Three-winding brushless DC motor with trapezoidal flux distribution

**Library:**Simscape / Electrical / Electromechanical / Permanent Magnet

The BLDC block models a permanent magnet synchronous machine with a three-phase wye-wound stator. The block has four options for defining the permanent magnet flux distribution as a function of rotor angle. Two options allow for simple parameterization by assuming a perfect trapezoid for the back emf. For simple parameterization, you specify either the flux linkage or the rotor-induced back emf. The other two options give more accurate results using tabulated data that you specify. For more accurate results, you specify either the flux linkage partial derivative or the measured back emf constant for a given rotor speed.

The figure shows the equivalent electrical circuit for the stator windings.

This figure shows the motor construction with a single pole-pair on the rotor.

For the axes convention in the preceding figure, the
*a*-phase and permanent magnet fluxes are aligned when rotor
angle *θ _{r}* is zero. The block supports a
second rotor-axis definition. For the second definition, the rotor angle is the
angle between the

The rotor magnetic field due to the permanent magnets create a trapezoidal rate of change of flux with rotor angle. The figure shows this rate of change of flux.

Back emf is the rate of change of flux, defined by

$\frac{d\Phi}{dt}=\frac{\partial \Phi}{\partial \theta}\frac{d\theta}{dt}=\frac{\partial \Phi}{\partial \theta}\omega ,$

where:

*Φ*is the permanent magnet flux linkage.*θ*is the rotor angle.*ω*is the mechanical rotational speed.

The height `h`

of the trapezoidal rate of change of flux profile
is derived from the permanent magnet peak flux.

Integrating $\frac{\partial \Phi}{\partial \theta}$ over the range 0 to π/2,

${\Phi}_{max}=\frac{h}{2}({\theta}_{F}+{\theta}_{W}),$

where:

*Φ*is the permanent magnet flux linkage._{max}*h*is the rate of change of flux profile height.*θ*is the rotor angle range over which the back emf that the permanent magnet flux induces in the stator is constant._{F}*θ*is the rotor angle range over which back emf increases or decreases linearly when the rotor moves at constant speed._{W}

Rearranging the preceding equation,

$h=2{\Phi}_{max}/({\theta}_{F}+{\theta}_{W}).$

Voltages across the stator windings are defined by

$$\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]=\left[\begin{array}{ccc}{R}_{s}& 0& 0\\ 0& {R}_{s}& 0\\ 0& 0& {R}_{s}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}\frac{d{\psi}_{a}}{dt}\\ \frac{d{\psi}_{b}}{dt}\\ \frac{d{\psi}_{c}}{dt}\end{array}\right],$$

where:

*v*,_{a}*v*, and_{b}*v*are the external voltages applied to the three motor electrical connections._{c}*R*is the equivalent resistance of each stator winding._{s}*i*,_{a}*i*, and_{b}*i*are the currents flowing in the stator windings._{c}$$\frac{d{\psi}_{a}}{dt},$$$$\frac{d{\psi}_{b}}{dt},$$ and $$\frac{d{\psi}_{c}}{dt}$$

are the rates of change of magnetic flux in each stator winding.

The permanent magnet and the three windings contribute to the total flux linking each winding. The total flux is defined by

$$\left[\begin{array}{c}{\psi}_{a}\\ {\psi}_{b}\\ {\psi}_{c}\end{array}\right]=\left[\begin{array}{ccc}{L}_{aa}& {L}_{ab}& {L}_{ac}\\ {L}_{ba}& {L}_{bb}& {L}_{bc}\\ {L}_{ca}& {L}_{cb}& {L}_{cc}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}{\psi}_{am}\\ {\psi}_{bm}\\ {\psi}_{cm}\end{array}\right],$$

where:

