BLDC

Three-winding brushless DC motor with trapezoidal flux distribution

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• Simscape / Electrical / Electromechanical / Permanent Magnet

Description

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.

Motor Construction

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 a-phase magnetic axis and the rotor q-axis.

Trapezoidal Rate of Change of Flux

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_max is the permanent magnet flux linkage.

• h is the rate of change of flux profile height.

• θ_F is the rotor angle range over which the back emf that the permanent magnet flux induces in the stator is constant.

• θ_W is the rotor angle range over which back emf increases or decreases linearly when the rotor moves at constant speed.

Rearranging the preceding equation,

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

Electrical Defining Equations

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_b, and v_c are the external voltages applied to the three motor electrical connections.

• R_s is the equivalent resistance of each stator winding.

• i_a, i_b, and i_c are the currents flowing in the stator windings.

• $$\frac{d{\psi }_a}{dt},$$$$\frac{d{\psi }_b}{dt},$$ 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_b, and ψ_c are the total fluxes linking each stator winding.

• L_aa, L_bb, and L_cc are the self-inductances of the stator windings.

• L_ab, L_ac, L_ba, etc. are the mutual inductances of the stator windings.

• ψ_am, ψ_bm, and ψ_cm are the permanent magnet fluxes linking the stator windings.

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

$L_{aa}=L_s+L_m\text{cos}(2{\theta }_r),$

$L_{bb}=L_s+L_m\text{cos}(2\left({\theta }_r-2\pi /3\right)),$

$L_{cc}=L_s+L_m\text{cos}(2\left({\theta }_r+2\pi /3\right)),$

$L_{ab}=L_{ba}=-M_s-L_m\cos \left(2\left({\theta }_r+\pi /6\right)\right),$

$L_{bc}=L_{cb}=-M_s-L_m\cos \left(2\left({\theta }_r+\pi /6-2\pi /3\right)\right),$

and

$L_{ca}=L_{ac}=-M_s-L_m\cos \left(2\left({\theta }_r+\pi /6+2\pi /3\right)\right),$

where:

• L_s is the stator self-inductance per phase — The average self-inductance of each of the stator windings.

• L_m is the stator inductance fluctuation — The amplitude of the fluctuation in self-inductance and mutual inductance with changing rotor angle.

• M_s is the stator mutual inductance — The average mutual inductance between the stator windings.

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.

Simplified Equations

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_q, and v₀ are the d-axis, q-axis, and zero-sequence voltages.

• P is Park’s Transformation, defined by

  $P=2/3\left[\begin{array}{ccc}\cos {\theta }_e& \cos \left({\theta }_e-2\pi /3\right)& \cos \left({\theta }_e+2\pi /3\right)\\ -\sin {\theta }_e& -\sin \left({\theta }_e-2\pi /3\right)& -\sin \left({\theta }_e+2\pi /3\right)\\ 0.5& 0.5& 0.5\end{array}\right].$

• 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_q, and i₀ are the 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_s + 3/2 L_m. L_d is the stator d-axis inductance.

• L_q = L_s + M_s − 3/2 L_m. L_q is the stator q-axis inductance.

• L₀ = L_s – 2M_s. L₀ is the stator zero-sequence inductance.

• T is the rotor torque. Torque flows from the motor case (block physical port C) to the motor rotor (block physical port R).

Variables

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).

Ports

Conserving

~ — Three-phase port

electrical

Expandable three-phase port.

n — Neutral phase

electrical

Electrical conserving port associated with the neutral phase

R — Motor rotor

mechanical

Mechanical rotational conserving port associated with the motor rotor

C — Motor case

mechanical

Mechanical rotational conserving port associated with the motor case

Parameters

Rotor

Winding type — Windings configuration

Wye-wound (default) | Delta-wound

Select the configuration for the windings:

• Wye-wound — The windings are wye-wound.

• Delta-wound — The windings are delta-wound. The a-phase is connected between ports a and b, the b-phase between ports b and c and the c-phase between ports c and a.

Back EMF profile — Back EMF profile

Perfect trapezoid - specify maximum flux linkage (default) | Perfect trapezoid - specify maximum rotor-induced back emf | Tabulated - specify flux partial derivative with respect to rotor angle | Tabulated - specify rotor-induced back emf as a function of rotor angle

Parameterization for defining the permanent magnet flux distribution as a function of rotor angle. Choose:

• Perfect trapezoid - specify maximum flux linkage to specify the maximum flux linkage for the permanent magnet and the rotor angle where the back emf is constant. The block assumes a perfect trapezoid for the back emf. This is the default value.

• Perfect trapezoid - specify maximum rotor-induced back emf to specify the maximum rotor-induced back emf and the corresponding rotor speed. The block assumes a perfect trapezoid for the back emf.

• Tabulated - specify flux partial derivative with respect to rotor angle to specify values for the partial derivative of flux linkage and the corresponding rotor angles.

• Tabulated - specify rotor-induced back emf as a function of rotor angle to specify the measured back emf constant and the corresponding rotor speed and angles.

Maximum permanent magnet flux linkage — Maximuym permanent magnet flux linkage

0.03 Wb (default)

Peak permanent magnet flux linkage with any of the stator windings. This parameter is visible only when is set to .

Dependencies

This parameter is visible only when you set the Back EMF profile parameter to Perfect trapezoid - specify maximum flux linkage.

Rotor angle over which back emf is constant — Rotor angle over which back emf is constant

pi / 12 rad (default)

Rotor angle range over which the permanent magnet flux linking the stator winding is constant. This angle is θ_F in the figure that shows the Trapezoidal Rate of Change of Flux.

Dependencies

This parameter is visible only when you set the Back EMF profile parameter to Perfect trapezoid - specify maximum flux linkage or Perfect trapezoid - specify maximum rotor-induced back emf.

