BLDC

Three-phase brushless DC motor with trapezoidal flux distribution

Since R2023a

• 

Libraries:
Motor Control Blockset / Electrical Systems / Motors

Description

The BLDC block implements a three-phase brushless DC (BLDC) motor with a trapezoidal back electromotive force that remains constant for a position range of 120 electrical degrees. The block uses the three-phase input voltages to regulate the individual phase currents, allowing control of the motor torque or speed.

By default, the block sets the Simulation type parameter to Continuous to use a continuous sample time during simulation. If you want to use the block for fixed-step simulation, set the parameter to Discrete. Then specify a Sample time (s) parameter.

Motor Construction

This figure shows a motor with a single pole pair.

The motor magnetic field due to the permanent magnets creates a trapezoidal rate of change of flux with the motor angle.

For the axes convention, the a-phase and permanent magnet fluxes align when the motor angle θ_r is zero.

Three-Phase Sinusoidal Model Electrical System

Instead of the standard dq reference frame, the block uses d’q’ reference frame, which is defined by these transformation equations.

$$v_{d\text{'}}=\frac{2}{3}\left(\left(v_a-R{i}_a-E_a\right)\cos {\theta }_e+\left(v_b-R{i}_b-E_b\right)\cos \left({\theta }_e-\frac{2\pi }{3}\right)+\left(v_c-R{i}_c-E_c\right)\cos \left({\theta }_e+\frac{2\pi }{3}\right)\right)$$

$$v_{q\text{'}}=-\frac{2\pi }{3}\left(\left(v_a-R{i}_a-E_a\right)\sin {\theta }_e+\left(v_b-R{i}_b-E_b\right)\sin \left({\theta }_e-\frac{2\pi }{3}\right)+\left(v_c-R{i}_c-E_c\right)\sin \left({\theta }_e+\frac{2\pi }{3}\right)\right)$$

$$\begin{array}{l}{E}_a=P{\omega }_m\frac{\partial {\psi }_a}{\partial {\theta }_e}\\ E_b=P{\omega }_m\frac{\partial {\psi }_b}{\partial {\theta }_e}\\ E_c=P{\omega }_m\frac{\partial {\psi }_c}{\partial {\theta }_e}\end{array}$$

$$\begin{array}{l}{i}_a=i_{d\text{'}}\cos {\theta }_e-i_{q\text{'}}\sin {\theta }_e\\ i_b=i_{d\text{'}}\cos \left({\theta }_e-\frac{2\pi }{3}\right)-i_{q\text{'}}\sin \left({\theta }_e-\frac{2\pi }{3}\right)\\ i_c=i_{d\text{'}}\cos \left({\theta }_e+\frac{2\pi }{3}\right)-i_{q\text{'}}\sin \left({\theta }_e+\frac{2\pi }{3}\right)\end{array}$$

The block implements these equations, expressed in the dq and d’q’ reference frame of the BLDC motor. All quantities in the motor reference frame are with respect to the stator phase A.

$$\begin{array}{l}{\omega }_e=P{\omega }_m\\ \frac{d}{dt}{i}_{d\text{'}}=\frac{1}{L_d}{v}_{d\text{'}}+\frac{L_q}{L_d}P{\omega }_m{i}_{q\text{'}}\end{array}$$

$$\frac{d}{dt}{i}_{q\text{'}}=\frac{1}{L_q}{v}_{q\text{'}}-\frac{L_d}{L_q}P{\omega }_m{i}_{d\text{'}}$$

$$T_e=1.5P\left((L_d-L_q)i_{d\text{'}}{i}_{q\text{'}}\right)+P\left(\frac{\partial {\psi }_a}{\partial {\theta }_e}{i}_a+\frac{\partial {\psi }_b}{\partial {\theta }_e}{i}_b+\frac{\partial {\psi }_c}{\partial {\theta }_e}{i}_c\right)$$

This table describes the variables used in these equations.

