Brushless direct current motor#

The Brushless direct current motor block is a model of a synchronous electric machine with a trapezoidal EMF shape and rotor position sensors.

The machine model is described by the following equations:

stator winding equations (1):

(1)#\[\begin{split}u_A=i_A \cdot R + \frac{di_A}{dt} \cdot L + E_A\\u_B=i_A \cdot R + \frac{di_B}{dt} \cdot L + E_B\\u_C=i_A \cdot R + \frac{di_C}{dt} \cdot L + E_C\end{split}\]

where:

\(i_A,i_B,i_C\) – current of phase A, B and C respectively, [A];

\(R\) – active resistance of the phase winding, [Ohm];

\(L\) – phase winding inductance, [H];

\(E_A,E_B,E_C\) – EMF of phase A, B and C respectively, [V].

equations EMF of windings (2):

(2)#\[\begin{split}E_A=\frac{p \cdot k_e}{2}\cdot \omega_r \cdot F(\theta_e)\\E_B=\frac{p \cdot k_e}{2}\cdot \omega_r \cdot F(\theta_e-\frac{2\cdot \pi}{3})\\E_C=\frac{p \cdot k_e}{2}\cdot \omega_r \cdot F(\theta_e-\frac{4\cdot \pi}{3})\end{split}\]

where:

\(p\) – number of pole pairs;

\(k_e\) – design constant EMF, [V/(rad/s)];

\(\omega_r\) – angular frequency of rotor rotation, [rad/s];

\( heta_e\) – electrical angle of rotation of the rotor, [rad];

\(F\) is a function of EMF, which realizes a trapezoidal shape depending on the electrical angle of rotation of the rotor, shown in Fig. 1 :

Fig. 1 Trapezoidal form of the EMF function#

electromagnetic torque on the shaft (3):

(3)#\[M_e=\frac{p \cdot k_t}{2} \cdot (i_A \cdot F(\theta_e)+i_B \cdot F(\theta_e - \frac{2\cdot \pi}{3})+i_C \cdot F(\theta_e - \frac{4\cdot \pi}{3}))\]

where \(k_t\) is the structural torque constant, [N\(\cdot\)m/A].

description of the mechanics of the engine (4):

(4)#\[\frac{d\omega}{dt}\cdot J_Я=M_{вр}-М_{тр}-М_{сн}\]

where:

\(J_Я\) – moment of inertia of the anchor, [N\(\cdot\)m\(^2\)];

\(M_{вр}\) – electromagnetic torque on the armature shaft, [N\(\cdot\)m];

\(M_{тр}\) – friction moment, determined by friction in the bearings and brush-collector apparatus, [N\(\cdot\)m];

\(M_{сн}\) – the moment of resistance created by the load on the armature shaft, [N\(\cdot\)m].

Note

This value is supplied to the unit from the outside via the automation port and/or the mechanics port.

the friction moment is determined by the formula (5):

(5)#\[M_{тр}=\omega_r \cdot K_{тр}\]

where \(K_{тр}\) is the friction coefficient.

1. Properties#

Property	Default Value	Description	Units of measurement
Number of pole pairs	2.0	–	–
Winding active resistance	2.875	–	Ohm
Winding inductance	0.0027	–	H
Torque constant	0.0042	–	N\(\cdot\)m/А
EMF constant	0.0042	–	V/(rad/s)
Rotor inertia	5.0E-4	–	kg\(\cdot\)m\(^2\)
Rotor friction coefficient	8.9E-5	–	N\(\cdot\)m/(rad/s)
Initial Condition Array [rotor angular velocity, rotor angle]	0.0; 0.0	–	rad/s; rad
Star connection of windings	On	Changes the electrical circuit of the stator windings and the number of block terminals	–

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Output torque	Input	The value of the external torque on the shaft from the load	\(N\cdot m\)	Automation
A+	Input	Beginning of phase A of the stator winding	-	Electronics
A-, available when selecting `Star connection of windings` - `Off`	Output	End of phase A of the stator winding	-	Electronics
B+	Input	Start of phase B of the stator winding	-	Electronics
B-, available when selecting `Star connection of windings` - `Off`	Output	End of phase B of the stator winding	-	Electronics
C+	Input	Start of phase C of the stator winding	-	Electronics
C-, available when selecting `Star connection of windings` - `Off`	Output	End of phase C of the stator winding	-	Electronics
N, available when selecting `Star connection of windings` - `On`	Output	Neutral connection point of the windings	-	Electronics
Angular velocity	Output	Connects to the angular velocity ports of blocks from the library Mechanics\Rotational motion	rad/s	Rotational motion

In order to see the connection capabilities of the block, you need to go to page Description of block ports