Maximum rotor-induced back emf — Maximum rotor-induced back emf

9.6 V (default)

Peak rotor-induced back emf into the stator windings.

Dependencies

This parameter is visible only when you set the Back EMF profile parameter to Perfect trapezoid - specify maximum rotor-induced back emf.

Rotor-induced back emf — Rotor-induced back emf

[0, -9.6, -9.6, 9.6, 9.6, 0] V (default)

Vector of values for the rotor-induced back emf as a function of rotor angle. The first and last values must be the same, and are normally both zero. For more information, see the Corresponding rotor angles parameter. First and last values are the same because flux is cyclic with period $2\pi /N$, where N is the number of permanent magnet pole pairs.

Dependencies

This parameter is visible only when you set the Back EMF profile parameter to Tabulated - specify rotor-induced back emf as a function of rotor angle.

Flux linkage partial derivative with respect to rotor angle — Flux linkage partial derivative with respect to rotor angle

[0, -.1528, -.1528, .1528, .1528, 0] Wb/rad (default)

Vector of values for the partial derivative of flux linkage (where flux linkage is flux times number of winding turns) with respect to rotor angle. The first and last values must be the same, and are normally both zero. For more information, see the Corresponding rotor angles parameter. First and last values are the same because flux is cyclic with period $2\pi /N$, where N is the number of permanent magnet pole pairs.

Corresponding rotor angles — Corresponding rotor angles

[0, 7.5, 22.5, 37.5, 52.5, 60] deg (default)

Vector of rotor angles where the flux linkage partial derivative or rotor-induced back emf is defined. Rotor angle is defined as the angle between the a-phase magnetic axis and the d-axis. That is, when the angle is zero, the magnetic fields due to the rotor and the a-phase winding align. This definition is used regardless of your block setting for rotor angle definition. The first value is zero, and the last value is $2\pi /N$, where N is the number of permanent magnet pole pairs.

Dependencies

This parameter is visible only when you set the Back EMF profile parameter to Tabulated - specify flux partial derivative with respect to rotor angle or to Tabulated - specify rotor-induced back emf as a function of rotor angle.

Rotor speed used for back emf measurement — Rotor speed used for back emf measurement

600 rpm (default)

Specify the rotor speed corresponding to the maximum rotor-induced back emf.

Dependencies

This parameter is visible only when you set the Back EMF profile parameter to Perfect trapezoid - specify maximum rotor-induced back emf or Tabulated - specify rotor-induced back emf as a function of rotor angle.

Number of pole pairs — Number of pole pairs

6 (default)

Number of permanent magnet pole pairs on the rotor.

Zero sequence — Zero sequence option

Include (default) | Exclude

Option to include or exclude zero-sequence terms.

• Include — Include zero-sequence terms. To prioritize model fidelity, use this default setting. Using this option:

  • Results in an error for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver (Simscape).

  • Exposes a zero-sequence parameter in the Impedances settings.

• Exclude — Exclude zero-sequence terms. To prioritize simulation speed for desktop simulation or real-time deployment, select this option.

Rotor angle definition — Reference point for the rotor angle measurement

Angle between the a-phase magnetic axis and the d-axis (default) | Angle between the a-phase magnetic axis and the q-axis

Reference point for the rotor angle measurement. The default value is Angle between the a-phase magnetic axis and the d-axis. This definition is shown in the Motor Construction figure. When you select this value, the rotor and a-phase fluxes are aligned when the rotor angle is zero.

The other value you can choose for this parameter is Angle between the a-phase magnetic axis and the q-axis. When you select this value, the a-phase current generates maximum torque when the rotor angle is zero.

Stator

Stator parameterization — Stator parameterization

Specify Ld, Lq, and L0 (default) | Specify Ls, Lm, and Ms

Choose Specify Ld, Lq, and L0 or Specify Ls, Lm, and Ms.

Stator d-axis inductance, Ld — Stator d-axis inductance

0.00022 H (default)

D-axis inductance.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0.

Stator q-axis inductance, Lq — Stator q-axis inductance

0.00022 H (default)

Q-axis inductance.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0.

Stator zero-sequence inductance, L0 — Stator zero-sequence inductance

0.00016 H (default)

Zero-sequence inductance.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0.

Stator self-inductance per phase, Ls — Stator self-inductance per phase

0.0002 H (default)

Average self-inductance of each of the three stator windings.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ls, Lm, and Ms.

Stator inductance fluctuation, Lm — Stator inductance fluctuation

0 H (default)

Amplitude of the fluctuation in self-inductance and mutual inductance of the stator windings with rotor angle.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ls, Lm, and Ms.

Stator mutual inductance, Ms — Stator mutual inductance

0.00002 H (default)

Average mutual inductance between the stator windings.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ls, Lm, and Ms.

Stator resistance per phase, Rs — Stator resistance per phase

0.013 Ohm (default)

Resistance of each of the stator windings.

Mechanical

Rotor inertia — Rotor inertia

0.01 kg*m^2 (default)

Inertia of the rotor attached to mechanical translational port R. The value can be zero.

Rotor damping — Rotor damping

0 N*m/(rad/s) (default)

Rotary damping.

Model Examples

BLDC Speed Control

Control the rotor speed in a BLDC based electrical drive. An ideal torque source provides the load. The Control subsystem uses a PI-based cascade control structure with an outer speed control loop and an inner dc-link voltage control loop. The dc-link voltage is adjusted through a DC-DC buck converter. The BLDC is fed by a controlled three-phase inverter. The gate signals for the inverter are obtained from hall signals. The simulation uses speed steps. The Scopes subsystem contains scopes that allow you to see the simulation results.

Open Model

References

[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.

Extended Capabilities

C/C++ Code Generation
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