CC0 · public domain AXF-1 · rev A/ outer-diameter sizing sweep/ self-consistent from §REL

AXF-1, how big should it be?

The sweep holds the electromagnetics fixed (B_g=0.40 T, current density, fill, axial stack) and scales only the outer diameter. At constant airgap shear stress the four quantities that decide the actuator each scale as a clean power of D, which on the log plot below are four straight lines fanning out from the 110 mm baseline. There is no electromagnetic optimum: the binding constraint is elsewhere.

CURVE

Everything scales as a power of diameter

log–log · normalized to 110 mm = 1×

Fig.2: Slope = scaling exponent. The payoff (torque, ∝D³) rises fast, but the dynamic cost (rotor inertia, ∝D⁴) rises faster: it is the steepest line on the chart. Km and mass rise together (∝D²): the motor constant is bought in proportion to added mass. All curves ride on B_g=0.40 T; a lower measured B_r shifts them down together without changing the slopes.

DATA

Absolute values at key diameters

joint torque after 8:1 × 0.85

OD	motor mass	T air (6)	T nom (9)	T peak (18)	Kₘ	TD peak	inertia
100 mm	0.70 kg	22.2	33.3	66.6	0.87	13.9	2.4e-4
110 mm	0.85 kg	29.6	44.3	88.7	1.06	15.3	3.5e-4
120 mm	1.01 kg	38.4	57.6	115.1	1.26	16.7	5.0e-4
130 mm	1.19 kg	48.8	73.2	146.4	1.48	18.1	6.8e-4
140 mm	1.38 kg	60.9	91.4	182.9	1.71	19.5	9.2e-4

Units, torque N·m · Kₘ N·m/√W · TD (motor torque density, peak) N·m/kg · inertia (rotor active) kg·m². Cooling column = continuous current density in A/mm². Nominal (9) is the datasheet operating point (σ ≈ 11.6 kPa, P_cu ≈ 38 W) and needs light forced convection; air (6) is passive-continuous.

READ

What the curve actually decides

▲ bigger is strictly better: electromagnetically

Torque ∝ D³, Km ∝ D², torque density ∝ D¹, all monotonic. There is no peak to find; the physics never says "stop." Doubling the payoff costs only a squared growth in mass.

It also wins on your ledger: Km rising means less copper loss per newton-metre, so a larger machine is more joules-efficient per unit torque. Energy-optimal points the same way as torque-optimal, up.

▼ what stops you is inertia and packaging

Rotor inertia ∝ D⁴, the steepest line. Reflected through the gearbox (×G² = 64), it dominates how fast the joint can accelerate. At 110 mm the reflected rotor inertia is ≈ 0.022 kg·m²; at 140 mm it is ~2.6× that. Past some diameter you add inertia faster than torque and the limb goes sluggish.

And the actuator sits at the joint, so its mass is distal load, every gram penalizes the limb it drives and the proximal joints that must swing it. The ceiling is a packaging + dynamics call, not a magnetics one.

The decision, as a curve

Hip / knee (needs torque, tolerates mass): 120–130 mm is the sweet spot: 130 mm buys 73 N·m nominal / 146 N·m peak and clears the Km ≥ 1.3 datasheet target (crossed at 122 mm) at 1.19 kg, if it packages into the thigh. Ankle / wrist / elbow (needs low distal inertia): stay at 100–110 mm and accept less torque, the D⁴ inertia term matters more distally than the D³ torque gain.

So AXF-1 is not one motor: it is one design scaled per joint: same poles, same coils, same magnets, same process, only D_o changes. That is the printability dividend, you re-slice, you don't re-engineer.

Correction the sweep forced. The rev-A datasheet targeted ≤ 0.60 kg motor mass at 110 mm. Self-consistent, the honest number is ≈ 0.85 kg (Cu 183 g + Fe 240 g + magnet 186 g + 40% structure). Motor-level torque density is therefore ~0.35× sintered, not the 0.55× first quoted, the sovereignty tax is larger than stated. The joint-level and Km stories are unchanged. Fold 0.85 kg into rev B before the RFQ goes out.