The fundamental motor torque formula defines the rotational force a motor can deliver at a given speed. In imperial units, it is expressed as T (lb-ft) = (HP × 5252) / RPM. In metric, it is τ (Nm) = P (Watts) / ω (rad/s), where ω is the angular velocity. However, for practical bench and jobsite sizing, you must calculate the required load torque using the linear-to-rotational translation: T = F × r (Force × radius), factoring in both friction and acceleration. Selecting the right drive is not just about matching peak power; it is about matching the torque curve to the mechanical load profile. Below, we break down the math, compare motor topologies, and detail the wiring and failure signatures you will encounter on the workbench.

The Core Motor Torque Formula and Sizing Rule of Thumb

To size a motor, you must first calculate the continuous and peak torque your mechanical system demands. The basic load torque formula is:

Ttotal = (Ffriction + Facceleration) × r

Where r is the radius of the drive pulley, gear, or lead screw effective radius.
Bench Rule of Thumb: Never blindly convert HP to kW and size a motor without looking at the RPM context. A 1/4 HP AC induction motor at 1750 RPM produces roughly 0.75 lb-ft of torque, while a 1/4 HP stepper at 100 RPM produces over 13 lb-ft. The power is identical, but the torque is vastly different. Always apply a 1.5x to 2.0x safety factor to your calculated continuous torque to account for unmodeled friction, voltage drops, and thermal derating.

Worked Load Example: Belt-Driven Linear Axis

Let’s size a motor for a CNC router X-axis moving a 20 kg gantry on a belt drive with a 0.05m (50mm) radius drive pulley. We need an acceleration of 0.5 m/s².

  1. Acceleration Force: Faccel = mass × acceleration = 20 kg × 0.5 m/s² = 10 N.
  2. Friction Force: Assuming a linear rail friction coefficient (μ) of 0.1, Ffriction = 20 kg × 9.81 m/s² × 0.1 = 19.62 N.
  3. Total Linear Force: Ftotal = 10 N + 19.62 N = 29.62 N.
  4. Required Torque: T = 29.62 N × 0.05 m = 1.48 Nm (peak during acceleration).

Applying a 2.0x safety factor, you need a motor capable of delivering at least 2.96 Nm of peak torque. A standard NEMA 23 stepper rated for 3.0 Nm holding torque is the baseline choice here, provided it doesn't lose torque at your target traverse speed.

Motor Type Selection: Matching the Load Profile to the Drive

Not all torque is created equal. Steppers and servos are absolutely not interchangeable; their torque curves dictate entirely different mechanical applications. The table below maps motor topologies to their specific load profiles and controller demands.
Motor Type Torque Curve Profile Required Controller / Driver Typical Cost (400W / NEMA 23 equiv) Ideal Load Profile
Bipolar Stepper Maximum torque at zero speed (holding); drops off sharply past 300-500 RPM due to back-EMF. Chopper drive (e.g., TB6600, TMC2209). Requires step/dir pulse signals. $25 - $60 Low-speed, high-holding-torque applications (3D printers, low-speed conveyors).
AC Servo Flat, constant torque curve up to rated RPM (usually 3000 RPM), then constant power drop-off. Closed-loop servo drive (e.g., Yaskawa Sigma-7). Requires high-res encoder feedback. $400 - $900 High-speed, high-dynamic-response loads (CNC spindles, robotic arms, pick-and-place).
BLDC (Inrunner) Broad, flat torque curve. High RPM capability but lower raw holding torque than steppers. FOC (Field Oriented Control) driver (e.g., ODrive, SimpleFOC). Needs Hall sensors or sensorless BEMF. $120 - $250 Continuous high-speed rotation with moderate load (drones, e-bikes, high-speed spindles).
AC Induction (VFD) Low starting torque (across-the-line), but stable running torque near synchronous speed. Variable Frequency Drive (VFD). Controls V/Hz ratio or uses vector control. $150 - $350 High-inertia, continuous-duty industrial loads (pumps, fans, heavy conveyors).

If your load requires holding a heavy mass stationary against gravity without a mechanical brake, a stepper or a servo with an electromagnetic fail-safe brake is mandatory. A BLDC or AC induction motor will back-drive and drop the load when power is removed.

