Stepper Motor Sizing: Torque, Speed and Holding Torque Calculations

Key Takeaway

Stepper motor sizing is a three-number problem: holding torque for the static case, pull-out torque at the working speed for the dynamic case, and acceleration torque for the transient. Size on the lowest point of the pull-out curve at speed, apply a factor of 2 margin, and never size on holding torque alone.

1. The Rule That Overrides Everything

A stepper motor converts a commanded number of pulses into an angular position and believes it is always exactly there. There is no measurement, so there is no error report. A 1.8 degree stepper with 200 microsteps gives 0.009 degrees of command resolution, which on a 20 mm lead screw is 20 x 0.009 / 360 = 0.0005 mm of command resolution. That number is excellent and it is worthless if the motor skipped 40 steps during a rapid move: you now have a 0.10 mm position error and a drive that reports no fault at all. Stepper motor sizing has one rule that overrides everything else: size on the pull-out torque curve at the working speed, never on the holding torque on the label.

Stepper motor sizing therefore has one rule that overrides everything else. Never size on holding torque alone. The holding torque on the datasheet is the torque the motor holds at standstill with all phases energised. It is the easiest number to read and it is the least useful one for a running axis. The number that decides the motor is the pull-out torque curve at your working speed.

Datasheet number	What it means	Use it for
Holding torque	Torque held at standstill, all phases on	Vertical axes, power-off holding check
Pull-in torque	Maximum torque at which the rotor can start and stay in sync from standstill	Start-up under load at low speed
Pull-out torque	Maximum torque the running motor can follow without losing sync	Running torque, this is the sizing number
Max torque per phase	Torque at the motor shaft at the rated phase current	Upper bound check

2. Converting the Load into Torque

Before any datasheet is opened, the load has to become a torque at the motor. For a screw drive the force at the nut becomes a torque through the lead: Before any of the torque numbers can be used, the force at the load has to become a torque at the motor, and that conversion is where stepper motor sizing starts.

T = F x L / (2 x pi x efficiency)

For a belt or a pulley the force becomes a torque through the radius:

T = F x r

A worked example: a packaging machine Z axis

Acceleration torque for 12 kg at 1.0 m/s^2: F_acc = 12 x 1.0 = 12 N, so T_acc = 12 x 0.010 / (2 x pi) = 0.019 Nm. Total peak torque = 0.239 + 0.019 = 0.258 Nm, call it 0.26 Nm with a margin. Every torque number in stepper motor sizing comes out of this one conversion, and getting the lead or the radius wrong here invalidates everything downstream.

Now the working speed: 0.4 m/s on a 10 mm lead is N = 0.4 / 0.010 = 40 rpm. That is slow enough that the pull-out torque at 40 rpm on a 1.8 degree NEMA 23 frame is close to the maximum torque per phase, which is exactly why steppers are used here.

3. Pull-Out Torque at Speed: The Number That Decides

Torque falls with speed because of the back EMF. The motor’s rotating magnetic field must overcome the voltage generated by the spinning rotor, and as the speed rises the voltage generated rises until the motor cannot develop more current at the available supply voltage. The result is a torque-speed curve that looks like this for a typical 1.8 degree 2.8 A frame: The pull-out curve is the heart of stepper motor sizing, because it is the only datasheet number that describes what the motor can actually do while it is turning.

Speed (rpm)	Typical pull-out torque	As a fraction of max torque per phase
0 to 100	Near maximum	95 to 100 percent
200	About 75 percent	75 percent
400	About 55 percent	55 percent
800	About 35 percent	35 percent
1200	About 25 percent	25 percent

Apply that to the packaging axis. If a 1.8 degree 2.8 A NEMA 23 motor has a maximum torque per phase of 1.1 Nm, then at 400 rpm it can produce about 0.61 Nm. At 1200 rpm it can produce about 0.28 Nm. The packaging axis needs 0.26 Nm and runs at 40 rpm, so the motor is comfortable with a pull-out margin of 1.1 / 0.26 = 4.2:1. That is a healthy margin.

