# Can I calculate the back EMF, motor constants and internal resistance of a DC motor based on these data?

I connected a small motor hub to the gearhead of the motor. Then, I measured the voltage, current and speed of turning using a tachometer. I am not sure if the added weight/inertia due to the 12g hub could be neglected in the calculations. Here is the data:

Supplied voltage: 11.1V

Measured voltage across the motor: 10.2V

Measured current in series with the motor: 1.2A

I also removed the gearhead and measured the voltage across the motor and current in series with the motor while the motor was stalled. I did 10 runs using an applied voltage of 5Volt (higher voltage caused smoke). Using R = average of measured V divided by average of measured I, I obtained a resistance of 0.1 Ohms. Is this the resistance of the coil inside the motor?

Based on these data, can I calculated the Back EMF constant, Torque constant and Voltage caused by the Back EMF? I believe the Back EMF constant is the same as the torque constant.

• You could have measured the coil resistance with a simple multimeter. Commented Apr 29, 2017 at 17:55
• How can be supplied voltage 11.1V and measured 10.2V? Commented Apr 29, 2017 at 18:16
• 11.1V sounds like the nominal voltage of a 3 cell Li-ion battery. Perhaps the battery wasn't charged and/or there was some loss in the wiring? Commented Apr 29, 2017 at 20:52
• I used an external power supply. Commented Apr 29, 2017 at 20:54
• So either the power supply wasn't actually delivering 11.1V, or there was loss in the wiring, right? To measure resistance you should use a current limiting device, eg. a 12V 12W light bulb should limit current to ~1A with a 12V supply, then volts across motor = resistance in Ohms. Commented Apr 29, 2017 at 21:00

$V_{supply} = V_{BEMF} + I\cdot R$

$V_{supply} = k_V\cdot\Omega + I\cdot R$

$k_V[Vs/rad] = \dfrac{V_{supply} - I\cdot R}{\Omega}$

$\Omega = \dfrac{\pi\cdot N_{mot}[rpm]}{30} = \dfrac{\pi\cdot p\cdot N_{out}[rpm]}{30}$

$k_v[Vs/rad] \approx k_t[Nm/A]$

• Thanks Marko. Am I right that Vsupply is 11.1V in this case? It seems that Trevor used the measured voltage across the motor rather than the supplied voltage to calculate the Back EMF. I guess the Current_Drop in his "Back EMF = Terminal_Voltage - Current_Drop" equation is actually voltage drop due to resistance? Commented Apr 29, 2017 at 18:51
• @questioner I have asked previously: What is supply voltage and motor voltage? They should be the same. Commented Apr 29, 2017 at 19:53
• Sorry I don't know the answer. Supplied voltage was 11.1V and the measured voltage across the motor was 10.2V. Perhaps somebody else could answer this? Commented Apr 29, 2017 at 20:21
• @questioner: Do you mean you set the external supply to 11.1V, or that you measured 11.1V at the terminals of the supply? Even when you're using a good name-brand lab supply you should measure the actual output voltage rather than trusting the meter on the supply. If you were seeing 11.1V at the terminals, then did you perhaps have a long run of thin wire to the motor? Commented Oct 10, 2018 at 20:40
• Gahh. Just noticed the date -- why did this pop up now? Commented Oct 10, 2018 at 20:41

You could have measured the coil resistance with a simple multi-meter.

However assuming your coil resistance is indeed $0.1\Omega$ then

At $1.2A$ Voltage drop due to resistance $= 1.2 * 0.1 = 0.12V$

Back EMF = Terminal_Voltage - Current_Drop = $10.2 - 0.12 = 10.08V$

At 78RPM Back EMF Constant = $10.08 / 78 = 0.129 V/rpm$

Torque constant can not be determined and is the amount of torque produced at the shaft for a given current applied.

Note: The above is not totally accurate since the coil resistance is really an impedance that changes as the motor speeds up. But it is a starting point.

• Thanks. When I connected the probes from the multi-meter directly to the two terminals from the DC motor, I got 0.8 Ohms. However, when I connected the probes from the multi-meter to the end of the two wires connected to those terminals, I got 0.5 Ohms. Using another multi-meter, I got 0.9-1.1 Ohms in both cases. Using another motor got different answers. What is going on here? Which value should I use? I read that depending on the orientation of the shaft, different answers could result. So, I used the method I mentioned in the original post as the orientation of the shaft were different. Commented Apr 29, 2017 at 18:17
• @questioner. If you moved the shaft during that time you can get different numbers as the brushes make different contact configurations. When using a multi-meter it is better to take a bunch of readings while turning it by hand. You should get two or three distinct values that you can average. Commented Apr 29, 2017 at 18:21
• Is the method mentioned in the original post valid? Isn't Back EMF constant the same as Torque constant? I read it from several books. Commented Apr 29, 2017 at 18:23
• @questioner like I mentioned in my answer... it gets complicated quickly. Commented Apr 29, 2017 at 18:23
• @questioner re torque constant, they are kind of related in that the faster you go the less torque you can get for a given voltage since the current goes down as back emf goes up. However, without knowing the actual NEwtons/M you can not calculate the torque. Commented Apr 29, 2017 at 18:26