An encoder counts pulses. A Hall sensor reads a magnet. A resolver is different: it's an analog rotating transformer that outputs absolute shaft angle as the ratio of two sine-wave voltages. No optics, no semiconductors on the shaft, no batteries to remember position — just copper windings in iron. That's why resolvers survive in places encoders won't: EV traction motors at 200°C, missile fins pulling 100g, aircraft flap actuators soaked in hydraulic fluid, steel-mill motors coated in scale.

How it works. The rotor carries one winding, excited by an AC reference (typically 5–10 kHz, 4–10 V RMS) fed through a rotating transformer (brushless resolvers) or slip rings (brushed). The stator carries two windings at 90° mechanical spacing. As the rotor turns, it magnetically couples into both stator windings, but by different amounts:

• Sine winding output: V_ref · sin(θ) · sin(ωt)
• Cosine winding output: V_ref · cos(θ) · sin(ωt)

The shaft angle θ is encoded as the ratio of the two envelope amplitudes. A resolver-to-digital converter (RDC) chip like the AD2S1210 demodulates the carrier and computes θ = arctan(sine/cosine), typically outputting 10–16 bits of absolute position plus a velocity estimate.

Why the ratio matters. Because position comes from a ratio, not an absolute amplitude, resolvers are immune to excitation voltage drift, temperature-induced winding resistance changes, and cable length. Drop the reference from 7 V to 5 V and both channels drop proportionally — θ is unchanged. Try that with a potentiometer or an analog Hall sensor and you'll get garbage.

Multi-speed resolvers. A "1X" resolver gives one electrical cycle per mechanical revolution — true absolute position. A "4X" or "16X" resolver has multiple magnetic pole pairs and gives multiple electrical cycles per turn, trading absolute-ness for finer resolution. Aerospace often stacks a 1X coarse resolver with an nX fine resolver on the same shaft to get both.

Rule of thumb — resolution. A 12-bit RDC gives 2^12 = 4096 counts per electrical cycle. On a 1X resolver, that's 360°/4096 ≈ 0.088° absolute. On a 4-pole-pair BLDC motor (common), electrical angle repeats every 90° mechanical — so 4096 counts covers 90°, giving 0.022° for commutation but requiring a separate coarse channel for absolute mechanical position.

Where you'll meet one. Tesla Model S drive units, F-16 stabilator actuators, CNC spindle feedback, and any BLDC servo where the environment would eat an optical encoder for breakfast. If you see a fat 6-wire cable coming out of a motor with R1/R2/S1/S2/S3/S4 labels, that's a resolver.