Now take a different machine: a 3 kg cross slide moving at 1200 rpm, which is a very ordinary CNC feed rate. With a 5 mm lead, F = 3 x 9.81 x 1.15 = 33.8 N, T = 33.8 x 0.005 / 5.6549 = 0.030 Nm. The torque is trivial. But at 1200 rpm the pull-out torque of that same motor is only 0.28 Nm, so the margin is 0.28 / 0.030 = 9.3:1. Also comfortable. The point of the table is that for the same motor, the margin changes by a factor of nearly 4 across the speed range, and the datasheet’s holding torque tells you nothing about it.

Applying the safety factor

For the packaging axis with a 3:1 factor on a vertical axis, the required pull-out torque is 0.26 x 3 = 0.78 Nm at 40 rpm. A 1.8 degree 2.8 A motor with 1.1 Nm max torque per phase has about 1.05 Nm at 40 rpm, so the margin is 1.05 / 0.78 = 1.35:1 on the pull-out curve with the factor already applied. That is correctly sized. The pull-out curve is the number that decides stepper motor sizing, because it is the only datasheet figure that describes what the motor can do while it is turning.

What it is not is correctly sized if you had used the holding torque of 1.1 Nm against the bare 0.26 Nm requirement and stopped there. That calculation gives a 4.2:1 margin, which looks better and is meaningless, because the motor can only pull 0.28 Nm at 1200 rpm and the same motor will be used on a different fixture next year at 1000 rpm.

4. Acceleration Torque and Inertia

The acceleration requirement is separate from the steady state and is compared against the pull-out torque, not added to it. The reason is that the pull-out torque is the total torque the motor can produce at that speed, and the acceleration torque has to come out of it. Acceleration is where stepper motor sizing is most often wrong, since the acceleration torque is compared against the pull-out torque rather than added to it.

For a screw drive the acceleration torque is:

T_acc = m x a x L / (2 x pi)

For a belt drive the equivalent inertia torque is:

T_acc = J x alpha

A fast indexing axis

The steady-state friction torque at speed is small, say 0.02 Nm. So the motor must produce 0.115 Nm during the acceleration. On a 2.8 A 1.8 degree frame with 1.1 Nm max torque, that is a 9.6:1 margin on torque. But the inertia check tells a different story. The reflected inertia of the 6 kg carriage through the 5 mm lead is: The acceleration check is where stepper motor sizing most often goes wrong, since the acceleration torque is compared against the pull-out torque rather than added to it.

J_reflected = 6 x (0.005 / (2 x 3.14159))^2 = 6 x (7.9577e-4)^2 = 6 x 6.3325e-7 = 3.80e-6 kg m^2

A 1.8 degree stepper of that size has a rotor inertia of roughly 5.4e-6 kg m^2, giving an inertia ratio of 3.80e-6 / 5.4e-6 = 0.70:1. That is very low, which is why stepper axes tolerate high accelerations that a servo of the same size would refuse. Stepper systems are inherently forgiving on inertia because the rotor inertia is comparable to the reflected load inertia.

Stepper motor sizing must still check the inertia, because the failure mode is different: instead of hunting, the axis loses steps at the end of the acceleration when the torque margin is gone.

5. The Speed Limit and Why It Is Electrical

The speed limit of a stepper is set by the back EMF curve and the supply voltage, and by the point at which the torque falls below what the load needs. It is not a mechanical limit. Speed is the other half of stepper motor sizing, and the number that limits it is electrical rather than mechanical.

The pull-out torque falls with speed because the motor voltage needed to drive the current at that speed rises. At the point where the required motor voltage exceeds the available supply voltage, the current falls and so does the torque. Two consequences follow:

• Raising the supply voltage raises the usable speed. Moving from a 24 V driver to a 48 V driver roughly doubles the maximum usable rpm on the same motor and frame, because the back EMF curve can be followed further before the current collapses.
• The torque-speed curve is set by the driver, not just the motor. A motor’s catalogue curve is usually shown at a reference voltage. A 24 V curve and a 48 V curve on the same motor are different curves.

For a packaging axis running at 40 rpm the supply voltage is irrelevant. For a CNC feed at 1200 rpm it decides whether the axis works at all. A 1.8 degree 2.8 A motor with a 24 V driver may be usable to about 600 rpm with useful torque, and to about 1200 rpm on 48 V. That is why every CNC retrofit that keeps a stepper on a feed axis replaces the driver with a 48 V unit at the same time as it replaces the motor.

