Stepper Motor Speed, Torque and Steps Calculator
Stepper Motor Speed, Torque and Steps Calculator
Steps per revolution, rpm and the linear travel per step for a leadscrew or a belt — the steps-per-millimetre figure every 3D printer and CNC build needs — plus the corner speed where the winding’s inductance stops the current reaching its setpoint and the torque starts to fall away.
stepper motor speed and steps
a 1.8° motor at 16× microstepping taking 8,000 pulses per second, driving a 20-tooth 2 mm belt pulley, from 24 V into 2.8 mH / 1.5 Ω at 1.7 A
Steps, millimetres and the inductive limit
rpm = pulse rate × 60 ÷ microsteps per revolution
steps/mm = microsteps per revolution ÷ mm per revolution
mm per revolution = leadscrew lead, or belt pitch × pulley teeth
felec = rev/s × full steps ÷ 4 corner: I √(R² + (2π f L)²) = Vsupply
τ = L / R Vrated = I × R
- step angle
- 1.8° for almost every NEMA 17 and NEMA 23 — 200 full steps per turn. It comes from the rotor’s 50 teeth and the two phases
- f elec
- the electrical frequency in one phase. A two-phase stepper takes four full steps to complete one electrical cycle, so a 200-step motor does 50 cycles per revolution
- L, R
- phase inductance and resistance, from the datasheet. L is what limits high-speed torque; R is what limits how hot the motor gets
- V supply
- the chopper drive’s supply, typically ten to twenty-five times the motor’s own rated voltage. It is the only lever that moves the corner speed
- holding torque
- the torque needed to pull the shaft out of position while stationary at rated current. An upper bound on what is available at any speed, and a bad estimate of it
Worked example
a 1.8° motor at 16× microstepping taking 8,000 pulses per second, driving a 20-tooth 2 mm belt pulley, from 24 V into 2.8 mH / 1.5 Ω at 1.7 A
360 ÷ 1.8° = 200 full steps per revolution, and at 16× that is 3,200 microsteps
8,000 pulses/s ÷ 3,200 = 2.500 rev/s = 150.00 rpm
The belt carries 2 × 20 = 40 mm per revolution, so steps/mm = 3,200 ÷ 40 = 80 and each microstep is 12.50 µm. The carriage moves at 100 mm/s
Electrically the motor is doing 2.500 × 200 ÷ 4 = 125.00 Hz per phase — four full steps to an electrical cycle
The winding is 2.8 mH and 1.5 Ω, so its L/R time constant is 1.87 ms and its own rated voltage is only 2.55 V — the 24 V supply is 9.4 times that, which is what makes the current rise fast enough to be useful
Corner speed: 1.7 A through √(R² + (2πfL)²) needs the whole 24 V at f = 798 Hz, which is 958 rpm or 638 mm/s. Below that the torque is near its low-speed value; above it, it falls
Holding torque 43 N·cm is 0.430 N·m, 4.38 kgf·cm or 60.9 oz·in — but that is the stationary figure, not what is available at 150 rpm
Steps per millimetre for the drives people actually build
| Drive | mm per revolution | Step angle | Microstepping | Microsteps per rev | Steps per mm | Travel per microstep |
|---|---|---|---|---|---|---|
| GT2 belt, 20-tooth pulley | 40.000 | 1.80° | 16× | 3,200 | 80.0000 | 12.500 µm |
| GT2 belt, 16-tooth pulley | 32.000 | 1.80° | 16× | 3,200 | 100.0000 | 10.000 µm |
| T8 leadscrew, 4 starts, 2 mm pitch | 8.000 | 1.80° | 16× | 3,200 | 400.0000 | 2.500 µm |
| T8 leadscrew, single start | 2.000 | 1.80° | 16× | 3,200 | 1,600.0000 | 0.625 µm |
| M5 threaded rod | 0.800 | 1.80° | 16× | 3,200 | 4,000.0000 | 0.250 µm |
| Ball screw, 5 mm lead | 5.000 | 1.80° | 8× | 1,600 | 320.0000 | 3.125 µm |
| Ball screw, 10 mm lead | 10.000 | 0.90° | 8× | 3,200 | 320.0000 | 3.125 µm |
| GT2 belt, 20-tooth, 0.9° motor | 40.000 | 0.90° | 16× | 6,400 | 160.0000 | 6.250 µm |
The arithmetic is exact; the torque is not
Half of this page is counting. A 1.8° motor has 200 full steps per revolution because its rotor carries 50 teeth and it has two phases; multiply by the driver’s microstepping and you have the microsteps per turn. Divide the pulse rate by that and you have the speed. Divide the microsteps per turn by the millimetres the mechanism moves per turn — the leadscrew’s lead, or the belt pitch times the pulley teeth — and you have steps per millimetre, which is the single number every 3D printer and CNC firmware asks for. None of that is approximate.
The other half is where designs fail. Each phase of a stepper is an inductance in series with a resistance, and torque is proportional to the current in it. At a standstill the current has all the time in the world to settle at whatever the driver sets. As the speed rises, the current has to be reversed in each phase faster and faster, and the inductance resists that: the voltage needed to drive a sinusoidal current of amplitude I at electrical frequency f is I·√(R² + (2πfL)²), and it rises without limit. When it reaches the supply voltage, the current can no longer reach its setpoint. That is the corner speed, and it is the single most useful number here: below it, torque is roughly the low-speed figure; above it, torque falls away.
