Six things a power split is not

A continuously variable transmission built as a power split sends part of an engine’s power through a variator, whose ratio can be set anywhere across a span, and the rest through fixed gears, and recombines the two in a planetary. It is an old idea, tractor and bus transmissions use it, and hybrid cars use its electrical form. Because it is old, it has gathered a set of things people say about it. Most are the natural extension of something true about a variator or about a gearset, and most turn out to be false or misapplied when a split is actually measured.

This essay collects six of them, as six things a hold is not collected six about parts held by contact. Each is answered with a number from the transmission essays linked below, which built the splits as solved machines rather than as diagrams. The variator throughout spans 3.333, a belt drive’s typical travel from 0.3 to 1, and the planetaries are simple ones with a ring of K times the sun’s teeth unless a second is named.

One: a split multiplies its variator’s span #

The picture behind the claim is a lever. A planetary adds two speeds, as two inputs and one output found a differential doing, so a small change in one input should move the output by more, and a variator of modest span should become a transmission of large span.

Every power split is on one curve measured it, on 1,994 splits over all twelve attachments and nine gearsets. The output’s span is $(1 + Ra)/(1 + a)$, where R is the variator’s own span and a the power its branch carries over the gears’ power. When both branches carry power forward — no circulation — that is a weighted mean of 1 and R, and it never reaches R. Held to a largest share S of the engine’s power, the best such split spans $R/(R(1 - S) + S)$: 2.703 for a variator of span 3.333 held to 0.9, and exactly R only when the variator carries everything and the gears nothing.

The reason is short. With both branches carrying power forward, the output’s speed is the gears’ contribution plus the variator’s, and only the variator’s part moves. A fraction of the output that does not move can only dilute the fraction that does, so the span sits between one and the variator’s own. The third branch, with the variator’s power running backwards, does worse again: at a variator rated for the engine’s own power it spans $2 - 1/R$, which is 1.700 for this variator, the number the circulating split of the power that goes round twice reached by a much longer route.

A split beats its variator’s span only on the branch where the variator carries more power than the engine, and there it gains $R - 1$ of span per unit of share. The figure above is the whole trade. A split is a way of sparing a variator. It is not a way of amplifying one, unless the variator is rated for more than the engine it serves.

Two: the belt decides where the output stands still #

A power split can bring its output to rest while the engine turns, a geared neutral, and it is natural to think of that as a setting of the variator: the belt is somewhere in its travel, and at that place the two branches cancel.

A bounded ratio made unbounded swept six hundred variator settings on four planetaries and three belt lengths and found the setting at which the output stops. In every case it is the planetary’s own tooth ratio K, to within 0.0033. Lengthening or shortening the belt by two per cent, which moves the variator’s whole travel, does not move it at all. The neutral belongs to the gearset. That is the planetary’s speed relation read at an output speed of nought: with the output on the carrier, the other two shafts must turn in the ratio of their teeth, and no belt appears in that statement. The variator decides only whether its travel reaches that setting, and since a simple planetary’s K is always above one, a gearset with a ring much more than twice its sun puts the neutral out of reach of a variator topping out at 1.

Three: near neutral the split gives any ratio it is asked for #

The same machine has a ratio that passes through infinity: as the variator approaches the neutral setting the output slows towards nought, so every overall ratio from very high to very low lies somewhere in its travel. The claim is that all of them are available.

They lie in the travel. They are not available, because the machine’s sensitivity to its own variator grows without bound on the way. The output’s ratio moves with the variator’s at a rate of $(1 + K)/\text{gap}^2$, where the gap is the variator’s distance from the neutral setting. That is 15 at a gap of 0.4, 240 at 0.1, 6,000 at 0.02 and 24,000 at 0.01. The closed form and a difference quotient on the solved ratio, which share none of the algebra, agree to $10^{-8}$.

A variator is uncertain in its own ratio by about one and a half per cent, from its belt length, its centre distance and its sheave positions. Far from neutral that becomes 1.1% at the output. At a gap of 0.02 it becomes 110% — more than the ratio itself. So the neutral is not a ratio a machine can be asked to hold, and nor is anything near it. It is a boundary the controller has to stay away from, by a margin that the gearset’s K and the variator’s tolerance decide together.

