How to calculate the current consumed by a brushless motor on a quadcopter

Question

I want to create a virtual quadcopter model, but I am struggling to come up with a satisfying model for the brushless motors & props.

Let's take an example, based on the great eCalc tool:

Let's say I want to know how much current is consumed by the motor in a hovering state. I know the mass of the quad (1500g), so I can easily compute the thrust produced by each motor:

Thrust = 1.5 * 9.81 / 4 = 3.68 N per motor

Thrust is produced by moving a mass of air at an average speed of V:

Thrust = 0.5 * rho * A * V²

Where rho (air density) is 1.225kg/m3 and A (propeller disk area) is PI * Radius² = 0.073m² (12" props). So I can compute V:

V = sqrt(Thrust / 0.5 / rho / A) = 9.07 m/s

All right, now I can calculate the aerodynamic power created by the propeller:

P = Thrust * V = 3.68 * 9.07 = 33.4 W

All right, now I can calculate the mechanical power actually produced by the motor. I use the PConst efficiency term from eCalc:

Pmec = Paero * PConst = 33.4 * 1.18 = 39.4W

Here, eCalc predicts 37.2W. It's not too far from my number, I imagine they use more sophisticated hypotheses... Fair enough.

From this post, I know that this power is also equal to:

Pmec = (Vin - Rm * Iin) * (Iin - Io)

Where I know Rm (0.08 Ohms) and Io (0.9 A). So, finally, my question: How do you calculate Vin and Iin from here? Of course, if I knew the rotation speed of the engine I could get Vin from:

n = Kv * Vin

Where Kv = 680 rpm/V. But unfortunately I don't know the rotation speed...

(Note that Vin is assumed to be averaged from the pulse-width-modulated output produced by the ESC)

Thanks for your help!

Answer 1

n = Kv * Vin only when there is no load on the motor. When there is a load on the motor, like from a propeller, you will need to be a bit more clever.

For a DC motor (which BLDC motors approximate by varying the Vin):

Torque = Kt * Vin / R - Ke * Kt * n / R

(Derivation example here: https://electronics.stackexchange.com/questions/90933/torque-relationship-to-speed-in-a-dc-motor)

Back the the propeller:

Paero = Thrust * Vp = torque * n

Vp = 0.5 * (Ve + Vac)

Since velocity of the aircraft at hover is 0, Vp becomes 0.5 * Ve

You actually used the Vp = 0.5 * (Ve + Vac) equation implicitly when you defined thrust. Those equations come from momentum theory. (Good read here: https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node86.html)

From there you can solve for Vin and then Iin. Do note, this is the current and voltage of the motor. The ESc will be pulling from a higher voltage and lower current (but same power ignoring ESC losses)

Answer 2

The missing piece was the expression of the pitch speed, which relates the parameter V with the rotation speed n and the pitch of the propeller:

V = n x pitch x (0.0254 / 60)

(The coefficient passes from in/min to m/s)

Knowing V and pitch, I can evaluate n, then (Vin - Rm * Iin), and finally Iin (since I know everything else).

By the way, I also recommend this link, which contains theoretical propeller formulas, corrected empirically.