# Drive voltage vs phase voltage of a BLDC motor

## Background

This video on using an oscilloscope to view the line-to-line phase voltages of BLDC motors demonstrates the "virtual" ground necessary for such measurements. I understand this idea: you need to make a virtual ground for the probes because the oscilloscope's channels are all grounded to the same point (earth).

However, I don't understand the video's results in the context of an electric motor model: $$v_{applied}(t) = 2i_{phase}(t)R + e_{l-l}(t) = 2i_{phase}(t)R_{phase} + k_{e}\omega$$ Where the units of $k_e$ make sense (I think it should to be defined in E-RMS/(rad/s), but I've seen so many different definitions). Sufficed to say, back-EMF = func(speed).

## Questions

Are the phase voltages seen in the video (snapshot below) equal to the voltage applied by the ESC ($v_{ac}(t)$)? Or are the measured voltages equal to the difference in the applied voltage minus the back-EMF ($2i_{a}(t)R_{phase}$)? Unless the virtual ground somehow negates the back-EMF, I don't see how these voltage measurements don't also include back-EMF.

Heck, I'm even more confused now about the sensorless speed control programs used in hobby ESCs: how do hobby ESCs measure back-EMF? If an ESC only energizes 2 phases at once (say, A is HI, B is LO) and measures EMF on the third phase (C), then why does the back EMF on phase C matter if it doesn't affect Vab?

I don't see these voltage measurements don't also include back-EMF.

They do include back-emf. The ESC was running at a low throttle level so it was applying PWM, and the scope was set to 'dot' mode so the high frequency PWM looks like random dots.

When PWM is applied the waveform alternates between voltage applied by the ESC when the PWM pulse is on, and back-emf during the 'off' part of the PWM period. Here is a trace taken from an analog scope which shows the effect more clearly. You can see the high frequency square wave PWM, and a sinusoidal back-emf during PWM off times.

During each commutation step two phases are powered, leaving the third phase floating. The scope is hooked across two phase so the waveform shows 1 step with both phases powered, 2 steps with one phase floating, then 1 step with both phases powered in the opposite (negative) direction. Voltage on the floating phase ramps up and down as it transitions from one supply rail to the other, creating a trapezoid drive waveform. At full throttle there is no PWM so this waveform is clearer:-

The ESC measures back-emf by creating a 'virtual center tap' representing the midpoint between the 2 powered phases. The floating phase generates a voltage relative to this point. When PWM is applied the midpoint voltage changes as the PWM pulse turns on and off, with some noise during the transitions. The PWM noise can be filtered out, or the ESC can only look for zero crossings between transitions when the signal is quieter.

The circuit below shows the basic principle of zero-crossing detection. Phases A and B are powered, with low-side PWM applied to phase B. Phase C is floating so its back-emf (generated relative the center tap) is available, but the center tap isn't. R1-R3 sum the 3 phase voltages to recreate the center tap. COMP1 compares the voltage on phase C to the virtual center tap voltage, switching output states during each zero crossing. To detect zero crossings on all 3 phases you need 3 comparators.

simulate this circuit – Schematic created using CircuitLab

In practice the comparator is usually powered by a lower voltage, so resistor voltage dividers are used to get within its common mode input voltage range. Capacitors can also be placed across the voltage divider outputs to filter out PWM noise.

Some MCUs have an internal comparator connected to an analog multiplexer with several inputs, allowing the single comparator to monitor all 3 phases. To filter out noise the comparator's output can be read several times in succession. This eliminates several external analog components, making the ESC more compact and cheaper to produce.

• Wow, did I misunderestimate the wee hobby ESC! So if the motor was running at its electrical-max speed at full throttle, then E = Vapplied and Vphase = 0 therefore the trapezoid would just be flat, correct? In your first waveform, there seems to some potential left between Vapplied and E. Also, are my steps correct? Feb 9 '18 at 16:33
• Also, thanks for taking the extra time to find some plots. I really appreciate visual aids! One more question, how would a power analyzer distinguish between Vapplied and E? Feb 9 '18 at 16:37
• Voltage swings below/above zero by about 0.6V due to voltage drop across the FET body diodes during flyback (PWM off time). This gives the appearance of higher voltage, but applied voltage = battery voltage during PWM on time (I didn't notice this effect at first, and centered the trace in the wrong place. It should be ~0.6V lower). Feb 9 '18 at 18:33
• Power analyzers are inserted at the DC input side of the ESC, so they measure battery voltage rather than motor voltage. This has the advantage of not having to analyze the peculiar 3 phase waveforms (which are difficult to measure accurately, particularly when PWM is applied). Disadvantage is the meter does not tell you what is happening to the motor at low throttle, eg. at half throttle (50% PWM) average motor voltage is half the supply voltage and average motor current is double the supply current. Feb 9 '18 at 18:50
• What if you had a 3-phase wattmeter? Would it distinguish between V and E? And just to be clear, your first waveform shows the applied PWM voltage and the resulting sinusoid EMF, right? And you're not capturing those on the same phase-combination, right? maybe AB for V-applied and CB for E. Feb 9 '18 at 19:01

A!: The voltage measured is the voltage applied by ESC. If the current is zero, then this voltage equals to BEMF voltage. Virtual ground does nothing, it doesn't negate the BEMF voltage, it's just a reference point. The BEMF voltage is included with current. Small BEMF -> large current, BEMF equal to the supply -> current is zero.

A2: The purpose of BEMF voltage measuring ass feedback information is used to detect when the rotor's magnet poles cross the stator slot. So to know when you have to switch the stator. Alternatively hall sensors are used. From the video is noted, that the motor is supplied with three phase sine wave voltage, not supplied just 2 phases at once and left third floating, which is the normal case in BLDC. In this case, voltages are sensed during OFF period of a PWM cycle.

P.S: voltages ae sensed - means they are detected by comparators not measured in magnitude, since this information is useless.