Electronic – BLDC motor poles versus Torque and RPM

First of all, Vref on the chip controls maximum power = maximum torque. When S1 on eval board is in TRQ mode, then VREF pin on CN4 is connected directly to Vref of the drived. In this mode, it will have very little effect if the motor is unloaded.

In order to control the speed of the motor while maintaining the torque you will need to wrap the whole thing into a control loop -- something measures motor speed, and changes power to motor correspondingly.

For the best performance, you will want to do it per-phase basis, delaying control inputs if speed is exceeded. However, this is quite involved, so I would instead use manufacturer-recommended analog way (see section 9, 'Tachometer'). This is already assembled for you on the eval board -- set S1 to SPEED mode, and VREF should vary motor voltage smoothly between 0 and max speed.

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.