Stepper Motors

i Learning Objectives

By the end of this chapter you should be able to:

Explain how a stepper motor converts digital pulses into discrete angular steps.
Describe variable-reluctance, permanent-magnet and hybrid steppers.
Calculate step angle, steps per revolution and shaft speed from the pulse rate.
Explain full-step, half-step and microstepping drive modes.
Interpret the pull-in and pull-out torque characteristics.

Section 88-1

Principle of Operation

A stepper motor has salient-pole stator windings that are energised one after another by a digital controller. Each pulse moves the rotor to a new position of minimum reluctance (or of magnetic alignment) and holds it there. The shaft position is therefore set by the number of pulses and the speed by their rate — without any feedback sensor.

⟳

Open-loop positioning

Pulses in → angle out

\[\theta = n\beta, \qquad N = \frac{\beta f_p}{360^\circ} \times 60~\text{rev/min}\]

\(\beta\) is the step angle, \(n\) the number of pulses and \(f_p\) the pulse rate in pulses per second. Provided no step is lost, position error does not accumulate.

Section 88-2

Types of Stepper Motor

Variable-reluctance (VR). A soft-iron toothed rotor with no magnet. With \(N_s\) stator teeth and \(N_r\) rotor teeth (\(N_s \ne N_r\)):

\[\beta = \frac{|N_s - N_r|}{N_sN_r} \times 360^\circ = \frac{360^\circ}{m N_r}\]

where \(m\) is the number of phases. VR motors have no detent torque when unexcited.

Permanent-magnet (PM). A magnetised cylindrical rotor; larger step angles (7.5°–15° typical), detent torque when unexcited, higher torque per ampere than VR.

Hybrid. An axially magnetised permanent magnet between two toothed soft-iron end caps, offset by half a tooth pitch. Combines the fine teeth of the VR type with the magnet of the PM type. For a two-phase hybrid with \(N_r\) rotor teeth:

\[\beta = \frac{360^\circ}{2mN_r} = \frac{360^\circ}{4N_r} \qquad (m = 2)\]

Table 88.1 — Comparison of stepper motor types.
Property	VR	PM	Hybrid
Rotor	Toothed soft iron	Permanent magnet	Magnet + toothed caps
Typical step angle	5°–15°	7.5°–15°	0.9°, 1.8°
Detent torque	None	Yes	Yes
Torque per unit size	Low	Medium	High
Direction depends on	Phase sequence only	Current polarity and sequence	Current polarity and sequence

1 Worked Example 88.1 — Step Angle and Speed

Problem. (a) A three-phase VR stepper has 12 stator teeth and 8 rotor teeth. Find its step angle. (b) A two-phase hybrid stepper has 50 rotor teeth and is driven at 1000 pulses/s in full-step mode. Find the step angle, steps per revolution and speed.

\[\text{(a)}\quad \beta = \frac{12 - 8}{12 \times 8} \times 360^\circ = 15^\circ \quad \left(= \frac{360^\circ}{3 \times 8}\right)\]

\[\text{(b)}\quad \beta = \frac{360^\circ}{4 \times 50} = 1.8^\circ, \qquad \frac{360}{1.8} = 200~\text{steps/rev}\]

\[N = \frac{1000}{200} = 5~\text{rev/s} = 300~\text{rev/min}\]

Section 88-3

Drive Modes

Wave drive (one phase on): simplest sequence A, B, C …; lowest torque.
Full-step, two phases on: AB, BC, CA …; same step angle, about 40 % more torque for a two-phase motor.
Half-step: alternates one and two phases on (A, AB, B, BC …); halves the step angle and doubles resolution.
Microstepping: sine/cosine-shaped currents in the two phases place the rotor at intermediate positions — 1/16 or 1/256 of a full step — giving smooth motion and reduced resonance.

Unipolar vs bipolar drivers. A unipolar driver uses centre-tapped windings and needs only one switch per half-coil; a bipolar (H-bridge) driver reverses the current in a whole coil, using all the copper and giving more torque from the same motor.

Section 88-4

Torque Characteristics

Holding torque: maximum torque the energised, stationary motor resists without moving.
Detent torque: holding torque with the windings unexcited (PM and hybrid only).
Pull-in curve: the maximum pulse rate at which the motor can start (or stop, or reverse) in step with a given load, without ramping.
Pull-out curve: the maximum torque the motor can deliver while already running at a given pulse rate. The region between the curves is the slew range, reached only by ramping the pulse rate up gradually.

Torque falls at high pulse rates because the winding inductance prevents current from reaching its full value in each short step; drivers use a high supply voltage with current chopping to overcome this.

Figure 88.1 — The motor can start instantly anywhere under the pull-in curve; it can reach the slew range only by accelerating gradually.

Applications, Merits and Limitations

3-D printers, CNC machines, printers and plotters, disk drives, camera lens and telescope positioning, robotics and medical pumps.

Merits: open-loop positioning, no cumulative error, full torque at standstill, easy digital interface. Limitations: may lose steps under overload or resonance without the controller knowing, low efficiency, limited high-speed torque, audible noise.

Section 88-5

Summary and Common Mistakes

Key Points

Each pulse produces one step \(\beta\); speed \(= \beta f_p / 360^\circ\) rev/s.
VR: \(\beta = 360^\circ/(mN_r)\); two-phase hybrid: \(\beta = 360^\circ/(4N_r)\).
Half-stepping halves \(\beta\); microstepping subdivides it further.
Pull-in curve bounds instant start; pull-out curve bounds running torque.

Common Mistakes

Applying the VR step-angle formula to a hybrid motor.
Forgetting to convert rev/s into rev/min.
Assuming a stepper can start at any pulse rate inside the pull-out curve.

Section 88-6

Chapter Review

Practice Problems

P88.1 A four-phase VR stepper has 8 stator teeth and 6 rotor teeth. Find the step angle and the pulse rate needed for 600 rev/min.

Show answer
\(\beta = 360/(4 \times 6) = 15^\circ\); 24 steps/rev; \(f_p = 24 \times 10 = 240\) pulses/s.
P88.2 A 1.8° hybrid stepper is driven in half-step mode. How many pulses turn the shaft through 90°?

Show answer
Half-step angle 0.9°; \(n = 90/0.9 = 100\) pulses.

Multiple-Choice Questions

MCQ 1. Which stepper has no detent torque?
(a) PM (b) hybrid (c) variable-reluctance (d) all have detent torque

Show answer
(c) — it has no magnet.
MCQ 2. A 200-step/rev motor driven at 400 pulses/s runs at:
(a) 60 (b) 120 (c) 240 (d) 400 rev/min

Show answer
(b) — 2 rev/s.

Conceptual Questions

Why does a stepper not need position feedback, and when can this fail?
Why does available torque fall at high stepping rates?
Compare unipolar and bipolar drives.

Looking Ahead

Add a position sensor to a permanent-magnet machine and switch its phases electronically according to rotor position, and the result is a motor with DC-motor characteristics but no brushes — the brushless DC motor of Chapter 89.