# Meaning of motor constants for three-phase PMSM

When looking at three-phase permanent magnet synchronous (brushless) motors, $K_T$ and $K_V$ numbers are often specified. Also, controllers for these motors typically specify maximum currents. However, I'm confused what these numbers are measuring. I can think of many "obvious" ways to measure voltage and current in a three-phase system like this:

• peak-to-peak for a single phase
• $abs(\text{peak})$ for a single phase (probably not for voltage)
• RMS for a single phase
• RMS for all three phases
• peak-to-peak line-to-line
• RMS line-to-line

It seems pretty obvious that the voltages specified on motor controllers are peak-to-peak line-to-line because that's what the input voltage corresponds, but the other three numbers (controller current, motor voltage, and motor current) don't have any similar constraints that are obvious to me.

The measurements could also be something different from those possibilities, and possibly not the same between current and voltage. What is the convention?

Also, the numeric values for $K_T$ change depending on the current waveform used to drive the motor. Converting between them is straightforwards, but what waveform are the specified values measured with? Does it depend on the flux linkage waveform of the specific motor or not?

If a specific example helps, consider the Turnigy AquaStar T20 3T. I have measured that particular motor's flux linkage (hooked it up to a scope and spun it with a drill), and it's very nearly proportional to $18.1 sin(\theta) - 2.65 sin(5 \theta)$ (that is, a sin wave with a pretty significant fifth harmonic). Does that mean the manufacturer-specified $K_V$ is for driving it with a matching current waveform proportional to $18.1 sin(\theta) + 2.65 sin(5 \theta)$ to get no torque ripple, or just plain sinusoidal current with some torque ripple, or just plain sinusoidal voltage with some (different) torque ripple, or something else?

I'm pretty sure this all applies to any synchronous motor (with any number of phases) too, but I'm excluding other kinds of synchronous motors from my question, and sticking with exactly three phases, to keep it more manageable.

I've found references for how all the math works, but I'm just having trouble deciphering what people actually mean when they say "this motor has a $K_V$ of 170 RPM/V". This answer guesses $K_V$ is RMS line-to-line, but 15% off seems like a lot and suspiciously close to $1 - \frac{\sqrt(3)}{2}$ which would make it peak line-to-line like that question says Wikipedia says. This other answer seems to completely ignore which waveform is being used to drive the sinusoidal vs trapezoidal motors, which seems like it affects the proportionality constants it's discussing.

• Partly it will come down to how trustworthy the manufacturer's specs are. To compare your measurements to the spec, it would be useful to know the speed you made them at, and load (if any). Re: drive waveforms; you can eliminate any suggestion that any driver would simulate a specific 5th harmonic component, but a trapezoidal waveform may happen to have some. – Brian Drummond Apr 15 '18 at 11:37
• Also the linked motor specifies Kv for both Y and Delta, reassuringly close to sqrt(3) apart. – Brian Drummond Apr 15 '18 at 11:39
• I'm aware that the manufacturer's specs aren't perfectly accurate, but it's hard to interpret them at all if I don't know what they're supposed to represent, which is what I'm trying to figure out. Also, I have made a motor controller which matches that 5th harmonic component to eliminate the torque ripple. Not sure if it matters a whole lot in the end, but it is definitely an interesting project. However, if you're saying that nobody else ever does that, that would answer the part of my question about what drive waveforms the typical numbers are measured with. – Brian Silverman Apr 15 '18 at 19:59