Make no mistake, this is not a quick read (355 pages), and it requires a basic background in electromagnetism, circuit theory, and electric machine theory. But I promise you that if you have at any point been frustrated by the inconsistent or incomplete treatment of electic motors found on the interwebs, or by the abstracted and overly complex theory in research papers and electric machine textbooks, this report is what you've been missing.

It's clear that the author put a tremendous amount of effort into this thesis, more than I can even comprehend.The explanations are clear, but not simplified. The notations are internally consistent and many (most?) variations found in other literature are explained and compared in tables, charts, and figures. Oh, the figures! They are all HAND DRAWN. And not like my hand drawn pictures.

Is that a jellyfish stuck inside an axial flux motor?

No, nothing like that. These have all the nice scale and readability of computer-generated graphics, but with the friendly this-isn't-as-complicated-as-it-sounds style of whiteboard sketches. Almost like a comic strip. About electric mtors. Look for yourself!

Anyway, I'm not telling you to read the whole thing (yes I am) but I did. In case you're busy watching football or eating Thanksgiving dinner (I will be, but I already read it), I can do a book review-style highlight of the most important concepts that I got out of it, without going into detail or ruining the ending.

1. Back EMF and torque are two sides of the same coin.

I sort-of knew this already. In a brushed motor, the back EMF constant (relating voltage produced at the open motor terminals to the speed of the motor) and the torque constant (relating torque produced to the current passing through the motor) are the same number. The simple explanation for this is that after all the electrico-magneto-mechanical interactions that happen inside the motor, the only thing that really matters is power conservation. Pushing current into a voltage (such as the back EMF) is positive power. This results in a positive torque (torque attempting to increase the speed of the motor).

The conversion is lossless. All the losses are taken into account by adding resistors to the electrical model and/or damping to the mechanical model. So if you put 500 electrical Watts into the back EMF, you will get 500 mechanial Watts out. If you divide by the speed of the motor, this gives you the torque. This is always true. At zero speed, you can take the limit of this to find that torque is still proportional to current. For negative power (generating), you have to be pulling negative current out of a positive back EMF (or pushing positive current into a negative back EMF).

Anyway, I'm not telling you to read the whole thing (yes I am) but I did. In case you're busy watching football or eating Thanksgiving dinner (I will be, but I already read it), I can do a book review-style highlight of the most important concepts that I got out of it, without going into detail or ruining the ending.

1. Back EMF and torque are two sides of the same coin.

I sort-of knew this already. In a brushed motor, the back EMF constant (relating voltage produced at the open motor terminals to the speed of the motor) and the torque constant (relating torque produced to the current passing through the motor) are the same number. The simple explanation for this is that after all the electrico-magneto-mechanical interactions that happen inside the motor, the only thing that really matters is power conservation. Pushing current into a voltage (such as the back EMF) is positive power. This results in a positive torque (torque attempting to increase the speed of the motor).

The conversion is lossless. All the losses are taken into account by adding resistors to the electrical model and/or damping to the mechanical model. So if you put 500 electrical Watts into the back EMF, you will get 500 mechanial Watts out. If you divide by the speed of the motor, this gives you the torque. This is always true. At zero speed, you can take the limit of this to find that torque is still proportional to current. For negative power (generating), you have to be pulling negative current out of a positive back EMF (or pushing positive current into a negative back EMF).

The simple DC motor model.

The thing made clear to me by James Mevey's thesis was that this model works just as well for a brushless DC motor. The power conservation through back EMF works the same way. You can prove this using a few different E&M and/or field theory tricks. (He does all of them, for good measure.) But intuitively it also makes sense that there needs to be something to push against, namely the back EMF, for power to be converted. In the limit as the back EMF and speed go to zero, torque is still proportional to current.

