Tachogenerators, Selsyns & Resolvers

Rotary electromechanical transducers reuse machine and transformer physics for measurement. A tachogenerator produces a signal proportional to angular velocity. A selsyn (synchro) or resolver encodes angular position in the amplitudes or phases of AC winding signals. Excitation, magnetic coupling, loading, phase error, and mechanical alignment govern all of them.

What each device measures #

These outputs are not interchangeable. Differentiating an angle sensor can estimate speed but amplifies measurement noise. Integrating a tachogenerator can estimate position only if offset and scale error are controlled; otherwise drift accumulates without bound.

Tachogenerator: speed to voltage #

For a DC tachogenerator with approximately constant flux,

$$v_o \approx K_t\omega - v_{\text{brush}} - i_oR_a.$$

The ideal transfer is $v_o=K_t\omega$. The intercept and slope errors come from brush contact, residual magnetism, temperature-dependent resistance and flux, commutation ripple, armature reaction, output loading, and bearing friction. Around zero speed, brush drop and friction can create a dead zone where output is not a reliable linear measure of speed.

An AC tachogenerator avoids a DC commutator in some constructions, but introduces excitation-amplitude, phase, residual-voltage, and demodulation errors. If frequency is used as the speed signal, counting over a fixed gate gives quantization that is worst at low speed; measuring period improves low-speed resolution but changes latency and estimator behavior.

Selsyns and synchros #

A selsyn is a self-synchronizing AC machine used as a transmitter, receiver, or control transformer. A common arrangement excites one winding and distributes angle-dependent voltages through three stator windings spaced by $120^\circ$. Connecting a transmitter and receiver produces currents and torque proportional to angular disagreement, driving the receiver toward alignment in indication mode.

The transmitter rotor sets the direction of its field, and the three stator axes sample that field. In a brushless implementation an auxiliary rotary transformer supplies rotor excitation without slip rings, but the angle encoding still comes from the relative winding geometry.

In control-transformer mode, the useful output is an error voltage rather than receiver torque. Near the null,

$$v_e \approx K_s\sin(\theta_t-\theta_r) \approx K_s(\theta_t-\theta_r),$$

so the pair behaves like an angular error detector over a limited linear region. Outside that region the sine relation must be retained; treating it as a global linear sensor creates ambiguous or weak control near large errors.

Selsyn accuracy depends on electrical symmetry, phase sequence, excitation, winding impedance, mechanical alignment, bearing friction in torque receivers, and the load connected to the signal windings. A wiring swap can reverse or offset the indicated angle without any component being defective.

Resolver: angle to sine and cosine #

A resolver is a rotating transformer. With sinusoidal excitation $v_r=V_r\sin \omega_c t$, its ideal secondary signals are

$$v_{sin}=K V_r\sin\theta\,\sin\omega_c t,$$

$$v_{cos}=K V_r\cos\theta\,\sin\omega_c t.$$

After synchronous demodulation, angle follows from

$$\hat{\theta}=\mathop{\mathrm{atan2}}(A_{\sin},A_{\cos}).$$

The ratio cancels common excitation-amplitude changes in the ideal case. Real errors come from gain mismatch, offset, quadrature error, harmonic distortion, phase shift, winding imbalance, eccentricity, temperature, and unequal analog channels.

Some resolvers use multiple magnetic pole pairs so one mechanical revolution produces several electrical cycles. That raises angle sensitivity and can act like an electrical gear, but the coarse absolute sector must then come from another track, startup knowledge, or a combined algorithm. Distributed excitation and orthogonal sine/cosine windings couple through rotor saliency; the electrical angle advances once per pole pair rather than once per mechanical revolution.

The resolver signal chain #

The sensor is passive, but the measurement system is not. It needs:

1. a stable carrier source and excitation amplifier;
2. two matched analog input paths with enough common-mode and fault tolerance;
3. simultaneous or timing-corrected sampling of sine, cosine, and usually the reference;
4. synchronous demodulation or a resolver-to-digital tracking loop;
5. angle unwrapping and a speed estimator;
6. diagnostics for loss of excitation, open/short wiring, amplitude imbalance, and out-of-range velocity.

Latency matters in a motor loop. Filtering the angle lowers noise but delays commutation and feedback; differentiating it raises noise. A tracking observer trades these explicitly and should be characterized as part of the closed-loop drive, not hidden inside a sensor driver.

Calibration and verification #

Mount the device to a traceable angular reference and sweep through at least one full mechanical revolution in both directions. Record raw channels, excitation, decoded angle, speed, direction, temperature, and time. Useful residual plots include:

• angle error versus shaft angle, revealing cyclic harmonics and eccentricity;
• sine/cosine locus, which should be a centered circle after normalization rather than an offset ellipse;
• zero-speed output and drift over temperature;
• scale and phase versus speed;
• direction reversal, exposing backlash and estimator lag.

For a tachogenerator, repeat the sweep with declared output loading. For a selsyn or resolver, verify phase order, polarity, zero alignment, insulation, and fault diagnostics. A plausible decoded angle is not proof that the wiring and sign conventions are safe for feedback.