Fan Motor Matching: Method

Motor Model and Operating Point

The motor as a DC machine

A brushless DC motor with its controller behaves, to first order, like a brushed DC machine. Its back-EMF is proportional to speed, E = n / K_v with K_v in rpm per volt; the current through the windings is set by the voltage left after the back-EMF, I = (V − E) / R_m; and the shaft torque is proportional to the current above the no-load current, Q_m = K_t (I − I₀). The torque constant and the speed constant are the same physical quantity: K_t = 60 / (2π K_v) N·m per ampere. The no-load current I₀ represents the iron and friction losses, the resistance the copper loss.

At a fixed voltage the motor's torque falls linearly with speed from the stall torque K_t (V/R_m − I₀) at zero speed to zero at the no-load speed V K_v. The controller's throttle scales the applied voltage.

The fan as a load

At fixed blade geometry the torque and thrust of a fan grow with the square of the rotor speed (the lift and drag of every section grow with the square of its velocity, while the angles of attack stay nearly the same). One reference point, the torque and thrust at one speed from the Ducted Fan Performance tool or a bench test, therefore fixes the whole load curve: Q = Q_ref (n / n_ref)². Reynolds and Mach effects bend the curve slightly; for a wide speed range, re-run the performance tool at the new speed.

The operating point

Setting motor torque equal to fan torque, referred to the same shaft through the gear ratio, gives a quadratic in the speed whose positive root is the operating point. From it follow the current, the electrical power V I, the shaft power Q n, the losses I² R_m and V I₀, and the motor efficiency. The thrust per electrical watt (grams per watt, the figure the drone and eVTOL community quotes) and the margin to the motor's or controller's current limit are reported with them.

Reading the charts

The torque-speed chart shows the motor line at the applied voltage and the fan parabola; a motor with a lower K_v steepens the line and moves the crossing to lower speed and current, a higher voltage shifts it right. The throttle sweep shows how speed, current and thrust rise toward full throttle; current climbs faster than thrust, so a fan sized to meet its thrust at full throttle leaves no current margin for maneuvering, and most designs set the hover point near 50 to 60 percent of full-throttle thrust.

Not included: gearbox losses, controller switching losses beyond the resistance entered, battery voltage sag with current (enter the loaded voltage), and the fall of K_v with winding temperature.