The fundamental torque in motor formula is expressed as T = P / ω (Torque = Power / angular velocity). In practical metric units, this translates to T (Nm) = (9550 × kW) / RPM. In imperial units, the formula is T (lb-ft) = (HP × 5252) / RPM. For example, if you are driving a 0.5 kW motor at 1500 RPM, the continuous output torque is exactly 31.83 Nm. However, calculating raw motor output is only the first step; matching that torque to your specific load profile, selecting the correct driver, and wiring the terminals correctly is where most bench and jobsite projects fail.
The Core Torque in Motor Formula and Sizing Rule of Thumb
When sizing a motor, you cannot rely on static friction alone. The total required torque is the sum of load torque (overcoming friction and gravity) and acceleration torque (overcoming inertia). The acceleration component relies on Newton's second law for rotation: T = J × α, where J is the moment of inertia (kg·m²) and α is angular acceleration (rad/s²).
Worked Load Example: Sizing a Conveyor Drive
Suppose you need to drive a flat belt conveyor moving a 50 kg payload at 0.5 m/s. The drive pulley has a radius of 0.1 m.
- Calculate Force: F = m × g = 50 kg × 9.81 m/s² = 490.5 N (assuming the belt is horizontal and friction coefficient is roughly 0.1, effective pulling force is ~49 N, but let's size for a 15-degree incline where gravity dominates: F = 50 × 9.81 × sin(15°) = 126.8 N).
- Calculate Static Load Torque: T = F × r = 126.8 N × 0.1 m = 12.68 Nm.
- Calculate Acceleration Torque: If the system must reach 0.5 m/s in 0.5 seconds, the linear acceleration is 1 m/s². Translating this to the pulley requires calculating the total reflected inertia of the belt, pulleys, and payload. Assuming a total reflected inertia J = 0.05 kg·m², and angular acceleration α = (1 m/s² / 0.1 m) = 10 rad/s². Acceleration Torque = 0.05 × 10 = 0.5 Nm.
- Total Peak Torque: 12.68 Nm + 0.5 Nm = 13.18 Nm.
- Apply Safety Factor: 13.18 Nm × 1.5 = 19.77 Nm.
You need a motor and gearbox combination capable of delivering at least 19.77 Nm at the output shaft. If you select a motor spinning at 3000 RPM, you would use a 60:1 gearbox to drop the speed to 50 RPM at the pulley, which multiplies the motor's required torque output by the gear ratio (minus efficiency losses).
Motor Type Comparison: Torque Curves and Drive Requirements
Selecting the right motor architecture depends entirely on the shape of your torque curve. Do not treat steppers and servos as interchangeable. A NEMA 23 stepper relies on open-loop magnetic detents for massive holding torque at zero RPM, while a NEMA 23-frame BLDC servo requires a closed-loop encoder to maintain position and delivers its peak torque dynamically at high speeds.
| Motor Type | Torque Curve Profile | Control / Driver Needs | Approx Cost (2026) |
|---|---|---|---|
| Bipolar Stepper | High holding torque at 0 RPM; drops off sharply after 1000 RPM due to back-EMF. | Open-loop chopper driver (e.g., TB6600, DM542T). Pulse/direction signals. | $25 - $60 |
| BLDC (Outrunner) | Flat continuous torque up to base speed, then constant power (torque drops as 1/RPM). | Sensorless or Hall-effect ESC. Requires 3-phase commutation and back-EMF zero-crossing detection. | $40 - $120 |
| AC Induction (NEMA) | Low starting torque, peaks at breakdown torque (usually ~1750 RPM for 4-pole 60Hz). | Direct-on-line (DOL) contactor or VFD for variable speed. Simple wiring. | $150 - $400+ |
| AC Servo | Constant peak torque (often 300% continuous) from 0 to rated RPM (e.g., 3000 RPM). | Closed-loop servo drive with high-resolution absolute encoder. EtherCAT/Modbus control. | $300 - $800+ |
Wiring and Terminal Identification: NEMA 23 Bipolar Stepper
When wiring a standard 4-lead NEMA 23 bipolar stepper motor to a chopper driver like the Texas Instruments DRV8825 or a commercial DM542T, correct phase pairing is critical. If you cross the phases, the motor will vibrate violently and stall.
| Driver Terminal | Motor Coil | Standard Wire Color | Verification Method |
|---|---|---|---|
| A+ | Coil A Start | Red | Short these two wires together. The motor shaft should become difficult to turn by hand (cogging). |
| A- | Coil A End | Blue | |
| B+ | Coil B Start | Green | Short these two wires together. The shaft should again exhibit strong magnetic resistance. |
| B- | Coil B End | Black |
Note: Wire colors vary by manufacturer (e.g., Wantai vs. DroidLabs). Always verify coil pairs with a multimeter in continuity mode before applying power.
