BLDC HDL

Three-phase brushless DC motor with trapezoidal flux distribution

Since R2023b

• 

Libraries:
Motor Control Blockset HDL Support / 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.

Motor Construction

This figure shows a motor with a single pole pair.

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

For the axes convention, the phase-a aligns with d-axis 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

Config — Block configuration signal

vector

Configuration signal for the BLDC HDL block containing the block configuration parameters. For more information about the constituent parameters, see the Config output port description in the BLDC Configuration block.

Data Types: single

PhaseVolt — Stator terminal voltages

1-by-3 array

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

Data Types: single

LdTrq/Spd — Load torque on motor or speed of motor shaft

scalar

Based on the mode of operation defined by the Port Configuration parameter available in the multiplexed Config input, the port supports one of these inputs:

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

• ω_m — Angular velocity of the motor in rad/s.

Data Types: single

Reset — Reset bus signal

scalar

Bus signal that the block uses to reset the internal integrators.

Data Types: Boolean

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 currents i_a , i_b , and i_c , in A of phases a, b, and c, respectively.

Torque — Motor torque

scalar

Motor torque, T_mtr, in N·m.

Speed — Motor speed

scalar

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

MtrElecAngle — Motor electrical angle

scalar

Electrical position of the motor, θ_e, in rad.

Data Types: single

Parameters

Sample time (s) — Sample time of block

1e-6 (default) | scalar

The fixed time interval (in seconds) between consecutive instances of block execution.

Extended Capabilities

HDL Code Generation

Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.

HDL Architecture

This block has one default HDL architecture.

HDL Block Properties

ConstrainedOutputPipeline	Number of registers to place at the outputs by moving existing delays within your design. Distributed pipelining does not redistribute these registers. The default is 0. For more details, see ConstrainedOutputPipeline (HDL Coder).
InputPipeline	Number of input pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see InputPipeline (HDL Coder).
OutputPipeline	Number of output pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see OutputPipeline (HDL Coder).

Version History

Introduced in R2023b