite unknown — carrier is now the known (input), sun is the unknown (output):

That 0.33:1 number is steeper than any production 4-speed actually uses in top gear — real boxes deliberately choose front-set tooth ratios that give a gentler overdrive, typically 0.70–0.75:1, because tripling output speed would put the final drive and tyres at an unusable rpm. The math is identical; production tooth counts are simply chosen to land the overdrive ratio somewhere useful rather than somewhere extreme. This is the same constraint-satisfaction mindset as the planet-count problem above: the kinematics tell you what's possible, the application tells you what's useful, and tooth-count selection is where those two meet.

Notice all three land 3rd gear at exactly 1.00 — that's not a coincidence, it's the direct lock-up mode you derived earlier, structurally guaranteed by the Willis equation whenever sun and ring of a set are clutched together. It costs nothing in engineering risk and always reads as exactly 1.00, which is why direct gear is the one ratio every Simpson-derived box shares regardless of manufacturer.

6-speed and 8-speed: more states, not new physics

Three planetary sets, then four. The Willis equation never changes — only how many valid grounding combinations the clutch hardware can select.

The Lepelletier architecture (ZF 6HP, GM 6L80) splits the job into two stages: a small, permanently-configured front planetary set that pre-reduces engine output, feeding a Ravigneaux compound set at the rear — two sun gears and two planet families sharing one ring and one carrier. More selectable member combinations from fewer total parts than stacking a third full Simpson-style set.

Modern 8-speed boxes (ZF 8HP and similar) push this to four planetary elements with enough clutches and brakes to select far more grounding combinations again. The tooth count grows modestly. The clutch count, valve body complexity, and control software grow substantially — that's where most of the added engineering effort actually goes.

How a Ravigneaux set actually works. A single Ravigneaux set replaces two full planetary sets with one ring, one carrier, and two suns of different sizes — a small sun and a large sun, each meshing with its own family of planets, but those two planet families mesh with each other as well as sharing the one ring and one carrier. The small sun's planets act as idlers that hand motion to the large sun's planets, which then mesh the ring. The result: you get every grounding combination a Simpson pair could give you, plus extras, from roughly 30% fewer total parts.

Because there are now two independent R values feeding the same ring/carrier pair, you get a richer menu of selectable ratios than two fully separate Simpson sets would give you for the same part count — which is exactly the efficiency gain that made Ravigneaux-based boxes (ZF 4HP, 5HP, 6HP families; most Aisin and GM Hydra-Matic units from the late 1990s onward) the dominant architecture industry-wide.

Why clutch count explodes faster than gear count. Each forward ratio needs a unique combination of which elements are grounded and which are driven. With 3 planetary elements and roughly 5 selectable clutches/brakes, you can combinatorially reach 6 clean ratios. Push to 4 planetary elements and the valve body needs roughly 5–6 shift elements engaging in carefully sequenced pairs (never more than 2 active at once, but which 2 changes every gear) to reach 8 or 9 ratios cleanly. The growth is the *control problem*, not the gear problem — this is precisely why 8 and 10-speed boxes lean so heavily on dedicated transmission control units doing real-time clutch pressure modulation, rather than the simpler hydraulic-only valve bodies that sufficed for 3 and 4-speed boxes.

Read the spread column as "overall spread," meaning top-gear ratio divided by 1st-gear ratio — it's the single number that tells you how wide a range of road speeds the box covers for one engine rpm window, completely independent of how many steps you take to get there. A wider spread with more steps is exactly the "engine stays on the power curve from a standstill to top speed" property you noticed in the tank.

Why the engine never leaves its sweet spot

The piece that ties tooth ratios back to the diesel engine itself.

A diesel engine — in your bus or that tank — produces peak torque across a narrow rpm band, typically a 600–900 rpm window. Outside that window, available torque drops off quickly in both directions: too low and the engine bogs, too high and you're burning fuel without proportional gain. The whole point of every ratio you've derived on this page is to keep the engine inside that narrow window as road speed sweeps from zero to maximum — a range that can be 50:1 or wider in road speed terms.

This is also exactly where the torque converter earns its place ahead of the planetary gearset, and why "torque translator" is a more honest name for the whole assembly than "gearbox." A torque converter is a fluid coupling that can itself multiply torque by a factor of 1.5–2.5:1 at low output speed (near a dead stop) and converges toward roughly 1:1 once output catches up to input speed. It covers the lowest, hardest-to-gear portion of the speed range — the moment of breaking static inertia, before any planetary ratio's reduction has fully taken effect — and then gets out of the way.

As road speed builds, the converter's multiplication fades toward 1:1 and the planetary box does the rest of the work, stepping up through 2nd, 3rd, and into overdrive — each shift timed by the control system to happen right as the engine reaches the top of its torque band, dropping it back to the bottom of that band in the next gear. Plotted against road speed, the result is a sawtooth: engine rpm climbs through the band, gearbox shifts, rpm drops back to the bottom of the band, climbs again. The engine spends its entire operating life riding that one narrow window — never far below peak torque, never uselessly high — while road speed itself climbs smoothly underneath it.