*ψ*,_{a}*ψ*, and_{b}*ψ*are the total fluxes linking each stator winding._{c}*L*,_{aa}*L*, and_{bb}*L*are the self-inductances of the stator windings._{cc}*L*,_{ab}*L*,_{ac}*L*, etc. are the mutual inductances of the stator windings._{ba}*ψ*,_{am}*ψ*, and_{bm}*ψ*are the permanent magnet fluxes linking the stator windings._{cm}

The inductances in the stator windings are functions of rotor angle, defined by

${L}_{aa}={L}_{s}+{L}_{m}\text{cos}(2{\theta}_{r}),$

${L}_{bb}={L}_{s}+{L}_{m}\text{cos}(2\left({\theta}_{r}-2\pi /3\right)),$

${L}_{cc}={L}_{s}+{L}_{m}\text{cos}(2\left({\theta}_{r}+2\pi /3\right)),$

${L}_{ab}={L}_{ba}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{r}+\pi /6\right)\right),$

${L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{r}+\pi /6-2\pi /3\right)\right),$

and

${L}_{ca}={L}_{ac}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{r}+\pi /6+2\pi /3\right)\right),$

where:

*L*is the stator self-inductance per phase — The average self-inductance of each of the stator windings._{s}*L*is the stator inductance fluctuation — The fluctuation in self-inductance and mutual inductance with changing rotor angle._{m}*M*is the stator mutual inductance — The average mutual inductance between the stator windings._{s}

The permanent magnet flux linking each stator winding follows the trapezoidal profile shown in the figure. The block implements the trapezoidal profile using lookup tables to calculate permanent magnet flux values.

The defining voltage and torque equations for the block are

$\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\\ {v}_{0}\end{array}\right]=P\left(\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]-N\omega \left[\begin{array}{c}\frac{\partial {\psi}_{am}}{\partial {\theta}_{r}}\\ \frac{\partial {\psi}_{bm}}{\partial {\theta}_{r}}\\ \frac{\partial {\psi}_{cm}}{\partial {\theta}_{r}}\end{array}\right]\right),$

${v}_{d}={R}_{s}{i}_{d}+{L}_{d}\frac{d{i}_{d}}{dt}-N\omega {i}_{q}{L}_{q},$

${v}_{q}={R}_{s}{i}_{q}+{L}_{q}\frac{d{i}_{q}}{dt}+N\omega {i}_{d}{L}_{d},$

${v}_{0}={R}_{s}{i}_{0}+{L}_{0}\frac{d{i}_{0}}{dt},$

and

$T=\frac{3}{2}N\left({i}_{q}{i}_{d}{L}_{d}-{i}_{d}{i}_{q}{L}_{q}\right)+\left[\begin{array}{ccc}{i}_{a}& {i}_{b}& {i}_{c}\end{array}\right]\left[\begin{array}{c}\frac{\partial {\psi}_{am}}{\partial {\theta}_{r}}\\ \frac{\partial {\psi}_{bm}}{\partial {\theta}_{r}}\\ \frac{\partial {\psi}_{cm}}{\partial {\theta}_{r}}\end{array}\right],$

where:

*v*,_{d}*v*, and_{q}*v*are the_{0}*d*-axis,*q*-axis, and zero-sequence voltages.*P*is Park’s Transformation, defined by$P=2/3\left[\begin{array}{ccc}\mathrm{cos}{\theta}_{e}& \mathrm{cos}\left({\theta}_{e}-2\pi /3\right)& \mathrm{cos}\left({\theta}_{e}+2\pi /3\right)\\ -\mathrm{sin}{\theta}_{e}& -\mathrm{sin}\left({\theta}_{e}-2\pi /3\right)& -\mathrm{sin}\left({\theta}_{e}+2\pi /3\right)\\ 0.5& 0.5& 0.5\end{array}\right]$

*N*is the number of rotor permanent magnet pole pairs.*ω*is the rotor mechanical rotational speed.$$\frac{\partial {\psi}_{am}}{\partial {\theta}_{r}},$$$$\frac{\partial {\psi}_{bm}}{\partial {\theta}_{r}},$$ and $$\frac{\partial {\psi}_{cm}}{\partial {\theta}_{r}}$$

are the partial derivatives of instantaneous permanent magnet flux linking each phase winding.