Matching the Motor to the Load Profile and Failure Signatures
Once you have applied the torque in motor formula and selected your hardware, you must monitor for failure signatures. Motors rarely fail silently; they give distinct electrical and acoustic warnings based on the NEMA MG 1 standards for thermal and mechanical limits.
Which Motor Fits Your Load Profile?
- High holding torque, low speed, precise indexing: Choose a Stepper. Ideal for 3D printer extruders, CNC Z-axes, and camera sliders. The open-loop nature means it holds position without drawing continuous feedback current, though it runs hot.
- High dynamic torque, high speed, rapid direction changes: Choose a BLDC Servo. Ideal for robotic arms, pick-and-place machines, and high-speed conveyors. The closed-loop controller dynamically adjusts current to match the exact torque demand, preventing stalls.
- Constant speed, high inertia, continuous duty: Choose an AC Induction Motor. Ideal for HVAC blowers, water pumps, and long warehouse conveyors. The rotor's slip naturally absorbs shock loads without stalling the drive.
Diagnostic Failure Signatures
- The 'Hum' or 'Buzz' (Mid-Band Resonance): Common in steppers between 100-300 Hz step rates. The rotor overshoots the magnetic detent and oscillates. Fix: Implement 1/16 microstepping on the driver, add a mechanical damper to the shaft, or use a driver with active anti-resonance algorithms (like the Leadshine DM556).
- Overheating (Winding Insulation Bake): If your motor casing exceeds 80°C, your continuous RMS current is likely set too high on the driver's DIP switches. Chopper drivers dissipate heat as I²R. Fix: Lower the driver's current limit to 70-80% of the motor's rated peak current, or enable 'idle current reduction' (often a 50% current drop when the step pulse stops).
- Stall (Missed Steps): The load has exceeded the motor's 'pull-out torque' curve at that specific RPM. Unlike a VFD-driven AC motor that will just draw more current and trip a breaker, an open-loop stepper will simply stop moving while the driver continues to pulse. Fix: Add a gearbox to multiply torque, or reduce the acceleration ramp in your firmware (e.g., lower the $120 acceleration setting in GRBL).
Frequently Asked Questions About Motor Torque Calculations
How does the torque in motor formula change when using a gearbox?
When you introduce a gearbox, the output torque is multiplied by the gear ratio (G), minus the mechanical efficiency (η) of the gear train. The modified formula is T_out = T_in × G × η. For example, a 2 Nm stepper motor driving a 10:1 planetary gearbox with 90% efficiency will yield 18 Nm of output torque. However, the output speed is simultaneously divided by the gear ratio. Always check the gearbox's maximum rated input torque; exceeding it will strip the planetary gears regardless of the motor's electrical limits.
Why does my calculated torque in the motor formula not match the stall torque on the datasheet?
Datasheets for steppers and servos typically advertise holding torque or peak stall torque, which is measured at zero RPM with maximum rated current applied. The standard torque in motor formula calculates dynamic continuous torque at a specific operating speed. Due to back-EMF (the voltage generated by the spinning rotor that opposes the supply voltage), a stepper motor that produces 3 Nm at 0 RPM might only produce 0.8 Nm at 2000 RPM. Always consult the manufacturer's Torque-Speed Curve graph, not just the headline spec.
Can I use the standard torque in motor formula for a closed-loop servo?
Yes, but with a crucial distinction. For a servo, the formula calculates the mechanical output required by the load. However, the servo drive electronics manage the internal commutation torque limits dynamically. A 400W AC servo rated for 1.27 Nm continuous torque can typically output 3.81 Nm (300% overload capacity) for short bursts (usually 3 to 5 seconds) to overcome static friction or sudden impact loads. You must size the servo based on the continuous RMS torque of your duty cycle, using the peak torque only for acceleration phases.
What happens to the torque in the motor formula if I increase the supply voltage?
Increasing the supply voltage to a stepper or BLDC motor does not increase its maximum holding torque—torque is strictly a function of current (T = k_t × I). However, increasing the voltage drastically alters the torque-speed curve. A higher voltage forces current into the inductive windings faster, pushing the 'corner frequency' higher. This means the motor can maintain its rated torque at much higher RPMs before back-EMF chokes the current. Running a 24V stepper on a 48V supply (with the driver's current limit unchanged) will yield significantly more torque at 1500 RPM, but zero additional torque at 0 RPM.
