The fundamental equation for torque of a motor in mechanical terms is τ = F × r (Torque = Force × radius). When translating electrical power to mechanical output, the governing equation is τ = P / ω (Torque = Power / angular velocity). In imperial shop terms, this translates to T (lb-ft) = (HP × 5252) / RPM. Knowing the math is only the first step; applying it to select the right motor topology, match the torque curve to your load profile, and wire the correct driver is where projects succeed or fail.
Below is a table-forward guide to sizing motors, understanding their torque delivery, and wiring the controllers that drive them.
Motor Type Selection Matrix: Torque Curves, Drivers, and Costs
Not all torque is created equal. A stepper motor delivers massive holding torque but loses it rapidly at speed, while a 3-phase AC induction motor delivers a flat torque curve up to its base speed. Use this matrix to select the right topology before running your sizing calculations.
| Motor Type | Torque Curve Shape | Peak vs. Continuous | Required Driver / Controller | Relative Cost (2026) |
|---|---|---|---|---|
| NEMA 23 Stepper (Bipolar) | High holding torque; drops sharply after 800-1200 RPM. | Peak = Holding. Continuous is ~50% of peak due to thermal limits. | Bipolar Chopper Drive (e.g., DM542T, TB6600). Requires step/dir pulses. | $ (Motor: $15, Driver: $20) |
| BLDC Outrunner (Hobby/Drone) | Flat constant torque to base speed, then constant power (drops). | Peak is 3x to 5x continuous for short bursts (limited by magnet temp). | FOC (Field Oriented Control) ESC (e.g., ODrive Pro, SimpleFOC). | $$ (Motor: $40, FOC: $60) |
| 3-Phase AC Induction (TEFC) | Flat constant torque up to nameplate base speed; steep drop at slip. | Breakdown torque is 200-300% of Full Load Amps (FLA) torque. | VFD (Variable Frequency Drive) with V/Hz or Vector control. | $$ (Motor: $80, VFD: $120) |
| Brushed DC (Permanent Magnet) | Linear drop from stall torque at 0 RPM to zero at no-load speed. | Stall torque is max, but continuous stall will burn the windings instantly. | H-Bridge (e.g., BTS7960) or simple PWM MOSFET switch. | $ (Motor: $10, Driver: $8) |
The Sizing Rule of Thumb & Worked Load Example
The Rule: Always size your motor for 150% to 200% of the calculated continuous load torque. This safety factor accounts for breakaway friction, inertia during acceleration, and voltage drops under load.
You are designing a small winch to lift a 20 lb (89 N) payload using a spool with a 2-inch (0.05 m) radius.
1. Calculate Load Torque: τ = F × r → 89 N × 0.05 m = 4.45 Nm (approx. 39.4 lb-in).
2. Apply Safety Factor: 4.45 Nm × 2.0 = 8.9 Nm required motor torque.
3. Selection: A standard NEMA 23 stepper (like the OmniStepper OS23-4) offers ~1.2 Nm holding torque—far too low. You need a NEMA 34 stepper, a high-torque BLDC with a 10:1 planetary gearbox (which multiplies torque by 10 while dividing speed by 10), or a 1/2 HP AC induction motor with a worm gear reduction.
Matching the Equation to Load Profiles
The equation τ = P / ω dictates how a motor behaves when the load demands change. Understanding your load profile prevents you from buying a motor that stalls under real-world conditions. For deeper physics on how magnetic fields generate this rotational force, refer to the All About Circuits DC motor torque chapter.
Constant Torque Loads (Conveyors, Hoists, Pumps)
In these applications, the torque required to move the load remains roughly the same regardless of speed. If you need 5 Nm to turn a conveyor belt at 100 RPM, you still need 5 Nm at 500 RPM. Because τ is constant, Power (P) must increase linearly as angular velocity (ω) increases. Best fit: AC Induction motors driven by a VFD, or BLDC motors operating below their base speed.
Constant Power Loads (Winders, Spindles, Traction)
In a winder, as the spool diameter grows, the RPM must drop to maintain constant material tension and line speed. Because Power (P) is constant and ω is dropping, the equation demands that Torque (τ) must rise. Best fit: BLDC or AC Induction motors utilizing field-weakening control, which allows the driver to trade top-end speed for low-end torque multiplication.
Wiring, Terminals, and Controller Demands
Miswiring a motor phase will result in violent vibration, immediate stalling, or a destroyed driver. Here is the terminal identification for the three most common topologies.