Wiring, Terminals, and Controller Demands

Miswiring motor phases is the fastest way to brick a driver or demagnetize a rotor. Terminal identification varies strictly by motor topology. Always refer to the NEMA MG-1 standard for industrial AC motors, and manufacturer datasheets for precision DC motors.

Stepper Motors (Bipolar)

Bipolar steppers have four wires, typically grouped into two coils (Phase A and Phase B). Terminals are labeled A+, A-, B+, B-. Bench verification: Use a digital multimeter (DMM) in resistance mode. You should read a low resistance (typically 0.5Ω to 3.0Ω) between A+ and A-, and between B+ and B-. There must be infinite resistance (OL) between any Phase A wire and any Phase B wire. If your stepper has 6 or 8 wires, you must identify and isolate the center taps, leaving them unconnected for bipolar chopper drivers.

BLDC Motors (3-Phase)

Brushless DC motors use three main power phases labeled U, V, and W. Swapping any two of these will reverse the motor's direction. In addition to the power phases, most hobby and industrial BLDCs include a 5-pin or 6-pin JST connector for Hall effect sensors (H1, H2, H3, 5V, GND). When wiring to an FOC controller like the ODrive robotics platform, ensuring the Hall sensor sequence matches the electrical phase sequence is critical; a mismatch will cause the motor to violently stutter or trip the driver's overcurrent protection.

AC Servo Motors

Servo wiring is the most complex, splitting into power and feedback domains. The power terminals are typically R, S, T (or L1, L2, L3) for the main AC input, and U, V, W for the motor output. The encoder cable carries differential pairs for A, B, and Z (index) channels, plus a 5V supply and shield. Never run the encoder cable parallel to the motor power cables without a grounded braided shield; the PWM switching noise from the servo drive (switching at 8-16 kHz) will induce errors in the encoder signals, causing position drift.

Diagnosing Failure Signatures: Hum, Overheat, and Stall

When a drive system fails, the physical symptoms tell you exactly which parameter is out of bounds. Here is how to diagnose the three most common bench and field failures.

1. Stepper Stall and the "Loud Hum"

Symptom: The motor emits a loud, low-frequency buzzing or humming sound, refuses to rotate, and the casing rapidly exceeds 80°C.
Cause: The demanded acceleration torque exceeds the motor's pull-out torque at that speed, or the driver current limit is set too low.
Fix: First, check your driver's Vref (reference voltage). For common drivers like the DRV8825, the RMS current limit is calculated as Vref = (I_rms × 8 × R_sense) / 1.414. If Vref is too low, the chopper drive is starving the coils. If Vref is correct, you must lower the acceleration ramp in your firmware (e.g., Marlin or Mach3) or add a mechanical reduction (gearbox) to multiply torque.

2. BLDC Desync and Overheat

Symptom: The motor runs smoothly at low RPM but suddenly stutters, emits a high-pitched whine, and stalls when a load is applied. The driver MOSFETs become too hot to touch.
Cause: Commutation desynchronization. The controller's FOC algorithm has lost track of the rotor's magnetic position relative to the stator coils, causing it to push current against the rotor's field rather than pulling it.
Fix: If using Hall sensors, verify the sensor wiring and check for loose crimps. If running sensorless, the back-EMF zero-crossing detection is failing at high load. Increase the "observer gain" or switch to a sensor-based FOC tuning profile. Consult the Texas Instruments motor drive portfolio documentation for specific FOC tuning registers on your IC.

3. Servo Oscillation (Hunting)

Symptom: The motor holds position but vibrates rapidly, emitting a high-frequency "chattering" sound, even when the load is static.
Cause: The derivative (D) gain in the servo drive's PID position loop is too high, or there is mechanical backlash in the coupling.
Fix: Access the drive's tuning software (e.g., Yaskawa SigmaWin+). Lower the derivative gain to dampen the high-frequency response. If the chatter persists, inspect the mechanical coupling between the motor shaft and the load; even 0.5mm of backlash in a flexible jaw coupling will cause a high-gain servo to endlessly hunt for the exact encoder tick mark.