Understanding the Equation for Torque of a Motor
Torque is the fundamental force that makes a motor turn, and it directly determines how much rotational work a motor can perform. Whether you are designing a robotic arm, selecting a drive for an electric vehicle, or simply trying to choose the right hobby‑scale motor for a model airplane, knowing the torque equation is essential for predicting performance, avoiding overload, and optimizing efficiency. This article breaks down the torque equation for electric motors, explains each variable, shows how to calculate torque in practical situations, and addresses common questions that engineers and hobbyists often encounter.
1. Introduction: Why Torque Matters
Torque (τ) is the rotational analogue of linear force. Practically speaking, in a motor, torque is what creates the shaft rotation that ultimately drives a load. While force pushes or pulls an object along a straight line, torque twists an object around an axis. The higher the torque, the greater the motor’s ability to accelerate a mass, climb a slope, or overcome friction And that's really what it comes down to..
Because torque is tied to power, speed, and current, understanding its equation helps you:
- Size the motor correctly for a given load.
- Predict acceleration and dynamic response.
- Estimate energy consumption and thermal requirements.
- Diagnose performance issues such as stall or overheating.
2. The Core Torque Equation for DC and Brushless Motors
For most electric motors—particularly brushed DC, permanent‑magnet brushless DC (BLDC), and synchronous AC motors—the torque can be expressed as:
[ \boxed{\tau = k_t , I} ]
- τ (tau) – Torque, measured in newton‑meters (N·m).
- kₜ – Torque constant, expressed in N·m/A (newton‑meters per ampere).
- I – Armature or phase current, measured in amperes (A).
2.1 What Is the Torque Constant (kₜ)?
The torque constant links electrical current to mechanical output. It is a property of the motor’s construction (magnet strength, winding geometry, and number of poles). For a given motor, kₜ is usually provided in the datasheet and can also be derived from the back‑EMF constant (kₑ) because:
[ k_t = \frac{k_e}{2\pi} ]
where kₑ is expressed in volts per radian per second (V·s/rad). In SI units, kₜ and kₑ have the same numeric value when expressed respectively as N·m/A and V·s/rad The details matter here..
2.2 Incorporating Efficiency and Speed
The simple linear relationship τ = kₜ·I assumes ideal conditions (no losses). In real applications, motor efficiency (η) and speed (ω) affect the actual torque that reaches the shaft. A more comprehensive expression is:
[ \tau = \frac{k_t , I , \eta}{1 + \frac{R_s}{\omega L_s}} ]
- η – Mechanical efficiency (dimensionless, 0–1).
- Rₛ – Stator resistance (Ω).
- Lₛ – Stator inductance (H).
- ω – Angular speed (rad/s).
At low speeds, the inductive term (\frac{R_s}{\omega L_s}) becomes significant, reducing effective torque. At high speeds, the term approaches zero, and the torque approximates the simple τ = kₜ·I·η.
3. Step‑by‑Step Torque Calculation
Below is a practical workflow for calculating motor torque in a typical design scenario.
3.1 Gather Motor Data
| Parameter | Typical Source | Units |
|---|---|---|
| Torque constant (kₜ) | Datasheet or calculation from kₑ | N·m/A |
| Rated current (Iₙ) | Datasheet | A |
| Efficiency (η) | Datasheet (often at rated point) | – |
| Stator resistance (Rₛ) | Datasheet | Ω |
| Stator inductance (Lₛ) | Datasheet | H |
| Desired speed (ω) | System requirement | rad/s |
3.2 Compute Base Torque
[ \tau_{\text{base}} = k_t \times I_{\text{operating}} ]
If you plan to run the motor at 80 % of its rated current, set (I_{\text{operating}} = 0.8 \times I_n) Simple as that..
3.3 Adjust for Efficiency
[ \tau_{\text{eff}} = \tau_{\text{base}} \times \eta ]
Typical efficiencies for small BLDC motors range from 0.70 to 0.90.
3.4 Include Speed‑Dependent Losses (optional)
If you need high accuracy at low speeds, calculate the inductive correction factor:
[ C = \frac{1}{1 + \frac{R_s}{\omega L_s}} ]
Then:
[ \tau_{\text{final}} = \tau_{\text{eff}} \times C ]
3.5 Example
- Motor: 200 W BLDC, kₜ = 0.12 N·m/A, Iₙ = 10 A, η = 0.85, Rₛ = 0.2 Ω, Lₛ = 0.5 mH.
- Desired operating point: 5 A at 3000 rpm (ω = 3000 × 2π/60 ≈ 314 rad/s).
- Base torque: (0.12 \times 5 = 0.60) N·m.
- Efficiency‑adjusted torque: (0.60 \times 0.85 = 0.51) N·m.
- Inductive factor: (C = 1 / (1 + 0.2/(314 \times 0.0005)) ≈ 0.99).
- Final torque: (0.51 \times 0.99 ≈ 0.505) N·m.
