Stepper Motor Sizing: Torque, Speed and Holding Torque Calculations

Key Takeaway

Stepper sizing is a three-step calculation: holding torque, dynamic torque, and speed. Size for holding torque first, verify dynamic torque at speed, and never select on holding torque alone.

Figure 1: Stepper motor sizing – torque, speed, holding torque

1. Holding Torque and the Safety Factor
2. Dynamic Torque: Pull-in and Pull-out
3. Speed: How Torque Falls with RPM
4. Mid-Resonance and the Torque Dip
5. Driver: Voltage, Current and Heat
6. Selection Procedure
Related Reading
Sources

1. Holding Torque and the Safety Factor

What holding torque is

Holding torque is the torque a stepper motor can produce when energized but not rotating. It is the torque that keeps the rotor in position against an external force when the motor is stopped. Holding torque is specified in the datasheet and is usually the largest number on the motor spec sheet, which is why it is the first number people look at.

Safety factor

Never select a stepper motor on holding torque alone. Apply a safety factor:

required_holding_torque = load_torque × safety_factor

For horizontal axes, use a safety factor of 1.5 to 2. For vertical axes (where gravity pulls against the motor when stopped), use 2 to 3. The safety factor covers load uncertainty, friction changes, temperature effects, and the fact that datasheet holding torque is measured under ideal conditions.

Worked example

A vertical axis lifts a 20 kg load through a 5:1 gearbox with a 10 mm lead screw:

Load torque at screw = (20 kg × 9.81 m/s² × 0.005 m) = 0.98 N·m
Reflected to motor through 5:1 gearbox = 0.98 / 5 = 0.196 N·m
Safety factor 2 for vertical = 0.196 × 2 = 0.392 N·m required holding torque

A stepper motor with 0.5 N·m holding torque would be adequate for this axis.

2. Dynamic Torque: Pull-in and Pull-out

Pull-in torque

Pull-in torque is the torque a stepper can produce when starting from rest and accelerating to a speed without losing steps. It is always lower than holding torque because the rotor must accelerate from zero speed while overcoming load inertia.

Pull-out torque

Pull-out torque is the torque a stepper can produce while already rotating, before it loses steps. It is usually higher than pull-in torque because the rotor is already spinning and has kinetic energy to help it through torque peaks.

The torque-speed curve

Stepper torque falls with speed. A motor rated at 1.0 N·m holding torque might produce only 0.4 N·m at 500 rpm and 0.1 N·m at 1000 rpm. Always check the torque-speed curve in the datasheet at the speed your machine needs, not the holding torque at standstill.

Acceleration torque

Acceleration torque = load inertia × acceleration. Add this to the load torque to get the total dynamic torque requirement:

total_torque = load_torque + (J_load × acceleration)

3. Speed: How Torque Falls with RPM

Why torque falls

Stepper torque falls with speed because of inductance. The motor coils are inductors, and as the switching frequency increases, the current cannot build up fast enough in each step. Less current means less torque. At high speed, the motor may produce only a fraction of its holding torque.

Voltage and current

Higher supply voltage helps the current build up faster in the coils, which extends the torque curve to higher speed. A 48 V driver will produce more high-speed torque than a 24 V driver on the same motor. Current limit also matters: running at the rated current produces maximum torque, but running above rated current overheats the motor.

No-load speed limit

Every stepper has a no-load speed limit where torque drops to zero. This is set by the inductance, the supply voltage, and the driver electronics. Do not select a motor that needs to run near its no-load speed; leave 30 to 50 percent margin.

4. Mid-Resonance and the Torque Dip

Mid-resonance

Stepper motors have a mid-resonance point, typically between 1 and 10 revolutions per second, where the torque drops sharply. This is a resonance between the motor inertia and the load inertia through the motor’s springiness (the magnetic coupling between rotor and stator). Operating at mid-resonance causes lost steps, vibration, and noise.

Avoiding mid-resonance

Use microstepping to move the resonance to a higher step frequency.
Add damping: a viscous damper on the motor shaft, or friction in the load.
Change the acceleration profile so the motor passes through mid-resonance quickly.
Change the gear ratio so the operating speed avoids the resonance band.

5. Driver: Voltage, Current and Heat

Supply voltage

Driver supply voltage determines how fast current builds up in the motor coils. Higher voltage = higher speed torque. Common stepper driver voltages: 24 V, 36 V, 48 V, 70 V. Match the driver voltage to the motor rating; exceeding the motor voltage rating can damage the insulation.

Current per phase

The driver current limit sets the maximum current per phase. Running at rated current gives rated torque. Running below rated current reduces torque and heat. Running above rated current overheats the motor and can demagnetize the rotor.

Heat rise

Stepper motors run hot because they draw full current even when stopped (to hold position). A motor rated at 40 °C rise can be too hot to touch at sustained rated current. Mount the motor where it can dissipate heat, and consider reducing idle current if the motor sits stopped for long periods.

6. Selection Procedure

Step-by-step

Determine the load torque.Calculate the torque needed at the load: friction, gravity (for vertical axes), and any process forces.
Determine the required speed.What speed must the axis move, and what acceleration is needed?
Select a gear ratio.Choose a gear ratio that lets the stepper run in its efficient speed range and multiplies torque at the load.
Calculate reflected load inertia.J_reflected = J_load / (gear_ratio)^2.
Size the motor holding torque.Holding torque > load torque × safety factor (1.5-2 horizontal, 2-3 vertical).
Check dynamic torque.At the required speed, the motor torque (from the torque-speed curve) must exceed load torque + acceleration torque.
Check mid-resonance.Ensure the operating speed does not fall in the motor’s mid-resonance band.
Select the driver.Match voltage and current to the motor; ensure the driver can supply the required current per phase.

Example: 500 mm linear axis

A belt-driven axis moves a 10 kg load at 0.5 m/s with a 20 mm pulley:

Load force = 10 kg × 9.81 = 98 N (if lifting) or friction force (if horizontal)
Torque at pulley = 98 N × 0.01 m = 0.98 N·m
Through 3:1 gearbox: motor torque = 0.98 / 3 = 0.33 N·m
Safety factor 2: required holding torque = 0.66 N·m
Select a stepper with ~1.0 N·m holding torque

Sources

Key Takeaways

Size for holding torque with a safety factor (1.5-2 horizontal, 2-3 vertical), but never select on holding torque alone.
Stepper torque falls with speed; check the torque-speed curve at your operating speed, not the holding torque at standstill.
Pull-in torque (starting from rest) is lower than pull-out torque (while rotating); acceleration torque adds to load torque.
Mid-resonance causes lost steps; avoid operating in the resonance band or use microstepping/damping.
Stepper drivers set current per phase; higher supply voltage extends high-speed torque, but exceeding motor voltage rating damages insulation.
Reflected load inertia = load inertia / (gear ratio)^2; use gearing to reduce inertia seen by the motor.