*3.3.1. Construction and operating principle*

The BLDC motor is also referred to as an electronically commuted motor. There are no brushes on the rotor and the commutation is performed electronically at certain rotor positions. In the DC commutator motor, the current polarity is reversed by the commutator and the brushes, but in the brushless DC motor, the polarity reversal is performed by semiconductor switches which are to be switched in synchronization with the rotor position. Besides of the higher reliability, the missing commutator brings another advantage. For the DC brushed motor the commutator presents also a limiting factor in the maximal speed. Therefore, the BLDC motor can be employed in applications requiring high speed (Jeon, 2000).

The BLDC motor is usually considered as a three-phase system and thus it has to be powered by a three-phase power supply. The rotor position must be known at certain angles, in order to align the applied voltage with the back-EMF. The alignment between the back-EMF and commutation events is very important.

A simple motor model of BLDC motor consisting of a three-phase power converter and a brushless DC motor is shown in Fig. 14.

### *3.3.2. Mathematical model of the BLDC motor*

Modeling of a BLDC motor can be developed in the similar manner as a three-phase synchronous machine. Since there is a permanent magnet mounted on the rotor, some dynamic characteristics are different. Similarly, the model of the armature winding for the BLDC motor is expressed as follows:

$$\mathbf{u}\_{\mathbf{a}} = \mathbf{R}\mathbf{i}\_{\mathbf{a}} + \mathbf{L}\frac{\mathbf{d}\mathbf{i}\_{\mathbf{a}}}{\mathbf{d}t} + \mathbf{e}\_{\mathbf{a}} \tag{4}$$

$$\mathbf{u}\_{\mathbf{b}} = \mathbf{R}\mathbf{i}\_{\mathbf{b}} + \mathbf{L}\frac{\mathrm{d}\mathbf{i}\_{\mathbf{b}}}{\mathrm{d}\mathbf{t}} + \mathbf{e}\_{\mathbf{b}} \tag{5}$$

$$\mathbf{u}\_{\mathbf{c}} = \mathbf{R}\mathbf{i}\_{\mathbf{c}} + \mathbf{L}\frac{\mathbf{d}\mathbf{i}\_{\mathbf{c}}}{\mathbf{d}\mathbf{t}} + \mathbf{e}\_{\mathbf{c}} \tag{6}$$

where *L* is armature self-inductance, *R* - armature resistance, *ua, ub, uc* - terminal phase voltages, *ia, ib, ic* - motor input currents, and *ea, eb, ec* - motor back-EMF.

**Figure 14.** BLDC motor model

330 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

commutation; instead of this they are commutated electronically.

in applications where space and weight are critical factors (Indu, 2008).

The Brushless Direct Current (BLDC) motor is rapidly gaining popularity by its utilization in various industries. As the name implies, the BLDC motor do not use brushes for

The BLDC motors have many advantages over brushed DC motors and induction motors. A few of these are: (1) Better speed versus torque characteristics; (2) High dynamic response; (3) High efficiency; (4) Long operating life; (5) Noiseless operation; (6) Higher speed ranges. In addition, the ratio of torque delivered to the size of the motor is higher, making it useful

The torque of the BLDC motor is mainly influenced by the waveform of back-EMF (the voltage induced into the stator winding due to rotor movement). Ideally, the BLDC motors have trapezoidal back-EMF waveforms and are fed with rectangular stator currents, which give theoretically constant torque. However, in practice, a torque ripple exists, mainly due to EMF waveform imperfections, current ripple, and phase current commutation. The current ripple follows up from PWM or hysteresis control. The EMF waveform imperfections result from variations in the shapes of slot, skew and magnet of BLDC motor, and are subject to design purposes. Hence, an error can occur between actual value and the simulation results. Several simulation models have been proposed for analysis of BLDC

The BLDC motor is also referred to as an electronically commuted motor. There are no brushes on the rotor and the commutation is performed electronically at certain rotor positions. In the DC commutator motor, the current polarity is reversed by the commutator and the brushes, but in the brushless DC motor, the polarity reversal is performed by semiconductor switches which are to be switched in synchronization with the rotor position. Besides of the higher reliability, the missing commutator brings another advantage. For the DC brushed motor the commutator presents also a limiting factor in the maximal speed. Therefore, the BLDC motor

The BLDC motor is usually considered as a three-phase system and thus it has to be powered by a three-phase power supply. The rotor position must be known at certain angles, in order to align the applied voltage with the back-EMF. The alignment between the

A simple motor model of BLDC motor consisting of a three-phase power converter and a

Modeling of a BLDC motor can be developed in the similar manner as a three-phase synchronous machine. Since there is a permanent magnet mounted on the rotor, some

**3.3. BLDC motor** 

motor (Jeon, 2000).

*3.3.1. Construction and operating principle* 

can be employed in applications requiring high speed (Jeon, 2000).

back-EMF and commutation events is very important.

brushless DC motor is shown in Fig. 14.

