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DC Motor Torque-Speed Linear Approximation

Estimates the torque a DC motor produces at a given speed using the standard linear approximation between stall torque and no-load speed.

Written by the Vorzep Team

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⟹ Solve⟷τ, τ_stall, ω, ω_no‑load

τ (torque)

N·m

τ_stall(stall)

N·m

ω (speed)

rad/s

ω_no‑load(no‑load)

rad/s

Please fix the errors above.

Solve for: τ τ_stall ω ω_no‑load

Presets:

Torque (τ)—

—

τ: —τ_stall: —ω: —ω_no‑load: —

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Torque Magnitude—

Low (< 0.5 N·m) Medium (0.5–1.5 N·m) High (1.5–3.0 N·m) Very High (> 3.0 N·m)

τ = τ_stall · (1 − ω / ω_no‑load) · Linear approximation for DC motor torque‑speed curve.

Variables

Symbol	Quantity	Unit
tau	Motor torque at the given speed	N.m
tau_stall	Stall torque (torque at zero speed)	N.m
omega	Current motor speed	rad/s
omega_no_load	No-load speed (speed at zero torque)	rad/s

What it means

A DC motor’s torque decreases linearly with speed, from stall torque at zero speed to zero torque at no‑load speed. This model is used for quick calculations in motor selection and for dynamic simulation. Understanding this helps in predicting motor performance under load.

Worked example

DC Motor Torque‑Speed – Two Detailed Examples

Real‑World

Scenario: A DC motor has a stall torque τ_stall = 10 N·m and no‑load speed ω_no_load = 100 rad/s. At a speed ω = 50 rad/s, the torque is τ = τ_stall × (1 - ω/ω_no_load) = 10 × (1 - 50/100) = 10 × 0.5 = 5 N·m. The engineer uses this linear approximation to determine the motor's torque output at a given speed, which is essential for sizing the motor for a specific application.

Parameter	Value
τ_stall (N·m)	10
ω (rad/s)	50
ω_no_load (rad/s)	100

1τ = 10 × (1 - 50/100) = 10 × 0.5 = 5 N·m

Result 5 N·m ✓ Torque at speed

Scenario: A motor with τ_stall = 20 N·m and ω_no_load = 200 rad/s operates at ω = 150 rad/s. The torque is τ = 20 × (1 - 150/200) = 20 × 0.25 = 5 N·m. This low torque at high speed indicates the motor is operating in the constant‑power region. The engineer uses this to select a gearbox that matches the load torque and speed requirements.

Parameter	Value
τ_stall	20
ω	150
ω_no_load	200

1τ = 20 × (1 - 150/200) = 20 × 0.25 = 5 N·m

Result 5 N·m ✓ High‑speed torque

Insight: The torque‑speed curve of a DC motor is approximately linear between the stall torque and the no‑load speed. The product of torque and speed (power) is maximum at half the no‑load speed, which is important for motor selection.

Common mistakes

DC motor torque‑speed linear approximation: τ = τ_stall × (1 − ω/ω_no_load).
τ_stall: Stall torque (at ω=0).
ω_no_load: No‑load speed (at τ=0).
Units: τ in N·m, ω in rad/s – ensure consistency.
Assumes: Linear region of the torque‑speed curve – valid for many brushed DC motors.

Applications

The DC motor torque‑speed linear approximation, τ = τ_stall·(1 − ω/ω_no_load), describes the linear relationship between torque and speed for a DC motor. This is used to select a motor for a given application, to predict performance, and to design speed controllers. Engineers use the stall torque and no‑load speed parameters to estimate the available torque at any speed, and to size gearboxes. This model is essential for preliminary design of robotic actuators and drive systems. Understanding this approximation helps in matching motor characteristics to load requirements.

Motor selection for robotic joints and mobile platforms
Estimation of available torque at operating speeds
Design of speed and torque control systems
Gear reduction selection for desired torque and speed
Performance analysis of DC motor driven systems