*i*,_{d}*i*, and_{q}*i*are the_{0}*d*-axis,*q*-axis, and zero-sequence currents, defined by$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]=P\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].$

*L*=_{d}*L*+_{s}*M*+ 3/2_{s}*L*._{m}*L*is the stator_{d}*d*-axis inductance.*L*=_{q}*L*+_{s}*M*− 3/2_{s}*L*._{m}*L*is the stator_{q}*q*-axis inductance.*L*=_{0}*L*– 2_{s}*M*._{s}*L*is the stator zero-sequence inductance._{0}*T*is the rotor torque. Torque flows from the motor case (block physical port C) to the motor rotor (block physical port R).

Iron losses are divided into two terms, one representing the main magnetizing path, and the other representing the cross-tooth tip path that becomes active during field weakened operation. The iron losses model, which is based on the work of Mellor [3].

The term representing the main magnetizing path depends on the induced RMS stator voltage, $${V}_{{m}_{rms}}^{}$$:

$${P}_{OC}\left({V}_{{m}_{rms}}^{}\right)=\frac{{a}_{h}}{k}{V}_{{m}_{rms}}^{}+\frac{{a}_{j}}{{k}^{2}}{V}_{{m}_{rms}}^{2}+\frac{{a}_{ex}}{{k}^{1.5}}{V}_{{m}_{rms}}^{1.5}$$

This is the dominant term during no-load operation. *k* is the
back emf constant relating RMS volts per Hz. It is defined as $$k={V}_{{m}_{rms}}^{}/f$$, where *f* is the electrical frequency. The first
term on the right-hand side is the magnetic hysteresis loss, the second is the eddy
current loss and the third is the excess loss. The three coefficients appearing on
the numerators are derived from the values that you provide for the open-circuit
hysteresis, eddy, and excess losses.

The term representing the cross-tooth tip path becomes important when a demagnetizing field is set up and can be determined from a finite element analysis short-circuit test. It depends on the RMS emf associated with the cross-tooth tip flux, $${V}_{{d}_{rms}}^{*}$$:

$${P}_{SC}\left({V}_{{d}_{rms}}^{*}\right)=\frac{{b}_{h}}{k}{V}_{{d}_{rms}}^{*}+\frac{{b}_{j}}{{k}^{2}}{V}_{{d}_{rms}}^{*2}+\frac{{b}_{ex}}{{k}^{1.5}}{V}_{{d}_{rms}}^{*1.5}$$

The three numerator terms are derived from the values you provide for the short-circuit hysteresis, eddy, and excess losses.

The block has four optional thermal ports, one for each of the three windings and
one for the rotor. These ports are hidden by default. To expose the thermal ports,
right-click the block in your model, select **Simscape** >
**Block choices**, and then select the desired block
variant with thermal ports: **Composite three-phase ports | Show thermal
port** or **Expanded three-phase ports | Show thermal
port**. This action displays the thermal ports on the block icon,
and exposes the **Temperature Dependence** and **Thermal
Port** parameters. These parameters are described further on this
reference page.

Use the thermal ports to simulate the effects of copper resistance and iron losses that convert electrical power to heat. For more information on using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

Use the **Variables** settings to specify the priority and initial target
values for the block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables (Simscape).

[1] Kundur, P. *Power
System Stability and Control.* New York, NY: McGraw Hill,
1993.

[2] Anderson, P. M.
*Analysis of Faulted Power Systems.* Hoboken, NJ: Wiley-IEEE
Press, 1995.

[3] Mellor, P.H., R. Wrobel, and
D. Holliday. “A computationally efficient iron loss model for brushless AC
machines that caters for rated flux and field weakened operation.”
*IEEE Electric Machines and Drives Conference*. May
2009.