Lq, Ld	q- and d-axis inductances (H)
R	Resistance of the stator windings (ohm)
iq', id'	q'- and d'-axis currents (A)
vq', vd'	q'- and d'-axis voltages (V)
ωm	Angular mechanical velocity of the motor (rad/s)
ωe	Angular electrical velocity of the motor (rad/s)
P	Number of pole pairs
Te	Electromagnetic torque (Nm)
Θe	Electrical angle (rad)
Va, Vb, Vc	Stator phase A, B, and C voltages (V)
ia, ib, ic	Stator phase A, B, and C currents (A)
Ea, Eb, Ec	Back EMF of stator phases A, B, and C (V/m/s)
Ψa, Ψb, Ψc	Total fluxes linking each stator winding (Wb)

Mechanical System

The motor angular velocity is given by:

$$\begin{array}{c}\frac{d}{dt}{\omega }_m=\frac{1}{J}\left(T_e-T_f-F{\omega }_m-T_m\right)\\ \frac{d{\theta }_m}{dt}={\omega }_m\end{array}$$

The equations use these variables.

J	Combined inertia of motor and load (kgm^2)
F	Combined viscous friction of motor and load (N·m/(rad/s))
θm	Motor mechanical angular position (rad)
Tm	Motor shaft torque (Nm)
Te	Electromagnetic torque (Nm)
Tf	Motor shaft static friction torque (Nm)
ωm	Angular mechanical velocity of the motor (rad/s)

Amplitude Invariant dq Transformation

The block uses these equations to implement amplitude invariant dq transformation to ensure that the dq and three phase amplitudes are equal.

$\left[\begin{array}{c}{v}_{sd}\\ v_{sq}\end{array}\right]= \frac{2}{3} \left[\begin{array}{ccc}\text{cos}({\Theta }_{da})& \text{cos}({\Theta }_{da}-\frac{2\pi }{3})& \text{cos}({\Theta }_{da}+\frac{2\pi }{3})\\ -\text{sin}({\Theta }_{da})& -\text{sin}({\Theta }_{da}-\frac{2\pi }{3})& -\text{sin}({\Theta }_{da}+\frac{2\pi }{3})\end{array}\right]\left[\begin{array}{c}{v}_a\\ v_b\\ v_c\end{array}\right]$

$\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]= \left[\begin{array}{cc}\text{cos}({\Theta }_{da})& -\text{sin}({\Theta }_{da})\\ \begin{array}{c}\text{cos}({\Theta }_{da}-\frac{2\pi }{3})\\ \text{cos}({\Theta }_{da}+\frac{2\pi }{3})\end{array}& \begin{array}{c}-\text{sin}({\Theta }_{da}-\frac{2\pi }{3})\\ -\text{sin}({\Theta }_{da}+\frac{2\pi }{3})\end{array}\end{array}\right]\left[\begin{array}{c}{i}_{sd}\\ i_{sq}\end{array}\right]$

The equations use these variables.

Θda	dq stator electrical angle with respect to the rotor a-axis (rad)
vsq, vsd	Stator q- and d-axis voltages (V)
isq, isd	Stator q- and d-axis currents (A)
va, vb, vc	Stator voltage phases a, b, c (V)
ia, ib, ic	Stator currents phases a, b, c (A)

Trapezoidal Rate of Change of Flux

The rotor magnetic field due to the permanent magnets create a trapezoidal rate of change of flux with the rotor angle. This figure shows the rate of change of flux for the three phases of a three-phase BLDC motor.

Examples

Six-Step Commutation of BLDC Motor Using Sensor Feedback

Use six-step commutation technique to control speed and direction of rotation of a three-phase BLDC motor.

• Open Model

Ports

Input

LdTrq — Load torque on motor

scalar

Load torque on the motor shaft, T_m, in N·m.

Dependencies

To enable this port, set the Mechanical input configuration parameter to Torque.

Spd — Motor shaft speed

scalar

Angular velocity of the motor, ω_m, in rad/s.