*3.3.2. Mathematical model of the BLDC motor* 

In the three-phase BLDC motor, the back-EMF is related to a function of rotor position and the back-EMF of each phase has 120º phase angle difference so the equation for each motor phase is as follows:

$$\mathbf{e}\_{\mathbf{a}} = \mathbf{K}\_{\mathbf{w}} \mathbf{f}(\theta\_{\mathbf{e}}) \,\alpha \tag{7}$$

$$e\_b = K\_\mathcal{W} f(\theta\_e - 2\,\pi/3)\omega \tag{8}$$

$$e\_c = K\_\mathcal{W} f(\theta\_e + 2\,\pi/3)\omega \tag{9}$$

where *Kw* is back EMF constant of one phase, *θ<sup>e</sup>* - electrical rotor angle, *ω* - rotor speed. The electrical rotor angle *θe* is equal to the mechanical rotor angle *θm* multiplied by the number of poles *p*:

$$
\Theta\_e = \frac{p}{2} \Theta\_m \tag{10}
$$

Total torque output *Te* can be represented as summation of that of each phase:

$$T\_e = \frac{e\_A i\_A + e\_B i\_B + e\_C i\_C}{\alpha} \tag{11}$$

The equation of mechanical part is represents as follows:

$$T\_e - T\_l = f \frac{d\alpha}{dt} + b\alpha \tag{12}$$

where *Tl* is load torque, *J* - rotor inertia, *b* - friction constant.

#### *3.3.3. Simulink model of the BLDC motor*

Fig. 16 shows the block diagram of the BLDC motor SIMULINK model in the rotor reference frame.

**Figure 15.** Simulink model of the BLDC motor

Fig. 16 shows detail of the BLDC motor block. Fig. 17a shows Simulink diagram of trapezoidal back-EMF and in Fig. 17b there is Simulink model of sinusoidal back-EMF. The trapezoidal functions and the position signals are stored in lookup tables that change their output according to the value of the electrical angle (Indu, 2008).

Unlike a brushed DC motor, the commutation of a BLDC motor is controlled electronically. To rotate the BLDC motor, the stator windings should be energized in sequences. In order to understand which winding will be energized following the energizing sequence, it is important to know the rotor position. It is sensed using Hall Effect sensors embedded into the stator. Most of the BLDC motors contain three Hall sensors embedded into the stator on the non-driving end of the motor. The number of electrical cycles to be repeated to complete a mechanical rotation is determined by rotor pole pairs. Number of electrical cycles/rotations equals to the rotor pole pairs. The commutation sequences are shown in Tab. 4.

**Figure 16.** Detailed overview of the BLDC motor block

Total torque output *Te* can be represented as summation of that of each phase:

*e*

The equation of mechanical part is represents as follows:

where *Tl* is load torque, *J* - rotor inertia, *b* - friction constant.

*3.3.3. Simulink model of the BLDC motor* 

**Figure 15.** Simulink model of the BLDC motor

output according to the value of the electrical angle (Indu, 2008).

equals to the rotor pole pairs. The commutation sequences are shown in Tab. 4.

frame.

2 *p*

*e i ei ei <sup>A</sup> A BB CC <sup>T</sup>*

*<sup>d</sup> TTJ b e l dt*

Fig. 16 shows the block diagram of the BLDC motor SIMULINK model in the rotor reference

Fig. 16 shows detail of the BLDC motor block. Fig. 17a shows Simulink diagram of trapezoidal back-EMF and in Fig. 17b there is Simulink model of sinusoidal back-EMF. The trapezoidal functions and the position signals are stored in lookup tables that change their

Unlike a brushed DC motor, the commutation of a BLDC motor is controlled electronically. To rotate the BLDC motor, the stator windings should be energized in sequences. In order to understand which winding will be energized following the energizing sequence, it is important to know the rotor position. It is sensed using Hall Effect sensors embedded into the stator. Most of the BLDC motors contain three Hall sensors embedded into the stator on the non-driving end of the motor. The number of electrical cycles to be repeated to complete a mechanical rotation is determined by rotor pole pairs. Number of electrical cycles/rotations

*e m* (10)

(12)

(11)

**Figure 17.** Trapezoidal (a) and simusoidal (b) model of the back-EMF


**Table 4.** Electrical degree, Hall sensor value and corresponding commuted phase in clockwise rotation of the rotor

#### *3.3.4. Mathematical and simulink model of the three-phase converter*

The converter supplies the input voltage for three phases of the BLDC motor. Each phase leg comprises two power semiconductor devices. Fig. 18 shows the scheme of the considered three-phase converter.

**Figure 18.** Modelled three-phase converter

Appropriate pairs of the switches (S1 to S6) are driven based on the Hall sensors input. Three phases are commutated in every 60° (el. degrees). The model of the converter is implemented using the equations:

$$U\_{an} = \mathcal{S}\_1 \frac{\upsilon d}{2} - \mathcal{S}\_4 \frac{\upsilon d}{2} - U\_f \tag{13}$$

$$U\_{bm} = \mathcal{S}\_3 \frac{\upsilon d}{2} - \mathcal{S}\_6 \frac{\upsilon d}{2} - U\_f \tag{14}$$

$$U\_{cn} = \mathcal{S}\_5 \frac{\upsilon d}{2} - \mathcal{S}\_2 \frac{\upsilon d}{2} - \mathcal{U}\_f \tag{15}$$

where *Uan, Ubn, Ucn* are line-neural voltages, *Ud* – the DC link voltage, *Uf* – the forward diode voltage drop.

Fig. 19a shows the Simulink model of the three-phase converter block. In the simulation we assumed an ideal diode with neglected voltage drop *Uf*. The *Commutation sequences* block was developed based on the commutation sequence shown in Tab. 4. Converter voltage waveforms that are switched according to the commutation sequences in Tab. 4 are shown in Fig. 19b.

**Figure 19.** Detailed overview of the three-phase converter (a) and voltage source waveforms (b)