But, unlike a brushed DC motor, the shape of the back EMF of each phase of a brushless motor is a function of rotor position and is periodic. It repeats every two poles, so for a 14-pole motor like on the scooter, it repeats seven times per revoltion. This base shape then gets scaled by the speed. It's as if the back EMF "constant" for a given phase is actually a periodic function of the rotor position. That's sort-of obvious, but the less obvious conclusion is that, taking the limit of the power conversion through the back EMF as speed approaches zero, the torque "constant" is the same period function of rotor position. So another way to look at it is that the torque produced by each phase is simply the base shape of this same rotor position-dependent function multiplied by the current going into the phase. This is also always true.

2. None of the above changes for trapezoidal (BLDC) vs. sinusoidal (BLAC/PMSM) motors.

Just by applying the very simple power-conserving back EMF idea, you can analyze any motor with any drive. I was wondering about the functional differences between BLDC and BLAC/PMSM. James Mevey says:

But, unlike a brushed DC motor, the shape of the back EMF of each phase of a brushless motor is a function of rotor position and is periodic. It repeats every two poles, so for a 14-pole motor like on the scooter, it repeats seven times per revoltion. This base shape then gets scaled by the speed. It's as if the back EMF "constant" for a given phase is actually a periodic function of the rotor position. That's sort-of obvious, but the less obvious conclusion is that, taking the limit of the power conversion through the back EMF as speed approaches zero, the torque "constant" is the same period function of rotor position. So another way to look at it is that the torque produced by each phase is simply the base shape of this same rotor position-dependent function multiplied by the current going into the phase. This is also always true.

2. None of the above changes for trapezoidal (BLDC) vs. sinusoidal (BLAC/PMSM) motors.

Just by applying the very simple power-conserving back EMF idea, you can analyze any motor with any drive. I was wondering about the functional differences between BLDC and BLAC/PMSM. James Mevey says:

"It is the author’s opinion that the difference between trap and sine BPMs is surrounded by more misunderstanding and confusion than any other subject in the field of brushless motor control..."I am inclined to agree. But this thesis makes it very clear that the only functional difference is the shape of the waveforms (both the back EMF and the drive waveform). BLDC refers to motors with more-or-less trapezoidal back EMF, common in small motors. Like this:

From the scooter motors, line-to-line and assumed line-to-neutral.

Why is it trapezoidal? This is entirely due to the geometry of the motor. It has nothing to do with the controller, which should be obvious (although for some reason it's not) since this measurement can be made with nothing attached to the motor at all. It has to do with the layout of magnets and coils and steel. The more you can see sharp transitions in magnets, coils, or steel, the more trapezoidal it becomes. This is common in small motors because it's easier to make a motor with concentrated windings and discrete, non-skewed magnets.

As a counter-example, larger motors often called "brushless AC" or "permanent magnet AC" tend to have skewed magnets and overlapped windings in a large number of slots, so their back EMF looks more sinusoidal. These motors can (should) be driven with a pure AC signal. If, for example, the back EMF and current are both sine waves that are in phase, and there are three balanced phases separated by 120º electrical, the power (torque) produced will be constant. Try adding together three sin^2 waves (current times backEMF) shifted by 1/3 period each. Or have Wolfram Alpha do it for you. There is clearly an advantage to having constant torque in things like large machines, electric vehicles, etc.

But what about BLDC motors with trapezoidal back EMFs? Well, this is where things get more interesting. You can drive them with a signal that looks like a square wave, except with gaps where the trapezoid has non-zero slope. Like this:

To the extent which you back EMF is truly a trapezoid with a 120º-wide top, you can split the drive into six equal-length segments, two positive, two negative, and two off. By doing so, you get zero current during the sloped part of the trapezoid and full current during the flat part. Multiplying back EMF by current still gives power (torque). Accounting for all three phases, offset by 1/3 period each, also gives constant power (torque). This is the basis of brushless DC motor control. It's very simple to implement in software or even just pure logic chips. Since it relies only on 60º position estimates, it can be easily accomplished with three hall-effect sensors or with many sensorless estimation methods.

