# Correct way of calculating BLDC torque for a given RPM and Current

A permanent-magnet 3-phase brushless DC electric motor is attached to a load. Motor is operated with vESC6.6 inverter, which is software-limited to 140A. Measured efficiency at maximum power is 81%.

Motor specs are:

Kv: 49 RPM\V
Operating voltage: 48V
Max RPM: 48V*49RPM\V=2352
Max Rated Current: 140A
Nominal Current: 70A
Resistance(R): 11 mOhm
Inductance(L): 54uH


The test results:

U: 43.1V
I: 140A
RPM: 769
P: 6035W


I then use two different formulas for estimating motor torque: $$Q_1=8.3*\frac{I}{Kv}=8.3*\frac{140}{49}=24\frac{N}{m}$$

$$Q_2=9.549*\frac{P}{RPM}=9.549*\frac{6035}{769}=75\frac{N}{m}$$

• Which value is correct?
• How can a 3-fold difference be explained?

*I took the formula for $$\Q_1\$$ from here. 8.3 is for 3-phases, not one.

• 24n.m x 769rpm = 1930W mechanical output power, but input power is 6035W. You say that efficiency is 81% at 140A so this doesn't compute. What were the measurements you used to get the 81% efficiency? Nov 2 '19 at 14:17
• For the sake of simplicity I have limited the calculation to Input Voltage and Input Current. As for the efficiency, I have measured the current and Voltage on a stalled motor. That gave me 20% drop in power. Nov 2 '19 at 14:28
• Torque is in N times m (or: Nm, $\text{N}\cdot\text{m}$ or N.m) (as @BruceAbbott also shows in a subtle way) Nov 2 '19 at 16:52
• @Bruce, while 24Nm doesn't compute, 75Nm does. I shouldn't have mentioned efficiency as the values given are on the controller input. The question now is - what is wrong with Q1 formula? Nov 2 '19 at 18:29

The 24 N.m figure implies a mechanical power output of 1930W and efficiency of only 32%, which is much too low if the motor is anywhere near 81% efficient at 140A. The error may be due to an incorrect assumption about phase current.

Motor current is only the same as controller input current when the motor is getting full input voltage. If the controller limits current by reducing motor voltage (which seems to be the case here) then the motor current will be higher than the supply current. Why? Power = voltage x current. Since (assuming negligible controller loss) input and output power are equal, as output voltage is lowered so output current must increase by the same ratio.

Motor current can be derived from Kt, rpm, and efficiency. 6034W in x 81% = 4888W out, or 61 N.m x 769rpm. Kt is the inverse of Kv, ie. 8.3/49 = 0.17 N.m/A, so motor current should be 61/0.17 = 359A, 2.56 x higher than the power supply current.

To check this figure we can calculate the expected motor voltage and derive controller input current from the voltage reduction ratio. Voltage drop inside the motor should be (neglecting inductance) 0.011Ω x 359A = 4V. 769rpm/49rpm/V = 15.7V back-emf, + 4V = 19.7V applied motor voltage. 43.1V/19.7V = 2.19, so controller input current should be 359A / 2.19 = 164A. That's only 17% higher than the measured current, which means we are 'in the ballpark'.

• Thank you Bruce. You have answered both of my questions. As a matter of fact there is 2 settings in VESC - "Max. motor current" was set to 140A, however I've had to go for an "Absolute Maximum Current" of 180A to avoid under-voltage errors. Nov 3 '19 at 8:06

From the motor specs, the following are device parameters (i.e. they don't change unless you pick a different motor)

• Kv: 49 RPM/V (speed constant)
• Kp: ?? N/A (torque constant)
• Resistance(R): 11 mOhm
• Inductance(L): 54uH

Parameters like power, efficiency are "useless" (unless you know how to use them). I.e. shown power and efficiency typically only apply for one working point (torque;speed point), where people not knowing how to use them, tend to apply 100% efficiency and full power even when the motor is stalling...

To calculate torque you need the missing Kp parameter, the torque constant.

• Unfortunately, Kp isn't provided by the vendor. Therefore, I am trying to measure the torque myself. The rest (R, L and Lambda) are measured by VESC controller for FOC operation. Clearly, biases involved. Nov 2 '19 at 14:32
• Assumin you provided all data, I guess you neither know the stalling current or max current and max torque? Nov 2 '19 at 15:38
• That is the whole problem. I dare not to go for an "unlimited current mode". I have stalled the motor at vendor recommended peak current (had 20 sec. to do the efficiency measurements). Nov 2 '19 at 18:04
• If you were able to determine the corresponding torque (at stall), you have one (torque; current) pair and one (torque; speed). Next, try to determine the no (mechanical) load current and speed and assume the torque there is zero. Worh these measurements yo can verify the speed constant and determine the torque constant Nov 2 '19 at 18:31

The relationship between the torque constant and the speed constant (Kv) in the article is correct. Ia is supposed to be armature current, not battery current. Was the armature current 140 Amps?