DC Motor Torque-Speed Linear Approximation

Mechanical power is P = τ · ω. Substituting the linear relation τ = τ_stall · (1 − ω/ω_nl) gives P(ω) = τ_stall · ω − (τ_stall/ω_nl) · ω². This is a downward parabola in ω. Its maximum occurs where dP/dω = 0, which gives ω = ω_nl / 2. At that point, τ = τ_stall / 2 and P = τ_stall · ω_nl / 4. Running a motor continuously at this point is usually thermally demanding because half the electrical input is dissipated as heat, but it is the most power-dense operating condition and is used for short bursts in traction and launch applications.

The motor constant K (sometimes written K_t or K_e in SI units, where the two are numerically equal) is the ratio of stall torque to stall current, or equivalently the back-EMF constant. It is the slope-related parameter that determines how much torque you get per amp of current, and how much voltage you need per rad/s of speed. A high-K motor produces large torque at low current but cannot reach high speed at a given voltage. A low-K motor reaches high speed but needs large currents to produce significant torque. Motor selection is largely a matter of choosing K to match the required torque-speed operating point.

Increasing supply voltage increases the no-load speed almost proportionally: ω_nl ≈ V / K_e. It also increases the stall current (I_stall = V / R), and therefore the stall torque (τ_stall = K_t · I_stall), though in practice stall torque is limited by the thermal capability of the winding. The net effect on the torque-speed line is that both endpoints move outward — the line shifts up and to the right. In a PWM-controlled drive, the duty cycle effectively scales the applied voltage, so the line slides smoothly between the origin and its full-voltage position. This is how speed control is achieved without changing the motor.

Stall torque is the theoretical maximum torque at zero speed — the y-intercept of the torque-speed line. It is usually limited by current and can only be sustained briefly because at zero speed there is no back-EMF, so the entire supply voltage drives current through the winding resistance, producing maximum heat. Rated torque is the torque the motor can deliver continuously without exceeding its thermal limits, and it is typically 10–30% of stall torque. Continuous operation above rated torque causes overheating and eventual winding failure. The calculator works with stall torque because that is the parameter that defines the torque-speed line, but always check the continuous rating before using a motor at a given operating point.

A gearbox with ratio N:1 (N revolutions of the motor per revolution of the output) multiplies output torque by N and divides output speed by N, ignoring losses. Applied to the torque-speed line, this shifts the x-intercept from ω_nl to ω_nl/N and raises the y-intercept from τ_stall to N · τ_stall. The slope therefore increases by a factor of N². A geared motor is effectively a different motor with a much steeper line, which is why gearboxes are so useful for applications needing high torque at low speed (robotics, wheelchairs, winches) and why direct-drive applications needing high speed at low torque (drones, fans) avoid them.

The linear model assumes constant winding resistance, constant magnet flux, negligible armature reaction, and no magnetic saturation. In practice, resistance rises with temperature (by about 0.4% per °C for copper), which reduces stall torque as the motor heats up. Armature reaction weakens the effective flux at high current, bending the line downward near stall. Magnetic saturation limits peak torque in high-performance machines. And at high speed, windage and bearing friction add a small speed-dependent torque loss. For first-pass sizing the linear model is excellent; for precision control, a more detailed model — or direct measurement — is preferred.