If they are simpler to make and to control, but yet still give constant ripple-free power, why not make all brushless motors this way? Well, for one, the purely trapezoidal back EMF is not realistic. It may not be as flat, for as long, as the 120º approximation. Things tend to smooth out into sinusoids if you let them. Which brings up the next point: what happens when you account for inductance?

That's the same square-wave drive, but now with a reasonable inductance factored in. Permanent magnet motors tend to have low inductance in general, but even so it still distorts the shape of the drive current at even modest speeds. It's not even a clean low-pass filter. That's because the drive inverter has flyback diodes start to conduct current during the "off" period. And on top of all that, the whole thing is phase-shifted so that current and back EMF no longer line up perfectly. The net result is a power (torque) fluctuation. The good news is that the fluctuation is at six times the commutation frequency, and so at high speed the system inertia easily damps it out. The bad news is it's still significantly lower than the torque produced at zero speed, since the low pass filter takes power out of the harmonics and moves the whole thing out of phase with the back EMF. The other bad news is that the diode conduction in the controller can start to cause more heating in otherwise highly-efficient MOSFET drives.

There is an obvious benefit to using sinusoidal drive, then. Everything, after passing through a low-pass filter, gets closer to looking like a sine wave anyway. You can also eliminate the diode conduction periods which throw the whole analysis off. And if your motor is designed to have sinusoidal back EMF, everything works nicely and all that's left are magnitude and phase shifts. To the extent which a trapezoidal motor can be thought of as having a fundamental sinusoidal component, some of the analysis used in sinusoidal drive can also be applied to trapezoidal motors, understanding that there will be harmonic distortion. But everthing still works within the back EMF power conservation analysis method for figuring out torque.

3. Field-oriented control is all about keeping current and back EMF in phase.

There are a lot of complicated ways to explain it, but the one that makes the most sense to me is that the goal of field-oriented control is to make up for the phase lag that comes in at high speed when the inductance start to become significant. Obviously for induction motors, there is more to it. But for permanent magnet motors, it's as simple as keeping the two sine waves (or whatever waves) of the back EMF and the current in phase as much as possible. This produces maximum torque, for reasons that should be obvious from the power conservation approach I keep mentioning.

The back EMF in a permanent magnet motor is fixed to the rotor position, always. This was another point of confusion for me, namely when it comes to field weakening. But field weakening is a lot simpler than I thought. It does not change the back EMF. The back EMF comes from the rotor magnets alone. What field weakening does is produce another field that fights the magnets in the air gap, reducing the flux to some degree. But this is all accounted for already! It's the inductor in the circuit! It's really that simple. But it only really makes sense if you look at things from the rotor reference frame, often called the d-q frame. (d = direct = in line with the magnet flux, q = quadrature = leading the magnet flux by 90º electrical). In this frame, there are only a few simple rules:

The first case is what things look like when the motor is spun up with no load. The voltage on the terminals (whether you apply it or not) is equal in magnitude to and in phase with the back EMF, which leads the magnets by 90 electrical degrees. (Keep in mind this whole picture is in electrical degrees, so you have to divide by the number of pole pairs to get the mechanical equivalent.) The second case is what happens when a load is applied to the motor without changing the relative position of the voltage. This would be like a simple hall-sensor or encoder based commutation that always fires the phases at the same position.,i.e. open-loop control. The problem is that at high speeds, current goes out of phase with back EMF, since some voltage is developed across the low pass filter. The sum of back EMF, voltage on the inductor, and voltage on the resistor is still equal to the voltage applied at the terminals, but this is a vector sum so things get tricky.

