CourseWiki/Engineering Fundamentals/Electrical Systems

DC Motor Torque, Speed, and Power

Use an ideal linear torque-speed curve to find a load point and see why peak output power occurs halfway to no-load speed.

More load torque means less speed on this curve

This model approximates a permanent-magnet DC motor at a fixed supply voltage with a straight line between two endpoints. Stall torque is the torque at zero speed. No-load speed is the speed at zero external shaft load. It is not a universal curve for induction motors, steppers, or regulated drives.

Let T_s be stall torque, n₀ no-load speed, and T the constant resisting load torque. The steady operating point is where motor torque equals load torque:

n = n₀(1 − T/T_s).

This is an assumed linear approximation, not a measured motor rating. It omits acceleration, current limiting, temperature changes, and supply voltage drop. At stall it predicts zero shaft power, not zero electrical input. A stall rating is not a continuous operating rating.

Manufacturer basis for the straight line

maxon's Motor Data and Operating Ranges, p. 10, states: Enhancing the load torque leads to a linear reduction of the speed. It identifies no-load speed and stall torque as the endpoints and specifies fixed applied voltage.

Explore the load point and the peak-power point; Interactive

Explore the load point and the peak-power point

Predict first: a model motor has 0.8 N·m stall torque and 3,000 rpm no-load speed. A constant load asks for 0.2 N·m. Does the motor run above or below half speed?

Normalized torque falls from one at stall to zero at no-load speed. Normalized output power peaks at half no-load speed. A load of one-quarter stall torque operates at three-quarters no-load speed.

Static example: n = 3,000(1 − 0.2/0.8) = 2,250 rpm. Angular speed is 2,250 × 2π/60 = 235.6 rad/s, so output power is 47.12 W. Peak power is 62.83 W at 1,500 rpm and 0.4 N·m.

Peak power is halfway, not at either endpoint; Practice

Peak power is halfway, not at either endpoint

Use P = Tω, with torque in N·m and angular speed in rad/s. Convert rpm using ω = 2πn/60. The definitions and units are on Power and Efficiency and Angular Motion.

At stall there is torque but no speed. At no load there is speed but no external load torque. Both endpoints therefore have zero shaft output power. Between them, torque times speed reaches a maximum.

Derive the halfway result from the straight line

Put x = n/n₀. Then T = T_s(1 − x), so P = T_sω₀x(1 − x). Since x(1 − x) = 1/4 − (x − 1/2)², the largest value is 1/4 at x = 1/2. Thus peak power = T_sω₀/4, at half no-load speed and half stall torque. The two endpoint maxima cannot occur at the same time.

Manufacturer check on peak power

Portescap's Brush DC Motor Basics, p. 1, point 4, places maximum mechanical power at half the stall torque. On the linear curve, this also means half no-load speed, as derived above.

Practice operating speed Practice output power

Peak efficiency needs electrical input as well as shaft output

Efficiency is output power divided by input power. Peak output power and peak efficiency answer different questions. The torque-speed line alone does not give electrical input or losses, so this explorer deliberately does not calculate efficiency.

For the brushed coreless DC motors described by Portescap, peak efficiency occurs at a lighter load and higher speed than peak output power. Its exact position needs loss or current data. No external load means no useful shaft output, even though a real motor draws current to overcome its internal losses. Consult the actual motor's efficiency and continuous-operation data before selecting a load point.

Manufacturer basis for the efficiency comparison

Portescap's Brush DC Motor Basics, p. 1, point 3, states: The maximum efficiency is obtained at high speed. Its accompanying torque plot places the efficiency peak below the torque of peak power; the descending speed line therefore puts it at higher speed.

Match the motor, reduction, and the same output load; Interactive; Practice

Match the motor, reduction, and the same output load

A gearbox changes the torque demanded at the motor shaft for the same output load. That can move the motor to a different point on its torque–speed curve. Predict both operating points before revealing them.

Static example: a motor has 0.8 N·m stall torque and 120 rpm no-load speed. The output load resists with 0.6 N·m. Compare direct drive with a 20-tooth driver, 40-tooth driven gear, and a stated 90% running mesh efficiency.

The same motor and output load, two connections
Connection	Torque required at motor	Motor speed	Output speed	Motor shaft power	Modeled output power
Direct	0.600 N·m	30.0 rpm	30.0 rpm	1.885 W	1.885 W
2:1 reduction, 90%	0.3333 N·m	70.0 rpm	35.0 rpm	2.443 W	2.199 W

Result: the reduction runs the output faster under this load, even though it divides motor speed. It reflects less load torque to the motor, so the motor reaches a higher speed. The gearbox did not create power: motor shaft power changed before the 10% modeled mesh loss.

This is a fixed-voltage steady-running comparison with a constant resisting torque. It does not establish starting ability, acceleration, thermal or continuous rating, back-driving, electrical input, or gear strength. At stall, speed and shaft output power are zero; if required motor torque exceeds stall torque, the model reports no forward steady operating point instead of assigning a fictitious speed.

Source and the standstill boundary

The running conversion follows maxon's gear conversion note: motor speed equals reduction multiplied by load speed, while motor torque equals load torque divided by reduction and gear efficiency. The note also requires checking actual gear torque and speed limits. A separate maxon standstill note explains that power efficiency is undefined when both input and output power are zero. This lesson therefore treats stall as a boundary; it does not calculate blocked-state efficiency or a permissible output torque.

Real loads may demand different torque at different speeds. Their operating point is an intersection of a load curve and a motor curve. Practice reflected load Practice output power

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DET 403 DET Electric Motors