Brushless DC Motor (BLDC)

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Description

This model represents a brushless DC (BLDC) motor with trapezoidal back electromotive force (EMF). During motoring operation, the torque and speed share the same sign. The rotor angle is defined as the angle between the a‑phase and the d‑axis, with its reference position set where the negative d‑axis current aligns with the positive a‑phase current.

Magnetic saturation is modeled using a piecewise‑linear approximation of the direct‑ and quadrature‑axis inductances. The stator windings are connected in a star configuration. To access the neutral‑point terminal, enable the Neutral Point option in the property panel.

Electrical model and equations

$v_a = R_s i_a + \frac{d}{dt} (L_{aa} i_a + L_{ab} i_b + L_{ac} i_c) + e_a$

$v_b = R_s i_b + \frac{d}{dt} (L_{ba} i_a + L_{bb} i_b + L_{bc} i_c) + e_b$

$v_c = R_s i_c + \frac{d}{dt} (L_{ca} i_a + L_{cb} i_b + L_{cc} i_c) + e_c$

where $v_a$ , $v_b$ , and $v_c$ are phase voltages. Phase inductances ( $L_{aa}$ , $L_{ab}$ , $L_{ac}$ ,..., $L_{cc}$ ) are computed from the incremental inductances in the rotor reference frame ( $L_{d}$ , $L_{q}$ ) specified in the property panel. When the phase currents fall between two defined saturation points, the model uses linearly interpolated inductance values.

Back EMF $e_a$ , $e_b$ , and $e_c$ have trapezoidal waveform defined as below.

$e_a = \omega_r \phi_m f_a(\theta_r)$

$e_b = \omega_r \phi_m f_b(\theta_r)$

$e_c = \omega_r \phi_m f_c(\theta_r)$

where $\omega_r = N_{pp} \Omega$ is the electrical speed of the rotor field, $\phi_m = \frac{K_e}{ N_{ pp} }$ is the permanent magnet flux linkage. $f_a(\theta_r)$ , $f_b(\theta_r)$ , and $f_c(\theta_r)$ are unit function corresponding to the trapezoidal induced emfs as a function of the rotor position. The figure below illustrates the typical BLDC trapezoidal back‑EMF waveforms.

Electromechanical equations

Electro-magnetic torque:

$T_e = (e_a * i_a + e_b * i_b + e_c * i_c)/\omega_r$

Mechanical rotational speed $\Omega$ :

$J \frac{d\Omega}{ dt} = T_e - B \Omega$

Reference

[1] Modeling and control of a brushless DC motor. MTech thesis, S. Rambabu, Department of Electrical Engineering National Institute of Technology Rourkela, 2007 https://core.ac.uk/download/pdf/53188902.pdf

[2] Simulation of BLDC Motor Speed PI Control in Simulink, Malcolm's Technical Blog Embedded,electronics,and other stuff https://malichao.wordpress.com/2015/11/18/simulation-of-bldc-motor-speed-pi-control-in-simulink/

Library

Electrical > Motors

Parameters

Property	Display Name	Parameter Type	Description
Rs	Rs - Stator Resistance [Ohm]	DoubleParameter	Stator Winding Resistance [Ohm]
Ld	Ld - Direct Axis Inductance [H]	DoubleMatrixParameter	Direct Axis Inductance [H]
Lq	Lq - Quadratic Axis Inductance [H]	DoubleMatrixParameter	Quadratic Axis Inductance [H]
Ke	Ke - Back-EMF Constant [V/(rad/sec)]	DoubleParameter	Back-EMF Constant
J	J - Rotor Inertia [kg.m²]	DoubleParameter	Rotor Inertia [kg.m²]
B	B - Rotor Friction Coefficient [N.m/(rad/s)]	DoubleParameter	Rotor Friction Coefficient [N.m/(rad/s)]
InitialSpeed	Rotor Initial Speed [rad/s]	DoubleParameter	Rotor initial speed [ rad/s]
InitialCurrent	Initial Current [A]	DoubleArrayParameter	Initial Current Vector [IA IB IC]
NPP	NPP - Poles pairs	IntParameter	Number of pole pairs
ExposeNeutralPoint	Neutral Point	BoolParameter	Expose Neutral Point (Boolean)

Pins

Property	Pin Name	Type	Description
Pin_A	A	Electrical	Phase A (Electrical)
Pin_B	B	Electrical	Phase B (Electrical)
Pin_C	C	Electrical	Phase C (Electrical)
Pin_R	R	RotationalMechanical	Rotor (Rotational Mechanical)
Pin_Angle	Angle	ControlOut	Rotor Angle in radians, electrical angle (Control)

Default Size

Width	Height
8	8