Stepper motor#

The Stepper motor block is a model of a synchronous brushless electric machine with a stepwise change in rotor position. The block implements two types of stepper motors:

reactive;
hybrid.

In hybrid stepper motor mode, the model equations depend on the number of phases. For m = 2 phases, the block is described by the following equations:

stator winding equations (1):

(1)#\[\begin{split}U_A=E_A+I_A \cdot R + \frac{dI_A}{dt}\cdot L\\U_B=E_B+I_B \cdot R + \frac{dI_B}{dt}\cdot L\end{split}\]

where:

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

\(I_A, I_B\) – current of phase A and B, respectively, [A];

R – active resistance of one phase winding, [Ohm];

L – inductance of one phase winding, [H].

equations EMF of windings (2):

(2)#\[\begin{split}E_A=-p \cdot \psi_m \cdot \frac{d\theta}{dt}\cdot sin(p \cdot \theta)\\E_B=-p\cdot \psi_m \cdot \frac{d\theta}{dt}\cdot sin(p \cdot (\theta-\pi/2))\end{split}\]

where:

p – the number of pole pairs, depends on the number of phases m and the motor step \(\Delta heta\), and is determined by the formula (3) :

(3)#\[p=\frac{360}{2\cdot m \cdot \Delta \theta}\]

\(\psi_m\) – maximum flux linkage from permanent magnets, [Wb];

\(\theta\) – rotor rotation angle, [degrees].

electromagnetic torque on the shaft (4):

(4)#\[M_e=-p \cdot \psi_m \cdot I_A \cdot sin(p\cdot \theta)-p \cdot \psi_m \cdot I_B \cdot sin(p\cdot (\theta- \pi/2))- T_{dm}\cdot sin(m\cdot2 \cdot p \cdot \theta)\]

where \(T_{dm}\) is the maximum holding moment, [N\(\cdot\)m].

When the number of phases m > 2, the block is described by the following equations:

stator winding equations (5):

(5)#\[U_i=E_i+I_i \cdot R + \frac{dI_i}{dt} \cdot L\]

where i is the phase ordinal number, i=1, 2, 3, 4, 5 (in this case, the windings have a letter designation, respectively – A, B, C, D, E).

equations EMF of windings (6) :

(6)#\[E_i=-p \cdot \psi_m \cdot \frac{d\theta}{dt} \cdot sin(p \cdot \theta - \frac{2\cdot \pi \cdot(i-1)}{m})\]

electromagnetic torque on the shaft (7) :

(7)#\[M_e=\sum^m_{i=1}(-p\cdot \psi_m\cdot I_i \cdot sin(p\cdot \theta-\frac{2\cdot \pi\cdot(i-1)}{m}))-T_{dm}\cdot sin(m\cdot 2 \cdot p \cdot \theta)\]

In reactive stepper motor mode, the machine equations are as follows:

stator winding equations (8):

(8)#\[U_i=I_i \cdot R + \frac{dI_i}{dt} \cdot L(\theta)\]

where:

i – phase ordinal number, i=1, 2, 3, 4, 5 (in this case, the windings have a letter designation, respectively – A, B, C, D, E);

\(L( heta)\) is the phase inductance, which changes depending on the rotor position angle \(\theta\) according to a sinusoidal law from the minimum value \(L_{min}\) to the maximum \(L_{max}\).

electromagnetic torque on the shaft (9):

(9)#\[M_e=\sum^m_{i=1}(\frac{I_i}{2}\cdot \frac{dL_i}{d\theta})\]

where \(\frac{dL_i}{ heta}\) is the change in inductance of the corresponding phase.

The mechanical equation of the block is the same for all types of stepper motor (10):

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

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];

\(M_{сн}\) – 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 torque is determined by the formula (11):

(11)#\[M_{тр}=\omega \cdot K_{тр}\]

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

1. Properties#

Property	Default Value	Description	Units of measurement
Type of stepper motor: Hybrid stepper motor (Permanent_magnet); Variable-reluctance stepper motor	Hybrid stepper motor (Permanent_magnet)	–	H
Number of phases: 2; 3; 4; 5	2	–	units
Active resistance of phase winding	0.7	–	Ohm
Winding inductance, available when selecting `Type of stepper motor` - `Hybrid stepper motor (Permanent_magnet)`	0.0014	–	H
Maximum flux linkage, available when selecting `Type of stepper motor` - `Hybrid stepper motor (Permanent_magnet)`	0.005	–	Wb
Maximum winding inductance, available when selecting `Type of stepper motor` - `Variable-reluctance stepper motor`	0.002	–	H
Minimum winding inductance, available when selecting `Type of stepper motor` - `Variable-reluctance stepper motor`	0.001	–	H
Step angle	1.8	–	degrees
Maximum detent torque	0.02	–	N\(\cdot\)m
Rotor inertia	1.2E-7	–	kg\(\cdot\)m\(^2\)
Rotor friction coefficient	0.001	–	N\(\cdot\)m/(rad/s)
Initial angular frequency	0.0	–	rad/s
Initial rotor rotation angle	0.0	–	degrees

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
External 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-	Output	End of phase A of the stator winding	-	Electronics
B+	Input	Start of phase B of the stator winding	-	Electronics
B-	Output	End of phase B of the stator winding	-	Electronics
C+	Input	Start of phase C of the stator winding, used for number of phases equal to 3, 4 and 5	-	Electronics
C-	Output	End of phase C of the stator winding, used for number of phases equal to 3, 4 and 5	-	Electronics
D+	Input	Start of phase D of the stator winding, used for number of phases equal to 4 and 5	-	Electronics
D-	Output	End of phase D of the stator winding, used for number of phases equal to 4 and 5	-	Electronics
E+	Input	Start of phase E of the stator winding, used when number of phases is equal to 5	-	Electronics
E-	Output	End of phase E of the stator winding, used when number of phases is equal to 5	-	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