1. Bipolar Stepper Motors (4-Wire)
Steppers have two independent coils. You must identify the pairs. Wire colors vary by manufacturer, so never trust the colors blindly.
- Testing: Set your multimeter to resistance (Ω). Probe the wires. Two wires will show a low resistance (typically 1Ω to 5Ω). That is Coil A (e.g., A+ and A-). The remaining two wires will show the same resistance; that is Coil B.
- Terminals: Connect Coil A to the driver's A+ and A-. Connect Coil B to B+ and B-.
- Direction Fix: If the motor spins backward, simply swap the two wires of one coil (e.g., swap A+ and A-). Do not mix A and B wires.
2. BLDC Motors (3-Phase + Hall Sensors)
Modern FOC (Field Oriented Control) drivers like those documented in the ODrive robotics documentation require precise phase and feedback wiring.
- Phase Wires (Thick): Labeled U, V, W. Connect these to the driver's U, V, W terminals. If the motor stutters or spins backward under FOC, swap any two phase wires (e.g., U and V).
- Hall Sensors (Thin): Usually 5 wires. 5V (Red), GND (Black), and signal lines H1, H2, H3 (often Green, Blue, Yellow). Connect these to the microcontroller or FOC driver's encoder inputs. Supplying 12V to the 5V Hall line will instantly fry the internal sensor ICs.
3. 3-Phase AC Induction (9-Lead Dual Voltage)
Standard TEFC (Totally Enclosed Fan Cooled) motors often have 9 leads (T1 through T9) in the peckerhead, allowing wiring for either 230V (Delta) or 460V (Wye).
- Low Voltage (230V Delta): Tie T1-T7 to L1, T2-T8 to L2, T3-T9 to L3. Tie T4-T5-T6 together.
- High Voltage (460V Wye): Tie T4-T7, T5-T8, T6-T9 together and insulate. Connect L1 to T1, L2 to T2, L3 to T3.
- Driver: Always use a VFD rated for the motor's FLA (Full Load Amps). Set the VFD's motor nameplate parameters (Hz, RPM, FLA) before running auto-tuning.
Failure Signatures: Diagnosing Hum, Overheat, and Stall
When the math on paper doesn't match reality on the bench, the motor will tell you what went wrong through acoustic and thermal signatures. Here is how to diagnose the big three.
The Stepper 'Scream' and Missed Steps
Symptom: The motor emits a high-pitched whine or hum, the shaft vibrates but doesn't turn, or it loses position during rapid acceleration.
The Cause: You have exceeded the motor's pull-out torque at that specific RPM. Remember the stepper torque curve: as RPM rises, back-EMF limits the current the chopper drive can push into the coils, causing torque to plummet.
The Fix:
1. Lower the acceleration ramp in your firmware (e.g., GRBL or Marlin).
2. Check the driver's DIP switches to ensure the RMS current limit matches the motor's datasheet (usually 70% of the peak current rating).
3. Increase the driver's DC supply voltage (e.g., from 24V to 48V) to overcome back-EMF faster, provided the driver's max voltage isn't exceeded.
BLDC Overheat and Demagnetization
Symptom: The motor smells like hot epoxy, loses torque permanently over time, or the ESC faults out on thermal shutdown.
The Cause: Continuous RMS current exceeds the thermal limit of the stator windings. In outrunner BLDCs, the permanent magnets are on the outer bell. If the stator gets too hot (typically >120°C for standard NdFeB magnets), the magnets suffer irreversible demagnetization.
The Fix: Measure the phase current with a clamp meter. If it's near the peak rating continuously, you must add forced air cooling (a fan blowing directly into the bell) or downgrade the continuous torque demand via mechanical gearing. Consult Texas Instruments' motor driver resources for calculating exact thermal dissipation requirements for your specific FOC bridge.
AC Induction Hum and Breaker Trips
Symptom: A loud, angry 120Hz hum, the motor shaft refuses to turn (locked rotor), and the branch circuit breaker trips within seconds.
The Cause: Single-phasing. One of the three AC legs is missing due to a blown fuse, a failed contactor pole, or a broken wire. The motor is attempting to run as a single-phase motor, which it cannot do from a dead stop, resulting in massive current draw on the remaining two legs.
The Fix: De-energize and lock out the panel. Use a multimeter to check phase-to-phase voltage at the motor terminals (T1-T2, T2-T3, T1-T3). All three readings must be within 2% of each other. If one reads 0V, trace back to the VFD output contactors or fuses. Never attempt to 'bump' the starter to force it to turn; you will burn the windings.