Thus, at 5 A and 3000 rpm the motor can deliver roughly 0.5 N·m of shaft torque.
4. Torque in AC Induction Motors
Induction motors do not have a fixed torque constant because torque is produced by the interaction of a rotating magnetic field and induced currents in the rotor. On the flip side, the torque can still be expressed using the air‑gap power (P_ag) and angular speed:
[ \tau = \frac{P_{ag}}{\omega_s} ]
- P_ag – Air‑gap power (the power transferred across the stator‑rotor air gap).
- ω_s – Synchronous angular speed (rad/s).
The air‑gap power is related to the slip (s) and mechanical output power (P_out):
[ P_{ag} = \frac{P_{out}}{1 - s} ]
Slip is defined as (s = \frac{\omega_s - \omega_r}{\omega_s}), where ω_r is the rotor speed. By measuring or estimating slip, you can compute torque for an induction motor without a direct torque constant Small thing, real impact. No workaround needed..
5. Scientific Explanation: Why Current Generates Torque
In a permanent‑magnet motor, the stator windings create a magnetic field when current flows. This field interacts with the permanent magnets on the rotor, producing a Lorentz force on each conductor:
[ \mathbf{F} = I (\mathbf{L} \times \mathbf{B}) ]
where L is the length vector of the conductor and B is the magnetic flux density. The forces on all conductors sum to a net torque about the motor shaft. Because the magnetic field strength (B) and geometry (L) are fixed for a given motor, torque varies linearly with current, giving rise to the simple τ = kₜ·I relationship.
In brushless AC (BLDC) or synchronous motors, the same principle applies, but the magnetic field rotates electronically, allowing precise control of torque and speed through pulse‑width modulation (PWM) of the phase currents.
6. Frequently Asked Questions (FAQ)
Q1: Can I use the torque equation for a stepper motor?
A: Stepper motors are essentially multiple small DC motors operating in sequence. The torque per phase can be approximated with τ = kₜ·I, but the overall torque also depends on microstepping, detent torque, and magnetic saturation. For high‑precision applications, refer to the manufacturer’s torque‑vs‑current curves Worth knowing..
Q2: What happens when the motor reaches its stall torque?
A: Stall torque is the maximum torque the motor can produce at zero speed, occurring when the current reaches its limit (often the rated current). At stall, the back‑EMF is zero, so the applied voltage appears across the winding resistance, causing the highest possible current—and thus the highest torque—according to τ = kₜ·I. Prolonged stall can overheat the windings.
Q3: How does gear reduction affect torque?
A: A gear train multiplies torque by the gear ratio (GR) while reducing speed by the same factor (ignoring losses). Effective output torque = τ_motor × GR × η_gear, where η_gear is the mechanical efficiency of the gearbox (typically 0.9–0.98).
Q4: Is torque constant the same as the motor’s “strength”?
A: kₜ reflects how efficiently a motor converts current into torque. A high kₜ means a small current yields a large torque, which is often desirable for low‑speed, high‑torque applications. On the flip side, overall “strength” also depends on voltage rating, thermal capacity, and speed capability Which is the point..
Q5: Can I increase torque by raising the supply voltage?
A: Raising voltage increases the no‑load speed because back‑EMF rises, but torque is primarily a function of current. Higher voltage allows the controller to reach higher currents faster (better dynamic response), yet the steady‑state torque remains limited by the current limit set in the driver.
7. Practical Tips for Maximizing Motor Torque
- Use a proper current controller – A PWM driver with current feedback (e.g., a field‑oriented control (FOC) board) ensures the motor receives the exact current needed for the desired torque.
- Maintain good cooling – Since torque production is current‑limited, higher currents generate more heat. Adequate heat sinks or forced air keep the motor within its thermal envelope, allowing sustained torque.
- Select the right winding configuration – Motors with more turns per phase have higher kₜ but lower back‑EMF, affecting speed. Balance torque needs against speed requirements.
- Minimize supply voltage ripple – Voltage fluctuations cause current ripple, which translates to torque ripple—undesirable in precision motion systems. Use decoupling capacitors and stable power supplies.
- Consider gear reduction – If the application demands torque beyond the motor’s native capability, a high‑efficiency gearbox is often more economical than oversizing the motor.
8. Conclusion
The torque equation τ = kₜ·I lies at the heart of motor selection, control, and performance analysis. By understanding each variable—torque constant, current, efficiency, and speed‑dependent losses—you can predict how a motor will behave under load, design appropriate control strategies, and avoid common pitfalls such as stall overload or insufficient acceleration. Whether you are working with DC, BLDC, or induction motors, applying the torque fundamentals enables smarter engineering decisions, better energy efficiency, and ultimately more reliable, high‑performance machines.
Remember: torque is not just a number on a datasheet; it is the bridge between electrical input and mechanical output. Mastering its equation empowers you to turn that bridge into a solid pathway for any motion‑controlled system.