The numbers make the margin concrete. Near the pole the output’s ratio grows as one over the gap and its sensitivity as one over the gap squared, so the output’s uncertainty, as a fraction of its ratio, grows as one over the gap. At 110% for a gap of 0.02 it is about 10% at a gap of 0.2 — so a controller that must hold the ratio to a tenth keeps the variator a fifth of a unit of ratio away from the neutral, and gives up everything the pole promised inside that. The unbounded span near the pole is real kinematics, and the tolerance of every real variator takes it back. Sliding the travel across the pole measured how much of a travel survives that margin: on its example, 75%, with the forward ratios spanning 4.77 rather than the unbounded span the kinematics promised.

Four: sliding the travel towards the pole is free span #

If the neutral is where the ratio becomes unbounded, then moving the variator’s travel closer to it should buy span for nothing. The kinematics agree: the output’s range grows as the travel approaches the pole, and on paper a variator of fixed span can be made to cover any output span at all.

It is not free, and the cost is paid in power. The power that goes round twice measured what each branch carries. Losslessness fixes a planetary’s torques at $1 : K : -(1 + K)$, and from that the variator’s branch carries $v/(K - v)$ of the engine’s power and the gear branch $K/(K - v)$. Both run away at the pole. The variator first carries the engine’s whole power at $v = K/2$ — exactly half the way to the pole, on all nine gearsets from K = 1 to 2.25 — and past that it carries more than the engine, the excess circulating back through the gears.

The ratio at which that happens barely depends on the gearset. At the crossing the machine’s overall ratio is $K/2(1 + K)$, which runs only from 0.250 to 0.346 across the nine gearsets. Every simple power split of this kind is overtaken by its own variator at roughly the same overall gear, whatever its teeth.

So a variator rated for the engine’s own power has its travel cut off at half the way to the pole. Its split spans 1.700 where the travel arithmetic alone, stopping only at the tolerance trim, promised 5.375. The two limits change places only at a variator rated for 6.25 times the engine’s power, and no machine is built that way. The span near the pole exists, and the variator pays for all of it.

Five: which shafts carry what is the design choice #

A simple planetary has three shafts and a power split puts three things on them, so there are six ways to attach the engine, the variator and the output, and twelve once the variator can sit at the input or the output. Transmission literature argues over which is best, and the claim is that the choice decides the machine.

It does not appear in the answer. The identity $(1 + Ra)/(1 + a)$ holds on every one of the twelve to $5 \times 10^{-15}$, and it has no K in it and no shaft names. Every attachment reaches every one of the three branches — no circulation, the variator’s branch running backwards, the variator overtaking the engine — and which branch a split is on is decided by where its travel sits and which way the variator turns.

The same machine shows it. With the engine on the ring, the variator on the sun and the output on the carrier, and K = 1.4, the variator turned backwards over a low stretch gives a span of 1.700. Turned forwards over a high stretch it spans 2.273 with no circulation. Turned backwards over a much higher stretch it spans 7.417 with the variator carrying up to 2.75 times the engine. Nothing about the gearset or the attachment changed between the three. The attachment decides the speeds and torques each shaft sees, which matters for bearings and for the variator’s own torque; it does not decide what the split can do.

The variator’s own ratio law does not decide it either. A ratio with no steps in it solved a belt variator’s pulley radii from its fixed belt length and found its ratio departing from the rule of thumb by 7.4% at full shift. None of that departure reaches the output’s span, which sees only where the variator’s travel starts and ends. The law in between decides how the output’s ratio is spread along the travel, and a controller needs it, but the span is set by R and a alone.

Six: a second planetary improves any split #

If one planetary lets a variator be spared, two should spare it more. A Simpson or Ravigneaux gearset has four shafts and two freedoms, and the claim is that building a split on one is better.

