Detection of BEMF

The next page is about the so-called BEMF. How to measure it and about the "consequences" for the commutation task.
Beforehand - the notion of BEMF (BackElectroMotivForce) is actually not right. Because not a force is detected, but a voltage. Since this term is used in almost all application reports, I will continue to use it.

What is the BEMF?

Before the theory a little experiment:
Turning an electric motor with open connections, it will take quite a long time until this comes to a standstill. This effect can be achieved by attaching an inertia (propeller) still further.
Now shorting the motor connections against each other, the motor could be twisted very hardly or is stopping abruptly if it was turnning.
The reason for this is a voltage - the result of self-induction.
For open connections no current can flow. When shorted, a current flows, which in turn generates a magnetic field that cause the opposite effect - the motor brakes.
Due to this phenomenon it is called BEMF. Meant is, as I said, the induced voltage.

The Theory

The voltage (UA, UB, UC), measured at the 3 phases, looks like this (idealized) :

In times of no current flowing in a phase, the BEMF voltage can be recognized.
This runs from Ubatt / GND -> 0 -> GND / Ubatt.
After 30 ° the BEMF-voltage passes through a point of symetrie.
The period measured from the start of BEMF voltage up to this point, is the time to be wait up to the next activation of this phase.
With (4) four times this time, you automatically get the duration, while the phase must be active.


From the measurement of the BEMF voltage the entire timing can be derived !

How can we now determine that point? A little math:
The theoretical considerations are based on the following (simplified) motor equivalent circuit (motor in star connection).

RL(A,B,C) is the strand resistance
L(A,B,C) is the inductance of the branch
ev(A,B,C) is the BEMF (induction voltage) to the branch
U(A,B,C) is the phase voltage relative to the star point

We are interested here in the voltage of phase (B), with no current flowing, while the other phases are connected to Ubatt (A) and GND (C).

For ease of calculation, a voltage Un is introduced. This is the voltage of the neutral point relative to GND.

1 for Phase A is: Un = Ubatt - RLA*i - LA di/dt ± evA
2 for Phase C is: Un = RLC*i + LC di/dt ± evC
3 under the assumption that
RLA = RLC
and LA = L C		 Un=Ubatt/2 ± (evA + evC) / 2
4 assuming a symmetrically design
so it could be said :
evA + evB + evC = 0		 Un=Ubatt / 2 ± evB / 2
5 we can say for the voltage of Phase B
relativ to GND
(without any current flowing through Phase B)

UB = Un ± evb

UB = Ubatt / 2 ± 3/2 evB

From equation 5 a simple method can be derived, to detect the point of symmetrie of the BEMF.

Because it is:  3/2 ev(A,B,C) = Ubatt /2 - U(A,B,C) = 0 for U(A,B,C) = Ubatt/2.

This is a simple comparison, which can be realized easily by a comparator circuitry .

What's missing now is an estimation how large the volatge U (A, B, C) can be. Because the equation says nothing about it.

As can be proved, with
6

ev(A,B,C) ~ NlrBω

 N is the turns per phase
l is the length of the rotor
r is the inner radius of the rotor
B is the magnetic induction
ω is the angular velocity of the rotor

Except for the angular velocity, all other values are specified through the design of the motor. And actually we get this information when we buy a motor . Usually an idle speed per volt is given. For example: 500 min-1/Volt. In other words, operating this motor with a voltage of eg 20 volts, we get an idle speed of 10,000 min-1. In this operating point the externally applied voltage equals the BEMF-voltage.
ev(A,B,C) = Ubatt

futher more:

UB = Ubatt / 2 ± 3/2 ev(A,B,C) = Ubatt / 2 ± 3/2 Ubatt;

That means the BEMF voltage moves with an amplitude of 3/2 Ubatt to Ubatt /2!
And so even exceeds the supply-voltage.

Really ?

Left is a picture of a real measurement to see. (Probe 1:10)

The course of the BEMF voltage is good to see.
When shutting off the stage, significant voltage fluctuations can be seen. But not as predicted, but with a different polarity. This is caused by the freewheeling diodes in the output stage, allowing the current in the phase to continue to flow. If the voltage drops below the forward voltage of the diodes, no current flows through the phase and we obtain the calculated behavior.

The zero-crossing of the BEMF voltage at Ubatt/2 in the middle between the commutation is good to see.

In summary:

The course of the BEMF voltage can be measured relatively easy.
The temporal connection of the BEMF voltage for commutation allows an adjustment of the timing.
From Equation 6 we found that the amplitude of the BEMF voltage depends of the speed of the motor.
Since this voltage is too small to be detected properly at startup, only a "forced" start comes into question.
Only when the BEMF voltage reaches a sufficient amplitude, we can automatically commutate.

Up to this point, we operate the motor at the full operating voltage. The motor is therefore at the maximum speed, which the required power allows. What's missing now is the possibility of power reduction. This could be found in the section on PWM.