The other speed issue is the resonance band. A 1.8 degree stepper has a strong mid-speed resonance, typically between 150 and 400 rpm depending on the frame, the load inertia and the microstep setting. In that band the rotor’s natural frequency coincides with the stepping frequency and the motor will stall with a full torque command. The band moves when the load inertia changes, which is why an axis that worked empty will not run with the fixture fitted.

6. Microstepping and What It Changes

Microstepping drives the two phases of a stepper with a sine and a cosine current waveform, so the rotor settles at an intermediate angle between the full steps. The command resolution improves by the microstep factor. Microstepping is the cheapest way to improve a stepper axis, and it changes two of the numbers that stepper motor sizing depends on.

Running through or below the band

Microstepping drives the two phases of a stepper with a sine and a cosine current waveform, so the rotor settles at an intermediate angle between the full steps. The command resolution improves by the microstep factor. The resonance band is the second half of stepper motor sizing, because a motor that loses sync is a motor that was sized wrong at the working speed.

On a 1.8 degree motor with 16 microsteps, the step angle is 1.8 / 16 = 0.1125 degrees. On a 20 mm lead screw, the command resolution is 20 x 0.1125 / 360 = 0.00625 mm. With 16 microsteps the rotor position is repeatable to a few percent of a step, so the effective positioning accuracy is a few microns rather than the 20 micron the command resolution suggests.

Three effects that matter for sizing

Microstepping is the cheapest way to improve a stepper axis, and it changes two of the numbers that stepper motor sizing depends on. The 0.9 degree step motor is the other way of buying resolution. It has twice the steps of a 1.8 degree motor on the same frame size, so the same command resolution with half the microstepping and therefore half the microstep rate. The trade-off is lower torque per phase, usually about 30 percent less, and a higher chance of losing steps under shock. Microstepping changes two of the numbers that stepper motor sizing depends on, and it is the cheapest improvement available on an existing axis.

7. The Driver: Voltage, Current and Heat

The driver is part of the motor as far as stepper motor sizing is concerned, and it makes three decisions that the motor datasheet does not: what voltage it runs from, what current it allows, and how it handles the phase energisation.

Supply voltage

The current setting on the driver is the RMS current per phase, and it is limited by two things: the motor’s rating and the driver’s own dissipation. A stepper motor’s rated current per phase is a thermal limit, not a peak limit, and running at the rated current continuously in a poorly ventilated cabinet will overheat the motor. The supply voltage is part of stepper motor sizing even though it appears on the driver’s nameplate rather than the motor’s, because it sets the speed ceiling.The rule is to derate to 70 to 80 percent of the rated phase current if the motor is in a hot enclosure or mounted next to a heat source. On a 2.8 A motor, run at 2.0 to 2.2 A. On a 1.8 A motor, run at 1.3 to 1.5 A. The driver display shows the actual RMS current, and if it is sitting at the setting continuously with no load, something is wrong with the wiring or the microstep setting. The current limit is a thermal number, and stepper motor sizing has to include the derating for the cabinet it will actually live in.

Heat

The thermal limit is usually specified as a winding temperature rise of 80 K over ambient at rated current. On a 40 degree C shop floor, that is a 120 degree C winding temperature. Motor insulation class F is rated to 155 degrees C, so there is margin, but only if the housing is actually dissipating. Mount stepper motors on a metal bracket or a cast plate, and never in a sealed plastic box. Splitting a long axis into two motors driven separately is often the cheapest stepper motor sizing decision available, because it halves the inertia each motor sees.

Split and parallel phasing

Two motors on one driver can be wired in series or in parallel. Series gives the higher speed capability because the back EMF per phase doubles; parallel gives the higher torque because the current per phase doubles at the same voltage. Splitting a long axis into two motors and driving them separately is better than either, because it halves the inertia each motor sees and it removes the backlash in the coupling between them.

8. Vertical Axes and the Power-Off Case

A vertical axis is the case where stepper motor sizing gets interesting, because the holding torque has to cover two separate situations: the running case with the driver powered, and the power-off case where the load must still be held.

Power-off holding

With a brake, the torque to release the brake is the brake coil’s holding torque plus the load. The safety factor on that number should be 3:1 or more, because a brake that has not fully released produces a torque spike that will cost steps every cycle. The power-off case is the second requirement, and stepper motor sizing has to cover it with a brake or with a different motor technology entirely.

