# How to calculate torque constant and back EMF constant of a BLDC motor?

I have a BLDC motor to run for a fan. The parameters are as follows:

• Electrical frequency measured: 61.936 Hz
• Pole pairs: 8
• Line voltage measured: 70.365 V RMS
• Line voltage peak: 99.51113732 V
• Phase voltage peak: 57.4527819 V

RPM = 60 * (Electrical frequency measured) / (Pole pairs) = 464.52

Back EMF constant = (Phase voltage peak) / RPM = 0.123735709

From the Mathworks website, in the article for surface mount PMSM, I found the following 2 equations:

• Back-emf---> λpm = (1/√3)(Ke/(1000P))*(60/2π)
• Torque constant ----> λpm = (2/3)*(Kt/P)

Where Ke = back EMF constant and Kt = torque constant.

In the equations, the back Emf constant has unit of Vpk_LL/krpm, where Vpk_LL is the peak voltage line-to-line measurement and krpm = 1000RPM.

I do not know how to properly use these equations, and I am getting a lot of wrong answers.

My expected torque is around 0.6 Nm and current around 0.25A, which should give a torque constant around 0.15 ~ 0.2.

• "Line voltage measured: 70.365 V RMS Line voltage peak: 99.51113732 V" - Its a DC motor, but you are powering it with AC voltage? "RPM = 60 * (Electrical frequency measured) / (Pole pairs) = 464.52" - what load did the motor have on it when you measured this frequency, and exactly how did you measure it? Commented Apr 29, 2022 at 23:19
• I was calculating back emf of the bldc motor. I used an ac motor, tied it to the bldc motor. So that, the ac motor gets the ac current supply, and that will in turn, rotate the bldc motor. Then I connected the oscilloscope to the phase wires and measured it Commented Apr 30, 2022 at 7:27
• OK, so the line voltage measured and peak are irrelevant. Just to confirm, the AC motor was running at 646.52 rpm, right? Will you be driving the BLDC motor with a trapezoid waveform (6 step commutation), or with sine waves? Commented Apr 30, 2022 at 8:48

It's not clear how you want to use those equations, what answers you are looking for. So maybe it would be helpful to describe the back emf and torque constants Ke and Kt and how they are related. Be aware that those equations, and the following discussion, describe a model of an ideal motor, neglecting all sorts of real motor characteristics

The product of Ke and speed is voltage, and the product of Kt and current is torque. The actual values of Ke and Kt depend on how speed, voltage, torque, and current are defined. I find it simplest to define speed in radians/sec, voltage in volts rms line-to-line, torque in Nm, and current in amps rms. Note that dimensionally, Ke and Kt are the same: volts / rad/s is equivalent to Nm / amp

Using these units, Kt = Ke * sqrt(3) for a three-phase motor.

For your motor then: 464.52 rpm is 48.64 rad/s. So Ke = 70.365 Vrms / 48.64 rad/s = 1.45 and Kt = 2.505 Nm/A. At 0.6 Nm load therefore the motor should draw 0.24 A rms.

• Sir i don't know what you mean when you said that i "should describe the torque and back emf constants, and how they are related "... I mean, how many different ways can they be related? More than one? I am clueless here. Secondly, u calculated Kt as 1.45, and then, immediately wrote Kt = 2.505.. u calculated it as 1.45, so where did 2.505 cone from? Even this is unclear to me Commented Apr 29, 2022 at 21:28
• @sukhbir1996 Oops, that should have been Ke. Fixed it. 2.505 is 1.45 times 1.732, the square root of 3 Commented Apr 29, 2022 at 22:21

Back EMF constant and torque constant are the same thing fundamentally. So you can convert one to the other easily without any additional information.

The back EMF constant is the (peak) induced line-to-line voltage divided by the mechanical rotation speed. This will give you the back EMF constant in units of "Vs". Mechanical and electrical rotation speed are linked together with the number of pole pairs. Electrical rotation speed and induced line-to-line voltage can be measured with an oscilloscope easily.

Dividing the back EMF constant in units of "Vs" by 2*Pi will directly give you the torque constant in units of "Nm/A" ("Nm/A" and "Vs" are the same unit).

One thing to watch out for is that there exist several different conventions for the back EMF constant. Sometimes the back EMF constant isn't noted in SI units or even flipped around (e.g. "kRPM per Volt").