Dependencies

To enable this port, set the Mechanical input configuration parameter to Speed.

PhaseVolt — Stator terminal voltages

1-by-3 array

Stator terminal voltages, V_a, V_b, and V_c, in V.

Output

Info — Bus signal

bus

The bus signal contains these block calculations.

Signal			Description	Variable	Units
IaStator			Stator phase current A	ia	A
IbStator			Stator phase current B	ib	A
IcStator			Stator phase current C	ic	A
IdSync			d' axis current	id'	A
IqSync			q' axis current	iq'	A
VdSync			d' axis voltage	vd'	V
VqSync			q' axis voltage	vq'	V
MtrSpd			Angular mechanical velocity of the motor	ωm	rad/s
MtrPos			Motor mechanical angular position	θm	rad
MtrTrq			Electromagnetic torque	Te	N·m
MtrHall			Hall sensor output	-	-
BackEMF			Back EMF generated in motor	Ea, Eb, Ec	V/m/s

PhaseCurr — Phase a, b, c current

1-by-3 array

Phase a, b, c current, i_a, i_b, and i_c, in A.

MtrTrq — Motor torque

scalar

Motor torque, T_mtr, in N·m.

Dependencies

To enable this port, set the Mechanical input configuration parameter to Speed.

MtrSpd — Motor speed

scalar

Angular speed of the motor, ω_mtr, in rad/s.

Dependencies

To enable this port, set the Mechanical input configuration parameter to Torque.

Parameters

Block Options

Mechanical input configuration — Select port configuration

Torque (default) | Speed

This table summarizes the port configurations.

Port Configuration	Creates Input Port	Creates Output Port
Torque	LdTrq	MtrSpd
Speed	Spd	MtrTrq

Simulation type — Select simulation type

Continuous (default) | Discrete

By default, the block uses a continuous sample time during simulation. If you want to generate code for single-precision targets, set the parameter to Discrete.

Sample time (s) — Sample time for discrete integration

25e-6 (default) | scalar

Integration sample time for discrete simulation, in s.

Dependencies

To enable this parameter, set Simulation type to Discrete.

Parameters

Number of pole pairs — Pole pairs of motor

4 (default) | scalar

Motor pole pairs, P.

Stator resistance per phase (Ohm) — Stator resistance in each phase

0.36 (default) | scalar

Stator resistance per phase, R_s, in ohms.

Stator d-axis inductance (H) — Stator inductance across d-axis

2e-4 (default) | scalar

Stator inductance across d-axis, L_d, in H.

Stator q-axis inductance (H) — Stator inductance across q-axis

2e-4 (default) | scalar

Stator inductance across q-axis, L_q, in H.

Permanent flux linkage constant (Wb) — Flux-linkage constant

0.0064 (default) | scalar

Permanent flux linkage constant, λ_pm, in Wb.

Inertia (kgm^2) — Rotor inertia

7.0616e-06 (default) | scalar

Rotor inertia, J, in kg.m^2.

Viscous damping (Nm/rad/s) — Viscous damping

2.6369e-06 (default) | scalar

Viscous damping, F, in N·m/(rad/s).

Static friction (Nm) — Static friction

0 (default) | scalar

Static friction, T_f, in N·m.

Initial Values

Initial d-axis current (A) — Initial current across d'-axis

0 (default) | scalar

Initial d-axis current, i_d', in A.

Initial q-axis current (A) — Initial current across q'-axis

0 (default) | scalar

Initial q-axis current, i_q', in A.

Initial rotor mechanical position (rad) — Initial mechanical position of rotor

0 (default) | scalar

Initial angular position of motor, θ_m0, in rad.

Initial rotor mechanical speed (rad/s) — Initial mechanical speed of rotor

0 (default) | scalar

Initial angular velocity of the motor, ω_m0, in rad/s.

Dependencies

To enable this parameter, set Mechanical input configuration to Torque.

Version History

Introduced in R2023a