The safety factor on a vertical axis

Combine the two: for a vertical axis with a holding brake, use the greater of 3 x the static torque (for the running case) and 2 x the load torque (for the brake release case). The packaging axis from section 3 needs 0.26 Nm peak, so 3 x 0.26 = 0.78 Nm at 40 rpm. The same axis without a brake, on a machine where the power never drops, can be sized at 2 x 0.26 = 0.52 Nm.

The cheaper and better answer for many vertical axes is a servo with a multi-turn absolute encoder. It costs more but it has a real holding brake, it knows its position at power-up, and it removes the entire class of lost-step failures. The comparison is in the servo drive selection guide and the broader context is in the industrial motion control guide.

9. Stepper Motor Sizing: What It Costs in India and the Decision

Item	Price range	Notes
NEMA 23 stepper, 1.8 deg, 2.8 A, 0.9 m	Rs 2,200 to 4,500	Local brands at the low end
NEMA 23 stepper, 1.8 deg, 2.8 A, 1.5 m	Rs 3,200 to 6,500	Long body frame
NEMA 34 stepper, 1.8 deg, 4 A	Rs 5,500 to 11,000	For higher thrust axes
Closed-loop stepper with encoder, NEMA 23	Rs 12,000 to 22,000	30 to 50 percent above open loop
Stepper driver, 24 to 48 V, 4.5 A, pulse train	Rs 3,500 to 8,000	Microstepping to 16 or 32
Closed-loop stepper driver	Rs 9,000 to 18,000	Alarm on lost step
Mounting bracket, aluminium	Rs 400 to 1,200	Do not skip the heat path
Ball screw, 16 mm, 10 mm lead, 500 mm	Rs 4,500 to 9,000	With machined ends and bearing blocks

A complete stepper axis for the packaging machine, with motor, driver, bracket, screw, bearing blocks and couplers, lands at roughly Rs 28,000 to Rs 52,000. The equivalent servo axis with a 750 W motor and drive lands at Rs 90,000 to Rs 1,50,000. The stepper is roughly a third of the cost, which is exactly why it is the right answer on a packaging machine where the cost of a dropped part is Rs 20 and the axis runs at 40 rpm. Stepper motor sizing and Indian prices together decide whether a stepper axis is viable at all.

When to move to a servo

Until one of those is true, a correctly sized stepper with a closed-loop driver is the cheapest reliable answer for an Indian shop, and the closed-loop version costs 30 to 50 percent more than the open-loop one while removing the only failure mode that matters. Stepper motor sizing against the cost of a servo is the decision, and the answer depends on whether a missed position costs more than the price difference.

Related Reading

• Industrial Motion Control: Complete Guide to Servo, Stepper and Axis Design
• Servo Drive Selection and Sizing: Power, Regeneration and Cable
• Inertia Ratio and Servo Oscillation: Causes, Limits and Fixes
• Motion Control Troubleshooting: Hunting, Overheating and Oscillation

Sources

1. Oriental Motor, Stepper Motor Technical Reference (pull-out torque characteristics)
2. Leadshine, Stepper Motor and Driver Selection Guide (supply voltage, current, microstepping)
3. Rockwell Automation, Stepper Motor Sizing and Selection Technical Reference
4. IEC 60034-1, Rotating electrical machines: Rating and performance
5. Krishna Electronics, Indian stepper motor and closed-loop driver catalogue

Key Takeaways

• Stepper motor sizing means never sizing on holding torque alone: it is the torque a stepper holds at standstill with all phases energised. It is not the running torque, and pull-out torque at the working speed is typically 40 to 70 percent lower.
• Apply a safety factor of 2 on the dynamic torque requirement for a horizontal axis and 3 to 4 for a vertical axis that must hold against gravity with the driver powered down.
• Torque falls with speed because of the back EMF. A stepper correctly sized at 500 rpm is 30 to 50 percent under-torqued at 1500 rpm on the same frame.
• The mid-speed resonance band around 200 to 400 rpm on a 1.8 degree stepper is where lost steps are most likely. Run below it, run above it, or use microstepping with closed-loop feedback.
• The driver supply voltage sets the top speed. Moving from 24 V to 48 V roughly doubles the maximum usable rpm on the same motor and frame.