BLDC Transfer Function Inductance Value

I realize a very similar question has been asked here, but I'm slightly confused about the derivation, specifically the quantities Lq and Rq which refer to the armature inductance and resistance. I'm using this transfer function to get the total loop gain of my BLDC closed-loop speed control system, so that I can model it and settle on some approximate PI coefficients using root-locus. The system uses trapezoidal control and gets speed measurements from hall sensors, feeding that speed measurement into a PI controller which updates a PWM duty cycle. So I need a gain that relates angular velocity to applied voltage:

Using this reference the transfer function is derived as follows:

$$V_{ab} = 2Ri + 2(L-M)\frac{di}{dt}+ (e_{a}-e_{b})$$ since $e_{a}=-e_{b}$ $$V_{ab} = 2Ri + 2(L-M)\frac{di}{dt}+ 2e_{a}$$ $$V_{ab} = R_{a}i + L_{a}\frac{di}{dt}+ K_{e}w(s)$$

where $K_{e}=\frac{2e_{a}}{w(s)}$ is the line back EMF constant, $R_{a}=2R$ is the line resistance and $L_{a}=2(L-M)$ is the line inductance.

With $i = \frac{T_{e}}{K_{t}}=\frac{\omega(s)B + J\omega(s)s}{K_{t}}$ then the equation becomes: $$V_{ab} = (R_{a} + sL_{a})\frac{\omega(s)B + J\omega(s)s}{K_{t}}+ K_{e}w(s)$$ and we get: $$\frac{\omega (s)}{V(s)} = \frac{K_{t}}{JL_{a}s^2 + (BL_{a} + JR_{a})s + (BR_{a} + K_{e}K_{t})}$$

My question is:

1. If the transfer equation derived above is correct, is $L_{q}$ in the post I mentioned above equivalent to the $L_{a}$ (the line inductance) in my equation? I'm not exactly sure what $L_{q}$ or $R_{q}$ are, though I do have them identified for my motor. I did read somewhere that line inductance changes as the rotor moves, which would seem to make this model totally useless for my purposes. Again, not trying to do FOC, just need trapezoidal speed control. Which inductance value do I use?

Thank you.

• I think it is a control problem to use a velocity gain function other than the motors no load linear kV/RPM or RPM/V. You want to control torque and measure current and have a target profile for max +/-acceleration, and +/- max velocity and desired result. – Tony Stewart Sunnyskyguy EE75 Sep 15 '18 at 2:21

A FOC control implements two PI controllers Q-Torque and D-Excitation. If there is no field weakening contrl, then the setpoint of d-axis is zero, because we don't want any additional flux in the rotor direction. The rotor itself is a permanent magnet that gives the excitation magnetic field.

The q-axis is responsible for torque. Here the similarity of brush DC comes into a play.

For six step control the hall sensors determine the switching sequence, so that stator magnetic field is almost at right angle with rotor magnetic field. The FOC with two PI controllers makes this even more precise, but for a good approximation let's say that six step method also keeps the right angle. Therefore, the inductance that the source will see it's dominantly the Lq.

You have to be aware that these inductivity could be refered with different methods. Can be a phase to neutral or phase to phase.

For your setup the valid La shall be the inductance of phase to phase when the rotor is locked at 90 degrees electrical angle from d-axis. The Ra shall be the phase to phase resistance.

You may find some details how to measure Lq and Ld, but pay attentiton if L is phase to phase or phase to neutral.

Determine PMSM parameters

• Thank you! That answers my question perfectly. I didn't realize that six-step even gets approximately close to making the magnetic fields orthogonal, but that makes sense looking at the torque waveforms as it's pretty much constant. I'm aware of the phase-phase versus phase-neutral distinction so I'll keep that in mind. Thanks! – Alexander Villa Sep 15 '18 at 22:11