A second planetary lets the variator carry a difference put every arrangement on four real two-planetary gearsets. With the fourth shaft free, nothing changes: all 3,588 such splits lie on the single-planetary curves, because a shaft that carries no torque adds a speed to the lever and no condition to the torques, which stay in the one proportion a simple planetary gives them. Holding the fourth shaft instead removes the split altogether, since two freedoms less a held shaft and the engine’s speed leave the variator nothing to set. The one new arrangement is the compound split, with the variator between two of the gearset’s shafts, and it can do better. Its variator carries the difference of two splits’ shares, and on the best lever it spans the square of $(\sqrt{R} + k)/(1 + k\sqrt{R})$, with $k = (1 - S)/(1 + S)$: 1.802 against one planetary’s 1.538 at a share of 0.5.

But the best lever has the variator’s two shafts exactly its share apart, centred between the engine and the output — at a quarter and three quarters for a share of 0.5, as the slider shows — and real gearsets do not. The four measured have their inner shafts about 0.22 apart and centred near 0.6. On them a compound split beats one planetary up to a share of 0.3, 1.320 against 1.266, and falls behind from 0.4: 1.423 against 1.538 at 0.5. So a second planetary helps a variator carrying a small share of the power, on the right gearset, and nothing else. Of the 1,562 compound splits the census built, 498 put their two halves’ shares in opposite directions, and every one of those is worse than a single planetary.

And one thing a split is #

The six share a shape. Each takes a property of one part — the variator’s span, the gearset’s addition, the pole’s infinity, the attachment’s variety, the second planetary’s extra shaft — and credits it to the machine. The measurements say the machine is governed by a much smaller set of numbers: the variator’s span, the share of the engine’s power the variator is rated to carry, and whether that power runs forwards or backwards.

Those three numbers also say which of the six claims a designer should have been asking instead. The first and fourth are the same question, how much power the variator may carry, asked about the span and about the travel. The second and third are about the gearset’s K, which places the neutral and scales the sensitivity around it. The fifth and sixth are about the shafts, which turn out not to matter on one planetary and to matter a great deal on two, where the lever positions set by the teeth decide whether the compound split helps at all.

What does buy span honestly is modes. A machine that changes attachment part way along its travel runs through several splits in turn, each sweeping its variator across the whole travel with no circulation and handing over at a matching ratio, so the spans multiply. With the variator held to 0.75 of the engine’s power, one mode spans 2.11, two span 4.43, three 9.33 and four 19.64. To match two modes’ 4.43 in a single split, the variator would have to carry 1.47 times the engine’s power, circulating. That is why multi-mode power splits exist, and why the second clutch in them is worth its weight: it replaces circulation with a handover.

What the six do not cover #

Losses. Every number here is for a lossless gearset and a lossless variator. A real variator loses a few per cent of what it carries, and a split that circulates power loses it more than once, which makes the circulating branches worse than these figures and the no-circulation ones slightly better by comparison.

Dynamics. The sensitivity near neutral is kinematic. A controller holding a ratio near the pole also meets the machine’s inertia and the variator’s response time, and the margin it needs is at least the kinematic one.

Electric variators. A hybrid car’s power split uses electric machines as its variator, whose span is not bounded the way a belt’s is and whose power can reverse freely. The identity and the curves still hold for it, since they are kinematic and lossless; what changes is that R is no longer a fixed property of the variator.

Still open: six things a cam drive is not #

Six claims each have now been answered about centres, shapes, strands, networks, holds, chains, joints, compiled linkages, bodies, measurements and power splits. Cam drives have their own, and nobody has put them to the test together: that a smoother motion law always runs more quietly, that a cam on a timing chain sees the law it was cut to, that a larger base circle always lowers the pressure angle enough, that a dwell is a stop.

The distinct argument would be six such claims about cam drives, each answered from the cam essays’ own measurements. The chain’s step at every tooth, which a cam on a chain steps at every tooth found falling only as one over the tooth count, and the clocking behind a bent joint are two of them already measured. The work would be finding the other four among what the field has measured, and stating each claim in the words someone would actually use.