Motors, torque-speed curves, and gearing

Requirement

A small vehicle has two demands that pull against each other. To start, especially on a slope, it needs enough torque at the wheel to overcome the force holding it back. To cruise, it needs enough wheel speed to reach a useful pace. A single motor spinning at a single voltage cannot deliver its largest torque and its largest speed at the same time, so the design task is to understand what the motor can actually offer across its range and then to choose a gear ratio that places the useful part of that range where the vehicle needs it.

Derivation

Two facts about a brushed permanent-magnet DC motor, each measurable without opening the case, generate the whole characteristic. First, the torque the motor produces is proportional to the current through it. Second, as the motor spins it generates a back-voltage proportional to its speed, opposing the supply. The electromagnetic reason for each belongs to the electromagnetics entries not yet written, and this entry defers that mechanism openly rather than faking it; the two proportionalities are taken as working facts.

Put them together with the circuit. The supply voltage is split between the resistance of the windings and the back-voltage: V equals I times R plus the back-voltage. At stall the motor is not turning, so the back-voltage is zero, all the voltage drives current through the resistance, the current is at its maximum, and so is the torque. This is the stall torque. At the other extreme, with no load, the motor speeds up until the back-voltage nearly equals the supply, current falls toward zero, and torque falls to zero; the speed there is the no-load speed. Between these two points torque falls in a straight line as speed rises, because torque tracks current and current falls linearly as the rising back-voltage eats into the supply. Mechanical power out is torque times speed, which is the product of a falling line and a rising one, a downward parabola that peaks halfway between stall and no-load, at half the no-load speed. So the motor makes its most power in the middle of its range, its most torque at the bottom, and its most speed at the top, and it can never have all three at once.

Design choices

Gearing resolves the conflict. The mechanism by which a gear train trades torque for speed at nearly constant power is developed in the entry on gears, belts, chains, and transmissions, which lists this entry as what it unlocks, so that result is used here rather than re-derived: a reduction gear multiplies torque and divides speed by the same ratio, leaving power almost unchanged, since power is torque times speed and the ratio cancels, minus a few percent lost to friction in the teeth per stage. Choosing the ratio is choosing where on the torque-speed line the motor sits when the vehicle is doing what matters. Too tall a ratio and the motor cannot make enough wheel torque to start on the grade; too short a ratio and the motor reaches its no-load speed before the vehicle reaches its target pace. The good ratio starts the vehicle with margin and cruises with the motor near the high-power middle of its curve, where it is also reasonably efficient.

Calculations

Take a vehicle of two kilograms on wheels of three centimeters radius, required to start on a ten-degree slope. The force pulling it back down the slope is its weight times the sine of ten degrees, about 3.4 newtons, so the torque needed at the wheel is that force times the wheel radius, about 0.10 newton-meters. If the chosen motor makes 0.02 newton-meters at stall, the gear reduction must be at least the ratio of these, about 5 to 1, just to hold on the slope; because stall is a burnout condition and not a place to operate, a real design picks more, perhaps 8 to 1, so the motor starts the climb comfortably below stall. The same ratio then fixes the top speed: the wheel turns at the motor's no-load speed divided by 8, times the wheel circumference, and that must clear the target cruising pace. If it does not, the two demands cannot both be met by this motor and a different motor or a two-speed transmission is needed.

Failure modes

Drives fail in characteristic ways. Running near stall burns the motor, because stall current is the maximum and nearly all of it becomes heat in the windings. A ratio too tall to start leaves the vehicle unable to climb; a ratio too short caps the top speed below the target. Gear losses stack multiplicatively across stages, so a three-stage reduction can quietly lose a tenth of the power. And the straight torque-speed line is the signature of the brushed permanent-magnet DC motor at fixed voltage specifically: induction motors, series-wound motors, and motors run from switching controllers have reshaped or nonlinear curves, and they stay out of scope here until the electromagnetics and power-electronics entries exist to treat them honestly.

Build with it

Take a different vehicle than the worked one, with its own mass, wheel radius, grade, and target cruising speed, and either a motor datasheet or two numbers you measure yourself: the no-load speed and the stall torque. Pick a gear ratio and then check three things with numbers shown. The wheel torque at the chosen ratio must exceed the grade force times the wheel radius at start. The wheel speed at the motor's no-load speed divided by the ratio must clear the target cruising pace. And the operating point during the climb must sit well below stall, so the motor is not asked to live where it overheats. Success is a ratio that passes all three checks, with the arithmetic for each written out.