Stepper motors and drive modes

Why it matters

Stepper motors position 3D-printer axes, small CNC routers, camera mounts, valves and lab automation without any position sensor: count the pulses and you know where the shaft is. That open-loop simplicity is only safe if the motor never misses a step, so a designer must understand step angle, drive modes and the torque–speed limits.

Key ideas

• Principle. A stepper converts each input pulse into a fixed angular increment, the step angle β. The shaft position is the number of pulses × β, and the speed is set by the pulse rate. The rotor aligns itself to the position of minimum magnetic reluctance (or to the field of the energised phases).
• Types.
  • Variable reluctance (VR) – soft-iron toothed rotor, no magnet. Step angle depends on the stator and rotor tooth numbers. No detent torque when unenergised.
  • Permanent magnet (PM) – magnetised rotor; larger step angles (typically 7.5°–18°), cheap, with detent torque.
  • Hybrid – axially magnetised rotor with toothed end caps; combines PM torque with fine VR teeth. The common two-phase hybrid has 50 rotor teeth and a 1.8° step (200 steps/rev). Most industrial steppers are hybrids.
• Windings and drivers. Unipolar windings (centre-tapped) need only one switch per half-winding but use half the copper at a time. Bipolar windings need an H-bridge per phase but give roughly 40 % more torque from the same frame. Modern drivers use a chopper (constant-current) drive: a supply voltage several times the rated winding voltage forces current up quickly, and PWM limits it to the rated value. This keeps torque up at higher step rates, because the winding time constant τ = L/R otherwise limits how fast current can rise.
• Drive modes (two-phase motor).
  • Wave drive – one phase on at a time; full step angle; least torque (one phase).
  • Full-step (two-phase-on) – two phases on at a time; same step angle as wave drive but about √2 times (≈41 % more) holding torque for equal phase current; rotor rests midway between poles.
  • Half-step – alternates one-phase-on and two-phase-on; step angle halved (0.9° for a 1.8° motor), so steps per revolution double; torque ripples between the two states unless current is compensated.
  • Microstepping – phase currents follow sine and cosine values in many small increments (e.g. 1/8, 1/16, 1/256 of a full step); very smooth motion and lower resonance. Positional accuracy per microstep is limited by detent torque and load, so microstepping improves smoothness more than true accuracy.
• Torque terms. Holding torque: maximum torque with windings energised at standstill. Detent torque: torque with windings off (PM and hybrid only). Pull-in torque: the load torque at which the motor can start and stop in step without ramping at a given pulse rate. Pull-out torque: the maximum torque the motor can deliver once running at that rate (pull-out curve lies above pull-in). The region between them can be used only with acceleration ramps.
• Losing steps and resonance. If load plus acceleration torque exceeds the available torque, the rotor slips poles and the controller's count is wrong with no warning. Mid-range resonance can also cause stalling. Remedies: speed ramps, microstepping, damping, margin of 30–50 % on torque, or closed-loop steppers with an encoder.
• Compared with servos. Steppers are cheap, have high torque at low speed and need no tuning; torque falls steeply with speed, efficiency is low (full current even at rest), and open-loop positioning can silently fail.

Formulas

β = 360° / (steps per revolution)

• β step angle (degrees).

β = 360° / (m · N_r) (variable reluctance) or β = (N_s − N_r)·360° / (N_s · N_r)

• m number of stator phases, N_r rotor teeth, N_s stator teeth (poles). Both give the same result for a standard VR motor.

β = 90° / N_r (two-phase hybrid, full step)

• N_r rotor teeth per end cap; 50 teeth give 1.8°.

θ = n · β

• θ angular displacement (degrees), n number of steps (pulses).

N = 60 · f_p · β / 360°

• N shaft speed (rpm), f_p pulse rate (steps/s).

x = p · θ / 360° (lead screw)

• x linear travel (m), p lead of the screw (m per revolution).

