Electronic – Viscous damping in BLDC

Driving a motor blind is a bad idea for several reasons:

1. It is inefficient. The most efficient way to run the motor is for the magentic field to be 90° ahead of the rotor. Put another way, the torque on the rotor is the cross product of the driving magnetic field and the rotor's magnetic orientation.

   With position feedback, the magnetic field can be kept near the optimum angle, which means the current goes to actually pushing the motor instead of holding it in place. Put another way, the amplitude is just what it needs to be to keep the motor spinning at the desired speed in the maximum torque configuration. When you don't know where the rotor is, you end up over-driving the motor.

   Another way to look at this is that the driving field has a sine and cosine component. Let's say the cosine is the part 90° ahead of the rotor and the sine part is where the rotor currently is. Any phase angle can be thought of as just a different mix of the sine and cosine components. However, only the cosine component moves the motor. The sine component only causes heating and represents wasted power.

2. Once you loose lock, the game is over. With a fixed drive, the drive angle will only be a little ahead of the rotor at low torque. As the speed increases (and the effective drive voltage automatically decreases due to back EMF) or the load increases, the fixed drive will get up to 90° ahead of the rotor.

   However, at this point it's right on the edge and any change will cause less torque. If the load on the motor increases, the rotor will get more than 90° behind, which causes less torque, which causes it to get even more behind. Over the next 1/4 turn of slip, the forward torque will decrease to zero. Then for the next 1/2 turn after that, the driving torque actually pushes the rotor backwards.

   At this point you're totally screwed. Remember you got into this predicament in the first place because the driving torque couldn't keep up with load, and you've just experienced a net negative drive over the last 3/4 turn. If the load is suddenly removed and if you're very lucky, the rotor might be able to speed up to sync up with the drive in the next 1/4 cycle, but certainly not if whatever condition caused the problem in the first place is still present.

   Once the rotor gets out of sync, the net torque over any one rotation is 0. The product of two sine waves of different frequency always averages to 0 regardless of the phase angle between them.

Brushless AC

Lets start with a brushless AC machine first & Field Orientated Control.

So BLAC machine's have their stator windings sinusoidally distributed (higher concentration close to the tooth, lower for the outer turns)

Now to control such machine is relatively complex and you can use a field orientated control ( F.O.C. ).Using Park's transform (Clark + rotation) the 3phase sinus stator current's can be transformed 1st into a rotating 2phase representation

Park Transforms - General

$$ I_{\alpha \beta 0 } = \frac{2}{3}\begin{bmatrix} 1 & \frac{-1}{2} & \frac{-1}{2} \\ 0 & \frac{\sqrt{3}}{2} &\frac{-\sqrt{3}}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} $$

or simply

\$I_\alpha = I_a \$

\$I_\beta = \frac{2I_b + Ia}{\sqrt{3}} \$

And these phasors can further be reduced to two DC quantities via a rotating frame of reference transform

$$ \begin{bmatrix} I_d\\ I_q \end{bmatrix} = \begin{bmatrix} Cos(\Theta ) & Sin(\Theta )\\ -Sin(\Theta ) & Cos(\Theta ) \end{bmatrix} \cdot \begin{bmatrix} I_\alpha\\ I_\beta \end{bmatrix} $$

These two DC terms (with Id usually controlled to 0, unless field weakening is desired) are simple inputs to two classic PI loops to control Id and Iq (the output of which is Vd & Vq).
This Vd & Vq, via inverse Clark&PArk produce the 3phase voltage that will be applied to the stator (insert SVM or SPWM)

Great, it works

Brushless DC & general Permanent Magnet machine equations

Thing is BLDC machines produce higher torque than a sinusoidally wound stator (downto the higher concentration of winding around the teeth to produce the flattened line-line profile). The downside is the torque ripple. This is mostly due to simplifying the control downto which equally limits the effective bandwidth of the controller.

A park-like transform can be applied to overcome some of these shortcomings as the aim is to produce a stator stimulus closer to the airgap profile - tpyically a Quazi squarewave (30-120-30) is superimposed onto a trapezoidal backEMF (60-60-60).

First of all you need the machine equation for a BLDC machine (which is the same for a BLAC machine). $$ \begin{bmatrix} V_a\\ V_b\\ V_c \end{bmatrix} = R_s \begin{bmatrix} i_a\\ i_b\\ i_c \end{bmatrix} + L\frac{\mathrm{d} }{\mathrm{d} t}\begin{bmatrix} i_a\\ i_b\\ i_c \end{bmatrix} + \begin{bmatrix} e_a\\ e_b\\ e_c \end{bmatrix}$$

\$e_a, e_b, e_c\$ are the 3 backEMF's that are generated for a rotating rotor. Via Faraday's law $$ \varepsilon = - \frac{\mathrm{d} \Phi _B}{\mathrm{d} t} $$

BackEMF is the rate of change of flux

$$ \frac{\mathrm{d} \Phi }{\mathrm{d} t} = \frac{\mathrm{d} \Phi }{\mathrm{d} \Theta}\frac{\mathrm{d} \Theta }{\mathrm{d} t} = \frac{\mathrm{d} \Phi }{\mathrm{d} \Theta}\omega $$

$$ e = \omega {\Phi}' $$

Thus: $$ \begin{bmatrix} e_a\\ e_b\\ e_c \end{bmatrix} = \omega_e \frac{\mathrm{d} }{\mathrm{d} \vartheta_e}\begin{bmatrix} \Phi_a\\ \Phi_b\\ \Phi_c \end{bmatrix} = \omega_e \begin{bmatrix} {\Phi}'_a\\ {\Phi}'_b\\ {\Phi}'_c \end{bmatrix}$$

As the total torque produced is the summation of the 3 phases producing torque (factoring in the pole-pair count):

Torque equation

\$T_e = P({\Phi}'_a i_a + {\Phi}'_b i_b + {\Phi}'_b i_b)\$

Now, the Park transform can be applied to voltage, currents and flux, thus:

$$ T_e = \frac{3}{2}P({\Phi}'_\alpha i_\alpha + {\Phi}'_\beta i_\beta + {\Phi}'_0 i_0) = \frac{3}{2}P({\Phi}'_d i_d + {\Phi}'_q i_q + {\Phi}'_0 i_0) = \frac{3}{2}P{\Phi}'_q i_q $$

For a BLAC machine \${\Phi}'_q\$ is a simple sinus profile (and thus a lookup table, CORDIC... can be used) but for a BLDC an appropriate lookup table of flux vs angle is required specific for the machine being used.