# Understanding back EMF measurement in brushed DC motors

I have been doing some preliminary research for a project and am trying to understand how sensor-less speed control of brushed DC motors can be done by inspecting the back EMF.

Where I am at with my understanding is that there are two key methods in use for this.

Measure Current & Voltage and do a calculation based on coil resistance.

I believe in this approach the back EMF is calculated by measuring the average voltage across the motor and using information about the average current and coil resistance to get the back EMF. i.e. $$V_{EMF} = V_{motor}-I_{motor} \times R_{coil}$$ This approach sounds pretty straight foward to implement but I found references that accuracy is compromised with changes in temperature due to resistance changes in the motor windings, this makes sense but I don't have a feel for what the temperature co-efficient of motor coils is, is it possible to just assume the temperature co-efficient of copper?

Measure the voltage across the motor while not being driven

This sounds like the more advanced approach but I don't quite understand how it works. I believe the idea is meant to be during the switch off period it is possible to directly measure the back-emf after waiting for "switching noise" to subside. What I don't understand is why the voltage drop from the coil resistance doesn't mess up this value.

I have been using the following resources for reference.

The last of these links portrays a basic uni-directional PWM control with a low side switching. Best of all the article includes some scope plots (if it is impolite to include there plot here let me know and I will remove it)

In the plot we can see the spike after turn off, and then back EMF being measured between the supply and motor terminal. I would have thought the only voltage we should see is the flyback diodes voltage drop. The only way I can understand this technique is if the current through the motor is dropping to zero in each PWM cycle, the "switching noise" being the time the current is flowing through the flyback diode is this typically what actually happens? I haven't really had any experience driving DC motors, but have had some experience in building test gear to measure solenoid opening times based on back EMF.

In Summary

• Does motor current really drop to zero each PWM cycle for a typical brushed DC motor.
• Am I understanding both methods correctly
• Are there any key criteria to select one method over the other (noting that I probably want to measure current anyway)
• Diagrams from elsewhere are usually OK under fair use IF a link to the source is provided. Jul 2 '21 at 10:18
• Winding resistance unimportant when not driven as no drive current so IR voltage minimal. There will be some effects from stray capacitance etc but usually relatively minimal Jul 2 '21 at 10:20
• Thanks @Russel McMahon, so I should be ok having referenced the source. So as per my comment on Andy's answer does this mean most motors will have a low time constant. Jul 2 '21 at 10:40

What I don't understand is why the voltage drop from the coil resistance doesn't mess up this value.

When you make the measurement you have to wait for the current to fall to zero as indicated by the spike in your diagram. After that point, the back-emf is valid.

Does motor current really drop to zero each PWM cycle for a typical brushed DC motor.

yes it will providing the energy stored in the leakage inductance of the motor has depleted (as per the end of the spike in your picture).

Are there any key criteria to select one method over the other (noting that I probably want to measure current anyway)

Both are used but, the back-emf measurement is a more reliable indicator for motor speed in my opinion.

• I guess following that, I would ask does this in general indicate that RL time constant of motor windings is generally much much lower than it is for solenoid coils where even with relatively low PWM frequencies current through the coil never drops to zero. Jul 2 '21 at 10:38
• Well, a DC solenoid is usually designed to run with its armature locked i.e. the solenoid being in its active position so, there can be no back-emf and, to prevent a hundred amps flowing the solenoid coil is designed to have a significant series resistance. Not so for a DC motor; DC motors are designed to have as little DC resistance as possible. Solenoids usually have much higher inductance also hence, a side-by-side comparison would seem that the time constant for a solenoid might be a 1,000 times longer than it is for a DC motor (ball-park estimation) @Hugoagogo Jul 2 '21 at 10:59
• Thanks Andy, the solenoid variety I was considering is the style used to position a spool inside a hydraulic control valve. Jul 2 '21 at 11:01
• That doesn't mean much to me. Jul 2 '21 at 11:05

[...] I don't have a feel for what the temperature co-efficient of motor coils is, is it possible to just assume the temperature co-efficient of copper?

By knowing coil resistance, and actively measuring motor current while running, back-EMF can be compensated...this is a kind of feedforward speed compensation.
Yes, coil resistance changes with temperature. But a potentially larger effect is that of the motor's brushes, whose resistance is in series with coil (copper) resistance. As the commutator wears, and as brushes wear down, total series resistance can change.
In its most common simple application, a fixed coil resistance is assumed. Nevertheless, this technique can work very well. An over-compensated motor will speed up as it is loaded more heavily - very odd.

This technique does have the advantage that the motor is driven 100% of the time - you needn't drop motor current to zero, take a back-EMF voltage measurement, then power-up the motor again. This can be a big advantage when variable motor speed is done with PWM.

• So does this only provide a benifit over the other method for very high PWM duty cycles. Jul 3 '21 at 0:56
• High PWM frequency does mean that you have to interrupt duty cycle to make a back-EMF measurement. Measuring motor current can be done without interruption, and some H-bridge driver chips allow you to add a current-sensing resistor. In either method, averaging multiple measurements (current in one case, back-EMF in the other) is a good idea. Jul 3 '21 at 5:06
• Thanks glen, I appreciate the both these answers, I ended up marking up the other answer because it more directly covered some of my final points. This was still very useful though, particularly with the comment on the changing brush resistance, I think ultimately I will design my prototype such that I can use both techniques and then evaluate from there. Jul 5 '21 at 0:02