Now here are two cases where the relative phase of voltage can be controlled (advanced). For demonstration purposes, this could be done by physically rotating the position sensor on the motor. Obviously for control, it would instead be done by adding some phase offset in software. The left-most case is field-oriented torque control. The voltage is advanced to make up for the current lagging behind it. The net result is current in phase with back EMF, the theme of this story, and ideal torque. The right-most case is what happens when you advance the voltage even more. Now, current goes out of phase the other way, so some torque is lost. But, the other components of voltage rotate out of the way a little, allowing the back EMF (speed) to grow without increasing the voltage applied. This is field weakening. You trade off torque for a bit more speed. The back EMF could even be higher than the voltage applied, extending the speed range of the motor without increasing the DC supply voltage. How much of this you can get away with depends on the inductance of the motor. Generally, surface-mounted magnet motors have low inductance and limited field-weakening capability. But "limited' might still mean you can get 50% or 100% more speed!

To some extent, I knew this already. But the great thing about seeing it consistently presented is that I now understand why it works. And even more cool, I see that it's all part of the same, simple model. Everything is accounted for without even diving into the magnetic domain. You just measure back EMF, draw some triangles, and come up with the performance of the motor. It's all become simpler to comprehend because it fits into a minimalist model.

Now I have some more solid footing to stand on for the full-AC version of my controller. I can start thinking about field-oriented control, how it will interact with different motors (trapezoidal, sinusoidal) at different speeds, etc. I guess at heart I am more interested in motor control than in motors themselves, but as I can see now you can't understand one without the other!

READ IT! Here's the link again.

As a counter-example, larger motors often called "brushless AC" or "permanent magnet AC" tend to have skewed magnets and overlapped windings in a large number of slots, so their back EMF looks more sinusoidal. These motors can (should) be driven with a pure AC signal. If, for example, the back EMF and current are both sine waves that are in phase, and there are three balanced phases separated by 120º electrical, the power (torque) produced will be constant. Try adding together three sin^2 waves (current times backEMF) shifted by 1/3 period each. Or have Wolfram Alpha do it for you. There is clearly an advantage to having constant torque in things like large machines, electric vehicles, etc.

But what about BLDC motors with trapezoidal back EMFs? Well, this is where things get more interesting. You can drive them with a signal that looks like a square wave, except with gaps where the trapezoid has non-zero slope. Like this:

To the extent which you back EMF is truly a trapezoid with a 120º-wide top, you can split the drive into six equal-length segments, two positive, two negative, and two off. By doing so, you get zero current during the sloped part of the trapezoid and full current during the flat part. Multiplying back EMF by current still gives power (torque). Accounting for all three phases, offset by 1/3 period each, also gives constant power (torque). This is the basis of brushless DC motor control. It's very simple to implement in software or even just pure logic chips. Since it relies only on 60º position estimates, it can be easily accomplished with three hall-effect sensors or with many sensorless estimation methods.

If they are simpler to make and to control, but yet still give constant ripple-free power, why not make all brushless motors this way? Well, for one, the purely trapezoidal back EMF is not realistic. It may not be as flat, for as long, as the 120º approximation. Things tend to smooth out into sinusoids if you let them. Which brings up the next point: what happens when you account for inductance?

That's the same square-wave drive, but now with a reasonable inductance factored in. Permanent magnet motors tend to have low inductance in general, but even so it still distorts the shape of the drive current at even modest speeds. It's not even a clean low-pass filter. That's because the drive inverter has flyback diodes start to conduct current during the "off" period. And on top of all that, the whole thing is phase-shifted so that current and back EMF no longer line up perfectly. The net result is a power (torque) fluctuation. The good news is that the fluctuation is at six times the commutation frequency, and so at high speed the system inertia easily damps it out. The bad news is it's still significantly lower than the torque produced at zero speed, since the low pass filter takes power out of the harmonics and moves the whole thing out of phase with the back EMF. The other bad news is that the diode conduction in the controller can start to cause more heating in otherwise highly-efficient MOSFET drives.

There is an obvious benefit to using sinusoidal drive, then. Everything, after passing through a low-pass filter, gets closer to looking like a sine wave anyway. You can also eliminate the diode conduction periods which throw the whole analysis off. And if your motor is designed to have sinusoidal back EMF, everything works nicely and all that's left are magnitude and phase shifts. To the extent which a trapezoidal motor can be thought of as having a fundamental sinusoidal component, some of the analysis used in sinusoidal drive can also be applied to trapezoidal motors, understanding that there will be harmonic distortion. But everthing still works within the back EMF power conservation analysis method for figuring out torque.

