Page Comparison

This block implements a brushless DC machine.

Image Removed

The BLDC block implements a three-phase Brushless DC motor (BLDC) model with resolvers and encoders. 

Equations & Characteristics

The BLDC machine shares the same equations as the permanent magnet synchronous machine.

General PMSM Solver Equation

The equation of the PMSM model can be expressed as follows:

I_{a b c}=\left[L_{a b c}(\theta)\right]^{-1}\left[\int\left(V_{a b c}-R I_{a b c}\right) d t-\psi_{a b c}\right]

where L_abc is the time-varying inductance matrix (global inductance for DQ and VDQ models), I_abc is the stator current inside the winding, R is the stator resistance and V_abc is the voltage across the stator windings. As for ψ_abc, it defines the magnet flux linked into the stator windings (for DQ and VDQ models), or the total flux (for the SH model),

Standard DQ Motor Characteristics

In normal conditions, the ideal sinusoidal stator voltages of the PMSM, back-EMFs, and inductances all have sinusoidal shapes. In the case of the BLDC, the back-EMFs are considered has trapezoidal. One can transform the equation using the Park transformation with a referential locked on the rotor position θ using (2a) and (2b).

\left[\begin{array}{l}
{V_{d s}} \\
{V_{q s}}
\end{array}\right]=\mathrm{T}\left[\begin{array}{l}
{V_{a s}} \\
{V_{b s}} \\
{V_{c s}}
\end{array}\right]

\mathrm{T}=\sqrt{2 / 3}\left[\begin{array}{cc}
{\cos (\theta)} & {\sin (\theta)} \\
{-\sin (\theta)} & {\cos (\theta)}
\end{array}\right]\left[\begin{array}{ccc}
{1} & {-0.5} & {-0.5} \\
{0} & {\sqrt{3} / 2} & {-\sqrt{3} / 2}
\end{array}\right]

The Park transform (also called ‘DQ’ transform) reduces sinusoidal varying quantities of inductances, flux, current, and voltage to constant values in the D-Q frame thus greatly facilitating the analysis and control of the device under study.

It is important to note that there are many different types of Park transforms and this often leads to confusion when interpreting the motor states inside the D-Q frame. The one used here presents the advantage of being orthonormal (notice the √3/2  factor). This particular Park orthonormal transform is power-invariant which means that the power computed in the D-Q frame by performing a dot product of currents and voltages will be numerically equal to the one computed in the phase domain. With this transform (and only this transform) the PMSM torque can be expressed by (3), where pp is the number of pole pairs.

T_{e}=p p\left[ I_{a b c}\cdot\frac{\partial \psi_{abc}}{\partial \theta_r} +\left(L_{d}-L_{q}\right) i_{d} i_{q}\right]

One may notice the absence of the 3/2 factor in (3), which is usually present in the PMSM torque equation when using non-orthonormal transforms. This is, again, because this model uses the orthonormal Park transform. Figure 1 explains the principle of the Park transform. Considering fixed ABC referential with all quantities ( V_bemf, motor current I) rotating at the electric frequency ω, if we observe these quantities in a D-Q frame turning at the same speed we can see that the motor quantities will be constant.

This is easy to see for the Back-EMF voltage V_bemf  that directly follows the Q-axis (because the magnet flux is on the D-axis by definition). In Figure 3, I leads  and the Q-axis by an angle called β (beta). The modulus of the vector I is called I_amp. In the figure below, θ is the rotor angle, aligned with the D-axis.

Image Removed

Mechanical Model Characteristics

The equation of the mechanical model in torque mode can be expressed as follows:

\omega_{r_{r}}=a \omega_{r z-1}+b\left(T_{e}-T_{L}\right)

a=\left(1-T_{S}\right) \frac{F_{v}}{J}

b=60 \frac{T_{r}}{2 \pi J}

where ω_r is the rotor speed, T_e is the electromagnetic torque, T_L is the torque command, F_vis the viscous friction coefficient, J is the inertia and T_s the sample time. There is a dead-zone implementation with the static friction torque if the electromagnetic doesn't exceed the static friction torque, the speed remains zero.

In speed mode, the rotor speed is directly set to the speed command ω_rc.

Resolver Characteristics

The equations of the resolver sensors can be expressed as follows:

\theta_{\text {res}}=\left(\theta_{\text {mee}}+\theta_{\text {offset}}\right) \cdot R_{p p}

\text { cosine signal }=\text { Excitation } *\left(\cos \left(\theta_{\text {res }}\right) * R_{k_{c a}}+\sin \left(\theta_{\text {res}}\right) * R_{k_{\text {st }}}\right)

\text { sine signal }=\text { Excitation } *\left(\cos \left(\theta_{\text {res }}\right) * R_{k_{\text {ce }}}+\sin \left(\theta_{\text {res }}\right) * R_{k_{\text {ge }}}\right)

where θ_res is the resolver angle, θ_mec is the mechanical angle of the machine, θ_offset is the angle offset, R_pp is the Number of pole pairs of the resolver and R_k are the resolver sine cosine gains.

