Hybrid Stepper Motor (2D)

How a hybrid stepper works

The rotor is a permanent magnet between two toothed cups (50 teeth each, the cups offset by half a tooth) so that one cup is magnetised north and the other south. The stator has eight poles with small teeth, wound in two phases A and B. Energising a phase pulls the nearest rotor teeth into line with the stator teeth; switching to the other phase moves the rotor a quarter of a tooth pitch, which is 360° / (4 x 50) = 1.8°, a full step. One electrical cycle of the currents is four steps and one tooth pitch (7.2°).

Torque and current

With electrical angle φ = 50 θ the flux linked by the phases is λA = Ψ cos φ, λB = Ψ sin φ, so the torque is T = Kt (-iA sin φ + iB cos φ) + Tdetent with Kt = 50 Ψ and the back-EMF is eA = -Kt ω sin φ, eB = Kt ω cos φ. If the current vector is (I cos φ*, I sin φ*) the torque is Kt I sin(φ* - φ): a restoring spring with stiffness Kt I 50, a holding torque Kt I and a natural frequency f = (1/2π) √(50 Kt I / J), which is why a lone step overshoots and rings. The detent term -Td sin 4φ holds the rotor with the power off.

Windings and the chopper drive

Each winding obeys L di/dt = v - R i - e. The driver compares the current with its order and applies the supply voltage with a fast current loop; the model integrates this implicitly with a voltage limit of ±V_s. At low speed the current follows the order; at high speed e and ωe L eat the voltage and the current can no longer reach the setting, so the pull-out torque falls with step rate. The pull-out curve uses the steady-state phasor relation V² = (R I + E)² + (ωe L I)² solved for I.

Full, half and microstepping

Full step drives both phases with ±I (vector length √2 I). Half step alternates one-phase and two-phase positions, so the torque alternates by 41 %. Microstepping drives sine and cosine currents so the rotor can rest between teeth: much smoother motion and less resonance, but the same current setting gives a vector of length I, and the holding stiffness near a microstep position is lower, so accuracy under load is limited by the detent and friction rather than the step size.

Resonance and lost steps

Because the rotor behaves like a lightly damped oscillator, stepping at rates near its natural frequency (divided by integer factors) can make the motion grow. If the load torque exceeds the pull-out torque the rotor slips behind the field and steps are lost; an open-loop stepper cannot know. The lead-screw carriage turns rotation into 8 mm per revolution; a constant load torque acts in the negative direction like a vertical axis.

Honest simplifications

Ideal sinusoidal flux, no magnetic saturation, no iron loss, no winding mutual coupling, constant resistance, the chopper as a proportional loop with a voltage limit (no switching ripple), friction as viscous plus smooth Coulomb, an end stop at 120 mm. Constants resemble a NEMA 23 motor (1.8 degrees, 0.45 N m/A, 0.9 ohm, 2.5 mH) but are not a datasheet.