How gear reductions work

October 16, 2026

Motors spin quickly and make little torque. A good brushless motor makes a couple of newton-metres at thousands of rpm, and most jobs want tens or hundreds of newton-metres at tens of rpm. So between almost every motor and its load sits a reduction: something that trades speed for torque.

This post covers what a ratio actually does (more than you'd think) and how gears mesh. Then it goes through the main ways to build a reduction: spur trains, planetaries, harmonic drives, cycloidal drives, and the options without gear teeth. It follows How electric motors work, and the motor equations from there come back here.

A ratio is a lever that keeps turning

A reduction of ratio N turns the output once for every N turns of the input. Power in equals power out, minus losses, so:

ω_{out} τ_{out} = ω_{in} / N = η N τ_{in}

η is the efficiency, from about 0.98 for a good gear mesh down to 0.5 or worse for some worm gears. That's the part everybody knows. A ratio also changes three things that are easy to forget, and they matter as much as torque:

Inertia goes up by N². Seen from the output, the motor's rotor has inertia N²·J_motor. At 100:1, a rotor that weighs almost nothing feels 10,000 times heavier.
Backdriving gets harder. To turn the output by hand you have to spin the motor N times faster against its friction, and push through the gearbox's own losses backwards. High ratios effectively lock the output.
Errors shrink, but backlash doesn't. Motor encoder resolution and motor-side errors get divided by N at the output. Play between teeth at the output stage passes straight through.

Spur gears

Two meshing gears behave like two circles rolling on each other without slipping. Those are the pitch circles, and every gear dimension starts from them.

pitch diameter ratio centre distance tooth force separating force d = m z N = z_{2} / z_{1} a = m (z_{1} + z_{2}) /2 F_{t} = 2 τ / d F_{r} = F_{t} tan 2 0^{\circ} m = module in mm, z = teeth tangential, does the work pushes the shafts apart

The module is the tooth size. Two gears must have the same module to mesh. Teeth are cut as involute curves. That makes the contact point move smoothly along a straight line, the line of action, tilted at the 20° pressure angle. It also keeps the ratio exactly constant through each tooth's contact.

Try a tiny pinion in the figure. The ratio goes up, but so does the tooth load, because the same torque now acts at a smaller radius. Below about 17 teeth, a standard 20° tooth also starts to get undercut at the root, which weakens it. Pinion size is usually what limits a gear stage.

A few more spur facts:

One mesh loses about 1–2%. Few ways to transmit power lose less.
Contact ratio is how many teeth share the load on average, typically 1.4–1.8 for spur gears. Load passes from one or two teeth to the next, which is a source of noise.
Helical gears cut the teeth at an angle so they engage gradually. They're quieter and carry more, at the cost of an axial thrust force. Bevel gears turn corners. Herringbone gears cancel the thrust by putting two helices back to back.
Backlash is the gap between teeth that lets a gear rock a little before the other one moves. Gears need some for lubrication and thermal growth. Reversing loads cross the gap every time.

A single spur stage is practical up to about 5–7:1. Beyond that the pinion gets too small or the gear too big, so you stack stages, and the ratios multiply. The hobby servo in the motors post uses four stages to get 5 × 4 × 3.17 × 3.17 ≈ 200:1.

Choosing a ratio

So what ratio should you use? Put the motor equations from the last post together with the N² inertia. The load's acceleration through a ratio N is:

α = \frac{η N τ _{motor}}{J _{load} + N ^{2} J _{motor}}

At low N, the motor's torque is the limit. At high N, the motor spends all its torque spinning up its own rotor. The acceleration peaks at the inertia match:

N^{*} = J_{load} / J_{motor}

But that ignores top speed. The output can never go faster than ω_motor / N, so a high ratio makes long moves slow. The ratio that finishes a move fastest balances the two, and it's usually below the inertia match.

Make the move short in the figure and the best ratio climbs toward the inertia match, because top speed stops mattering. Make it long and the best ratio drops. Also check the static case: can the motor hold the load at all? Holding torque at the output is η·N·τ_continuous.

Planetary gears

A planetary stage puts a sun gear in the middle, planet gears around it on a carrier, and a ring gear with internal teeth around the outside. Hold one of the three, drive another, and the third is the output.

The general relation between the three speeds is the Willis equation:

\frac{ω _{sun} - ω _{carrier}}{ω _{ring} - ω _{carrier}} = - \frac{z _{ring}}{z _{sun}}

Hold the ring, drive the sun, take the output from the carrier, and the ratio is 1 + z_ring / z_sun. With 12 and 48 teeth that's 5:1, same direction. Two constraints decide which tooth counts work. The planets have to fit, so z_ring = z_sun + 2·z_planet. And they have to sit evenly around the sun, so (z_sun + z_ring) must divide by the number of planets.