This block implements a brushless DC machine.

...

The BLDC block implements a three-phase Brushless DC motor (BLDC) model with resolvers and encoders. 

Equations & Characteristics

The BLDC machine shares the same equations as the permanent magnet synchronous machine.

General PMSM Solver Equation

The equation of the PMSM model can be expressed as follows:

I_{a b c}=\left[L_{a b c}(\theta)\right]^{-1}\left[\int\left(V_{a b c}-R I_{a b c}\right) d t-\psi_{a b c}\right]

where L_abc is the time-varying inductance matrix (global inductance for DQ and VDQ models), I_abc is the stator current inside the winding, R is the stator resistance and V_abc is the voltage across the stator windings. As for ψ_abc, it defines the magnet flux linked into the stator windings (for DQ and VDQ models), or the total flux (for the SH model),

Standard DQ Motor Characteristics

In normal conditions, the ideal sinusoidal stator voltages of the PMSM, back-EMFs, and inductances all have sinusoidal shapes. In the case of the BLDC, the back-EMFs are considered has trapezoidal. One can transform the equation using the Park transformation with a referential locked on the rotor position θ using (2a) and (2b).

\left[\begin{array}{l}
{V_{d s}} \\
{V_{q s}}
\end{array}\right]=\mathrm{T}\left[\begin{array}{l}
{V_{a s}} \\
{V_{b s}} \\
{V_{c s}}
\end{array}\right]

\mathrm{T}=\sqrt{2 / 3}\left[\begin{array}{cc}
{\cos (\theta)} & {\sin (\theta)} \\
{-\sin (\theta)} & {\cos (\theta)}
\end{array}\right]\left[\begin{array}{ccc}
{1} & {-0.5} & {-0.5} \\
{0} & {\sqrt{3} / 2} & {-\sqrt{3} / 2}
\end{array}\right]

The Park transform (also called ‘DQ’ transform) reduces sinusoidal varying quantities of inductances, flux, current, and voltage to constant values in the D-Q frame thus greatly facilitating the analysis and control of the device under study.

It is important to note that there are many different types of Park transforms and this often leads to confusion when interpreting the motor states inside the D-Q frame. The one used here presents the advantage of being orthonormal (notice the √3/2  factor). This particular Park orthonormal transform is power-invariant which means that the power computed in the D-Q frame by performing a dot product of currents and voltages will be numerically equal to the one computed in the phase domain. With this transform (and only this transform) the PMSM torque can be expressed by (3), where pp is the number of pole pairs.

T_{e}=p p\left[ I_{a b c}\cdot\frac{\partial \psi_{abc}}{\partial \theta_r} +\left(L_{d}-L_{q}\right) i_{d} i_{q}\right]

One may notice the absence of the 3/2 factor in (3), which is usually present in the PMSM torque equation when using non-orthonormal transforms. This is, again, because this model uses the orthonormal Park transform. Figure 1 explains the principle of the Park transform. Considering fixed ABC referential with all quantities ( V_bemf, motor current I) rotating at the electric frequency ω, if we observe these quantities in a D-Q frame turning at the same speed we can see that the motor quantities will be constant.

This is easy to see for the Back-EMF voltage V_bemf  that directly follows the Q-axis (because the magnet flux is on the D-axis by definition). In Figure 3, I leads  and the Q-axis by an angle called β (beta). The modulus of the vectorI is called I_amp. In the figure below, θ is the rotor angle, aligned with the D-axis.

...

Trapezoidal back-EMF Characteristics

The main difference between the PMSM and the BLDC lies in the shape of the Back EMF voltage. The BLDC has a trapezoidal back EMF shape that is parametrized with λ_m the permanent flux linkage and H the back EMF flat area in degree.Image Removed

...

The electromotive force is constructed from a cosine table as described in the following equations:  

Sine \; Output = [Sin.Sin*sin(pp(\theta_m - \theta_{Offset})) + Sin.Cos * cos(pp * \theta_m - \theta_{Offset}))] * Excitation

Cosine \; Output = [Cos.Sin*sin(pp(\theta_m - \theta_{Offset})) + Cos.Cos * cos(pp * \theta_m - \theta_{Offset}))] * Excitation

Sine \; Output = sin(\theta_m) * Excitation