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Fan Motor Matching

Operating point of a ducted fan on a brushless DC motor: where the motor's torque-speed line meets the fan's torque curve. The fan is described by its torque at one reference speed (from the Ducted Fan Performance tool or a test), scaled with the square of the speed, and the motor by its speed constant Kv, winding resistance Rm and no-load current I0 at the supply voltage, with the throttle setting as the fraction of that voltage. The tool solves the operating speed, current, electrical and shaft power, efficiency and the thrust at that speed, checks the motor's current limit, and charts the two torque curves and the throttle sweep so the margin to full throttle can be read.

  • Brushless DC motor model: torque = Kt (I − I0), Kt = 60/(2π Kv), I = (Vthrottle − n/Kv)/Rm

  • Fan torque Q = Qref (n/nref)², thrust T = Tref (n/nref)² from one measured or computed point

  • Operating speed, current, electrical and shaft power, motor efficiency, thrust per watt, current-limit margin

  • Throttle sweep from 30 to 100 percent with charts of speed, current and thrust

Fan Motor Matching

Unit System: /

Fan Reference Point




Motor and Supply








Motor Model and Operating Point | Method page
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Fan Motor Matching

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 / Kv with Kv in rpm per volt; the current through the windings is set by the voltage left after the back-EMF, I = (V − E) / Rm; and the shaft torque is proportional to the current above the no-load current, Qm = Kt (I − I0). The torque constant and the speed constant are the same physical quantity: Kt = 60 / (2π Kv) 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 Kt (V/Rm − I0) at zero speed to zero at the no-load speed V Kv. 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 = Qref (n / nref)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 Rm 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 Kv 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 Kv with winding temperature.



References

Hanselman, D. C. (2003). Brushless Permanent Magnet Motor Design, 2nd ed. The Writers' Collective, Cranston, RI.

Drela, M. (2006). DC Motor / Propeller Matching. Lecture notes, MIT 16.01/16.02 Unified Engineering, Cambridge, MA.

Johnson, W. (1980). Helicopter Theory. Princeton University Press, Princeton, NJ.

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