Planetaries are the default reduction for good reasons:

Load sharing. Three or four planets split the load, so a planetary carries a lot of torque for its size.
Coaxial. Input and output are on the same axis, so it bolts straight onto a motor.
Low loss. Around 95–97% efficiency per stage.
Stackable. Three to ten to one per stage. Two stages give 25:1, three give 125:1.
Backdrivable at low ratios. This is the basis of quasi-direct-drive actuators, which pair a big motor with a single 6–10:1 stage.

The cost is backlash. Each stage has a few arcminutes of play: about 3–10 for precision gearboxes, much more for cheap ones. That adds up across stages.

Harmonic drives

A harmonic drive (strain wave gear) gets a huge ratio from one stage by bending a gear. An elliptical plug, the wave generator, spins inside a thin flexible toothed cup, the flexspline. The ellipse pushes the flexspline's teeth into mesh with a rigid internal ring, the circular spline, at two places on opposite sides. The ring has two more teeth than the flexspline. So each turn of the ellipse walks the flexspline back by two teeth.

ratio = \frac{z _{flex}}{z _{circular} - z _{flex}} output reversed

The figure uses 50 and 52 teeth so you can see them, which gives 25:1. Real ones use 60 to 320 fine teeth for 30:1 to 160:1 in a single stage.

Why they are in so many robot joints:

No backlash. The wave generator preloads the flexspline into the ring, and many teeth engage at once, about a third of them.
Huge ratio in one compact stage. It's light and often has a hollow centre for cables.
Precise. Positioning repeats to within arcseconds.

What you trade for that:

Springy. The flexspline twists measurably under load, and the stiffness isn't linear.
Limited life. The flexspline flexes twice per input turn, so its fatigue life and the wave generator bearing set the drive's life.
Higher losses. Efficiency is roughly 60–85%, worse when cold and at low speed.
Not backdrivable in any useful way.

Cycloidal drives

A cycloidal drive rolls a lobed disc around a ring of pins. An eccentric on the input shaft pushes the disc around the inside of the ring. The disc has one lobe fewer than there are pins, so each input turn rolls it back by one lobe.

ratio = lobes = pins - 1 output reversed

The disc wobbles as well as turning. To take off only the turning, output pins pass through holes in the disc that are bigger than the pins by twice the eccentricity. Real drives use two discs 180° apart so the wobble cancels out.

Many lobes are in contact with pins at once, and they roll rather than slide. That makes cycloidal drives very stiff, with backlash under an arcminute, and able to take big shock loads. RV reducers, which put a planetary stage in front of a cycloidal stage, are the standard reduction in the base and shoulder joints of large industrial robots. The downsides are weight, cost, and the vibration from that eccentric motion.

Without gear teeth

Worm gears. A screw driving a gear, with ratios of 10:1 to 100:1 in one stage at right angles. At small lead angles the friction is high enough that the load can't turn the motor (self-locking): the output holds with the power off. Efficiency runs from about 40% to 90%, mostly sliding friction and heat.
Belts. A timing belt is quiet and cheap, loses only about 2%, and has very little backlash, but it stretches under load and is practical up to about 5:1 per stage. Belts are also good at moving a heavy motor away from where you don't want its weight.
Cable capstans. A cable wrapped around a small drum and a large one. Zero backlash and very backdrivable, with ratios up to about 10–20:1. Used in haptics and research robots.
Lead screws and ball screws. These turn rotation into linear motion. Force F = 2π·η·τ / lead. Ball screws have over 90% efficiency and are backdrivable. Plain lead screws often have 20–50% efficiency and are self-locking.
Direct drive. No reduction, just a much bigger motor. No backlash and perfect backdrivability, but expensive and heavy for the torque.

Side by side

	Spur / helical train	Planetary	Harmonic	Cycloidal / RV	Worm	Belt
Ratio per stage	up to ~7:1	3–10:1	30–160:1	10–100:1	10–100:1	up to ~5:1
Backlash	some	3–10 arcmin (precision)	~0	under 1 arcmin	some, grows with wear	~0, but stretches
Efficiency	97–99% / mesh	95–97% / stage	~60–85%	~80–90%	~40–90%	~98%
Backdrivable	yes at low ratio	at low ratio	no	no	usually not	yes
Strength	cheap, simple	compact, high torque	precise, light	stiff, shock-proof	holds without power	quiet, cheap

These are typical ranges, and every one has exceptions.

Rules of thumb

Start from the motor's speed and the load's speed. The ratio is roughly the motor's comfortable speed divided by the speed you need. Then check torque, heat, and inertia.
Check reflected inertia. If the motor's reflected inertia N²·J_motor is far bigger than the load's, most of the motor's effort goes into spinning itself.
Precision comes from the last stage. Backlash and stiffness at the output stage go straight to the load. Errors earlier in the train get divided by the ratio after them.
Decide whether the load should be able to push back. A low-ratio planetary lets a robot feel contact through its motor current. A harmonic or worm drive holds position against anything, and is blind to contact.
Efficiency is heat. A 70% gearbox passing 100 W makes 30 W of heat inside a sealed box.

The next post puts motors and reductions together into robot arm joints.