3. Field-oriented control is all about keeping current and back EMF in phase.

There are a lot of complicated ways to explain it, but the one that makes the most sense to me is that the goal of field-oriented control is to make up for the phase lag that comes in at high speed when the inductance start to become significant. Obviously for induction motors, there is more to it. But for permanent magnet motors, it's as simple as keeping the two sine waves (or whatever waves) of the back EMF and the current in phase as much as possible. This produces maximum torque, for reasons that should be obvious from the power conservation approach I keep mentioning.

The back EMF in a permanent magnet motor is fixed to the rotor position, always. This was another point of confusion for me, namely when it comes to field weakening. But field weakening is a lot simpler than I thought. It does not change the back EMF. The back EMF comes from the rotor magnets alone. What field weakening does is produce another field that fights the magnets in the air gap, reducing the flux to some degree. But this is all accounted for already! It's the inductor in the circuit! It's really that simple. But it only really makes sense if you look at things from the rotor reference frame, often called the d-q frame. (d = direct = in line with the magnet flux, q = quadrature = leading the magnet flux by 90º electrical). In this frame, there are only a few simple rules:

- Back EMF leads magnet flux by 90º. That is, it is always on the q-axis.
- Voltage leads current by some amount determined by the motor inductance and resistance.
- Torque is proportional to the amount of current in phase with back EMF, i.e. the dot product.

The first case is what things look like when the motor is spun up with no load. The voltage on the terminals (whether you apply it or not) is equal in magnitude to and in phase with the back EMF, which leads the magnets by 90 electrical degrees. (Keep in mind this whole picture is in electrical degrees, so you have to divide by the number of pole pairs to get the mechanical equivalent.) The second case is what happens when a load is applied to the motor without changing the relative position of the voltage. This would be like a simple hall-sensor or encoder based commutation that always fires the phases at the same position.,i.e. open-loop control. The problem is that at high speeds, current goes out of phase with back EMF, since some voltage is developed across the low pass filter. The sum of back EMF, voltage on the inductor, and voltage on the resistor is still equal to the voltage applied at the terminals, but this is a vector sum so things get tricky.

Now here are two cases where the relative phase of voltage can be controlled (advanced). For demonstration purposes, this could be done by physically rotating the position sensor on the motor. Obviously for control, it would instead be done by adding some phase offset in software. The left-most case is field-oriented torque control. The voltage is advanced to make up for the current lagging behind it. The net result is current in phase with back EMF, the theme of this story, and ideal torque. The right-most case is what happens when you advance the voltage even more. Now, current goes out of phase the other way, so some torque is lost. But, the other components of voltage rotate out of the way a little, allowing the back EMF (speed) to grow without increasing the voltage applied. This is field weakening. You trade off torque for a bit more speed. The back EMF could even be higher than the voltage applied, extending the speed range of the motor without increasing the DC supply voltage. How much of this you can get away with depends on the inductance of the motor. Generally, surface-mounted magnet motors have low inductance and limited field-weakening capability. But "limited' might still mean you can get 50% or 100% more speed!

To some extent, I knew this already. But the great thing about seeing it consistently presented is that I now understand why it works. And even more cool, I see that it's all part of the same, simple model. Everything is accounted for without even diving into the magnetic domain. You just measure back EMF, draw some triangles, and come up with the performance of the motor. It's all become simpler to comprehend because it fits into a minimalist model.

Now I have some more solid footing to stand on for the full-AC version of my controller. I can start thinking about field-oriented control, how it will interact with different motors (trapezoidal, sinusoidal) at different speeds, etc. I guess at heart I am more interested in motor control than in motors themselves, but as I can see now you can't understand one without the other!

READ IT! Here's the link again.