Two things follow immediately. First, the supply voltage is the lever. A chopper drive exists precisely so that the supply can be ten to twenty-five times the motor’s own rated voltage (which is just rated current × phase resistance, and is typically only two or three volts); the chopper switches to keep the current at the setpoint and the high voltage is what makes the current rise fast. Doubling the supply roughly doubles the corner speed. Second, low inductance beats high torque for a fast machine. Two motors with the same holding torque but inductances of 2 mH and 8 mH have corner speeds a factor of four apart, and above the corner it is the low-inductance motor that wins by a mile. This is why a big high-torque motor often makes a machine slower rather than faster.
And now the honest part. The corner speed is a first-order estimate from R + jωL and nothing else. It does not include the back-EMF, which subtracts from the available voltage and grows with speed; it does not include the chopper’s own switching behaviour; it does not include resonance, which can dump a stepper out of synchronism at a few hundred rpm even where there is plenty of torque. Pull-out torque — the torque actually available at a given speed — comes from the manufacturer’s measured speed-torque curve, not from any formula on this page or anywhere else. Those curves are made by spinning the motor up to a known speed with a particular drive at a particular voltage and loading it with a brake until it stalls, which is why they are always published against a stated supply voltage and driver. Use the corner speed to see whether a design is in the flat part of the curve or the falling part, then get the curve.
One more caveat, because it is the commonest misunderstanding in hobby machine building: microstepping is smoothness, not resolution. The torque holding the rotor at a microstep is the holding torque times the sine of the angle from the nearest full-step detent, so at 1/16 stepping one microstep from the detent has about 10% of full torque behind it, and at 1/256 about 0.6%. Friction in the machine will absorb several microsteps’ worth of position error without the driver ever knowing. Fine microstepping makes a motor quiet and smooth; it does not make a machine accurate.
For the torque, power and speed relationship at the shaft in the general case — and the same N·m, kgf·m and lbf·ft conventions used here — see the motor power, torque and speed calculator. For what the driver’s switching costs in heat, the BLDC inverter loss calculator covers the same bridge topology, and the heatsink calculator sizes the cooling. For the step pulse train itself, the PWM duty cycle calculator. Use the figures from your part’s datasheet; typical values vary widely between manufacturers.
Frequently asked questions
How do I calculate steps per mm?
Microsteps per revolution divided by millimetres of travel per revolution. A 1.8° motor at 16× microstepping is 3,200 microsteps per turn; on a GT2 belt with a 20-tooth pulley the carriage moves 40 mm per turn, so it is 80 steps/mm. On a T8 leadscrew with an 8 mm lead it is 400 steps/mm.
What is the corner speed of a stepper motor?
The speed above which the supply can no longer push rated current through the winding’s impedance before the step is over, so the torque starts to fall. It is the frequency at which I·√(R² + (2πfL)²) equals the supply voltage, converted to rpm. A 1.8° motor with 2.8 mH and 1.5 Ω at 1.7 A from 24 V corners at about 958 rpm.
Why does stepper torque fall off at speed?
Because the winding is an inductance. Torque follows current, and reversing the current in an inductance takes voltage: the faster the steps, the less of the rated current the drive can establish before the step ends. Back-EMF makes it worse again. A higher supply voltage is the fix, and a lower-inductance motor is the other one.
Can I calculate pull-out torque?
No, and neither can anyone else from a datasheet’s static figures. Pull-out torque at a given speed depends on the motor, the driver, the supply voltage, the microstepping scheme and the load inertia, and it is measured rather than computed — a brake is applied at each speed until the motor loses synchronism. Get the manufacturer’s speed-torque curve for your supply voltage and read it.
Does more microstepping make my machine more accurate?
No. It makes it smoother and quieter. The torque available to hold a microstep falls with the sine of the microstep angle, so at 1/16 a single microstep from the detent commands only about 10% of full torque and friction will swallow it. Positional accuracy comes from the screw or belt, the bearings and the structure, not from the driver’s step resolution.
What supply voltage should I use for a stepper?
As high as the driver and the motor’s heating allow — that is what raises the corner speed. LinuxCNC’s stepper guide gives 32 × √(inductance in mH) volts as a working ceiling, so about 54 V for a 2.8 mH motor, with some headroom below the driver’s own maximum. The motor’s own rated voltage is irrelevant with a chopper drive: it is only rated current × phase resistance, typically two or three volts.
Related calculators
References
- Oriental Motor. Speed – torque curves for stepper motors (technology note). Defines pull-out torque as “the maximum torque that the stepper motor can supply to a load at any given speed”, and states how the curves are produced: “spinning a step motor up to a known speed and then gradually applying torque to the output shaft with a brake and measured with a torque transducer”, repeated three times per point. They are measured, not calculated — which is why this page refuses to compute one.
- STMicroelectronics. AN460: Stepper motor drive considerations — common problems and solutions. “At higher speeds the winding inductance limits the rate of change of the current and the current can no longer reach its full value in each step”; V = L di/dt; and why chopper control “improves the current rise time in the motor and improves the torque at high speeds”.
- LinuxCNC. Stepper motor information (integrator manual). The supply-voltage rule of thumb used here: “take the square-root of the winding inductance quoted in the motor datasheet (expressed in milli-Henries) and multiply by 32”, with approximately 10% headroom below the driver’s maximum rating.
- Ningbo Leison / Usongshine 17HS4401 datasheet (NEMA 17, 40 mm). Step angle 1.8° ±5%, two phase, rated voltage 2.6 V dc, rated current 1.7 A per phase, phase resistance 1.5 Ω ±10%, phase inductance 2.8 mH ±20% at 1 kHz, holding torque 43 N·cm, rotor inertia 54 g·cm². These are the page’s defaults.
- Acarnley PP. Stepping Motors: A Guide to Theory and Practice, 4th ed. IET, 2002. Chapters 3 and 6 — the hybrid motor’s four full steps per electrical cycle, the static torque/angle curve behind the microstepping caveat, and why the high-speed torque limit is a voltage problem rather than a current one.