Torque costs heat, even when nothing moves

A brushless joint motor makes torque in proportion to current: τ = Kt·I, where Kt is the torque constant in N·m per ampere. The current also heats the windings at I²R, where R is the winding resistance. When a humanoid holds a crouch, its knee motors do no mechanical work at all, yet they keep burning power in the copper for as long as the pose lasts.

Eliminate current from those two equations and a single figure falls out. The copper loss needed for a torque τ is P = (τ/Km)², with Km = Kt/√R. Wensing and colleagues at MIT, who used it to size the MIT Cheetah’s actuators, describe Km² as torque squared per unit of ohmic power loss. Its unit is N·m/√W. Doubling Km quarters the heat for the same torque; at the same heat, it doubles the torque.

The number a rewind cannot change

Kt on its own is a poor way to compare motors, because it depends on how many turns of wire sit in each slot. Wind the same stator with twice the turns of half-section wire and Kt doubles, but each turn is now thinner and the total wire is twice as long, so resistance rises four times. Kt/√R stays exactly where it was. That is why motor makers can offer one frame in several windings for 24 V or 48 V buses with different Kt values, while Km stays nearly constant across them.

What does change Km is the copper and magnetic design itself: how much of the slot is filled with copper, the magnet grade, the air-gap radius and the stack length. Alva Industries, a Norwegian motor maker, attributes its own high Km figures to winding fill factor and geometry. A related caution from the same source matters for anyone reading a humanoid actuator datasheet: if Kt is quoted per RMS ampere, the resistance used must be the three-phase equivalent, 1.5 times the line-to-line value, not the line-to-line figure a multimeter reads.

Bigger diameter beats longer stack

Kollmorgen’s TBM2G frameless motors, sold for robot joints, are one of the few humanoid-relevant ranges that publish Km for every size. At the 26.3 mm stack, Km climbs from 0.128 N·m/√W for the 50 mm frame to 0.802 N·m/√W for the 115 mm frame. The diameter grows 2.3 times; Km grows 6.3 times, roughly in proportion to diameter to the power of 2.2.

That is steeper than the idealised scaling. In the model used by Wensing et al. and later by Urs et al., Km² grows with the cube of gap radius times stack length, so Km would rise with radius to the power of 1.5 at fixed length. Wensing’s fit to a real motor catalogue gave Kt²/R rising with radius to the power of 4.1, or Km with roughly radius squared, close to what the TBM2G data shows. Stack length helps less. On the 115 mm frame, going from an 8.2 mm to a 26.3 mm stack (3.2 times longer) raises Km 2.6 times, from 0.310 to 0.802.

For a humanoid designer this is the core argument for large, flat motors in hips and knees: at the same copper loss, a wider motor gives far more torque than a longer one. The cost shows up elsewhere. Rotor inertia rises steeply with radius, and Urs et al. propose dividing rotor inertia by Km² as a responsiveness metric, so a joint that runs cool is not automatically quick. Their two T-Motor examples show the gap: the small RI50 reaches 0.12 N·m/√W with a 0.67 ms rating on that metric, while the larger U8 reaches 0.23 N·m/√W but rates 2.3 ms.

The last column of Table 1 is a useful sanity check. Dividing each motor’s continuous stall torque by its Km and squaring the result gives the copper heat it is rated to carry continuously, from about 25 W for the 50 mm frame to 57 W for the 115 mm frame. Bigger motors have more surface to shed heat, but the actuator housing, the reducer and the robot’s shell decide whether that heat actually gets out. Our explainer on peak versus continuous torque follows that heat path.

What the gearbox does to Km

Seen from the joint output, a reducer with ratio N multiplies torque by N for the same current, while the copper loss stays the same. Km at the joint is therefore N times the motor’s Km, before gearbox losses. Urs et al. use exactly this, N·Km, as the effective motor constant of an actuator. The consequence is large, because heat scales with the square.

Holding 50 N·m through a 100:1 strain-wave reducer costs this motor about 1.4 W of copper loss. Through a 9:1 planetary, the kind used in quasi-direct-drive legs, the same motor would need about 176 W and more than double its continuous torque rating. That is why quasi-direct-drive designs choose large-diameter motors with high Km to begin with, and why their makers worry about heat while standing still. The gear ratio explainer covers the impact and back-drivability side of the same trade.

Temperature erodes Km while the robot works

Copper resistance rises with temperature. Haydon Kerk Pittman’s worked example uses a coefficient of 0.0040 per °C: a winding heated from 25 °C to 125 °C has 40 percent more resistance. Because Km depends on the square root of R, that alone cuts Km by about 15 percent, so the same torque now costs about 40 percent more heat, which heats the winding further. Magnets also weaken as they warm, lowering Kt; the size of that effect depends on the magnet material and grade, and the same document notes the real change is smaller when magnets run cooler than the windings.

Two practical points follow. A datasheet Km should state the winding temperature it applies to, and many do not; Kollmorgen’s brochure lists Km without stating conditions. And a joint that spends a shift holding loads will run at a lower Km in the afternoon than its datasheet promises, which control software that estimates torque from current has to account for.

Reading a humanoid actuator datasheet for Km

• Look for Km first, then Kt. Frameless motor ranges such as TBM2G publish Km. Integrated actuator modules usually do not.
• Check the conventions before estimating it. The CubeMars AK80-9 lists Kv 100 rpm/V, Kt 0.105 N·m/A and 170 mΩ line-to-line. An ideal Kv of 100 rpm/V corresponds to about 0.095 V·s/rad, so the quoted Kt already reflects a convention or measurement choice. Compute Km only when you know whether Kt is per RMS or peak ampere and which resistance it pairs with.
• Ask at what temperature. A Km quoted at 25 °C flatters a joint that runs at 100 °C.
• Multiply by the ratio, then square. N·Km gives the joint-level figure; (τ/(N·Km))² gives the copper heat for a holding torque, before gearbox losses.
• Remember speed. Km ignores iron losses. For fast joints, ask for losses at speed or an effective motor constant.

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