T_required = J · α + T_load, α = Δω / Δt

• J total inertia referred to the motor shaft (kg·m²), α angular acceleration (rad/s²), T_load friction and load torque (N·m).

Worked examples

Example 1 (standard). A 1.8° stepper in half-step mode drives a lead screw of 5 mm lead. Find the linear resolution, the pulses needed to move 30 mm, and the time taken at a pulse rate of 2000 pulses/s.

1. Half-step angle β = 1.8°/2 = 0.9°, so steps per revolution = 360/0.9 = 400.
2. Linear resolution = p / 400 = 5 / 400 = 0.0125 mm per pulse.
3. Pulses for 30 mm = 30 / 0.0125 = 2400 pulses.
4. Speed N = 60 × 2000 / 400 = 300 rpm = 5 rev/s, so linear speed = 5 × 5 = 25 mm/s.
5. Time = 30 / 25 = 1.2 s (ignoring the acceleration ramp).

Answer: 0.0125 mm per pulse, 2400 pulses, 1.2 s.

Example 2 (GATE level). A three-phase VR stepper has 12 stator teeth and 8 rotor teeth. It must run at 600 rpm, reaching that speed from rest in 0.1 s with a constant acceleration. Total inertia at the shaft is 2 × 10⁻⁴ kg·m² and friction torque is 0.05 N·m. Find the step angle, the pulse rate at 600 rpm, and the minimum pull-out torque needed with a 50 % safety margin.

1. β = (N_s − N_r)·360°/(N_s·N_r) = (12 − 8) × 360/(12 × 8) = 15°. Check: 360/(m·N_r) = 360/(3 × 8) = 15°.
2. Steps per revolution = 360/15 = 24.
3. Pulse rate f_p = N × steps/rev / 60 = 600 × 24/60 = 240 pulses/s.
4. ω = 2π × 600/60 = 62.83 rad/s; α = 62.83/0.1 = 628.3 rad/s².
5. T_acc = J·α = 2 × 10⁻⁴ × 628.3 = 0.1257 N·m; T_required = 0.1257 + 0.05 = 0.1757 N·m.
6. With 50 % margin: 1.5 × 0.1757 = 0.264 N·m, which must lie under the pull-out curve at 240 pulses/s.

Answer: β = 15°, f_p = 240 pulses/s, pull-out torque ≥ about 0.26 N·m.

Common mistakes

• Assuming full-step (two-phase-on) mode has a smaller step angle than wave drive. It has the same step angle and more torque; only half-step and microstepping reduce the angle.
• Forgetting to double the steps per revolution in half-step mode when converting pulses to angle or distance.
• Treating microstep resolution as true positional accuracy.
• Sizing on holding torque. Available torque at operating speed (from the pull-out curve) is much lower.
• Starting a high-inertia load at a high pulse rate without a ramp; the motor stalls and the position count is lost.
• Mixing up rpm and rev/s, or degrees and radians, in the pulse-rate and acceleration calculations.

For GATE ME

Expect numericals on step angle from tooth and phase numbers, steps per revolution, pulses for a given angle or lead-screw travel, pulse rate for a given speed, and torque needed to accelerate an inertia. Conceptual MCQs test drive modes (which ones halve the step, which gives most torque), holding versus detent torque, pull-in versus pull-out characteristics, and why steppers can work open loop. Practise lead-screw and half-step conversions until they are automatic.

Quick check

1. What is the step angle of a motor with 200 full steps per revolution in half-step mode?
2. A VR motor has 3 phases and 6 rotor teeth. What is its step angle?
3. How many pulses per second give 120 rpm on a 1.8° motor in full-step mode?
4. Which drive mode gives the highest holding torque at the same phase current: wave or two-phase-on full step?
5. Which torque exists only in PM and hybrid motors with the windings off?

Answers: 1. 0.9°. 2. 20°. 3. 400 pulses/s. 4. Two-phase-on full step. 5. Detent torque.