Asynchronous motor with squirrel-cage rotor (nonlinear model)#

The Asynchronous motor with squirrel-cage rotor (nonlinear model) block represents a model of a three-phase electrical machine with a squirrel-cage rotor, accounting for parameter nonlinearity.

Note

The stator windings are connected in a star configuration;
The block’s mathematical model is described in the d-q coordinate system, oriented along the rotor flux linkage vector and rotating synchronously with it.

The model is described by the following equations:

stator winding equations (1), (2)

(1)#\[u_{sd}=\frac{d\psi_{sd}}{dt}+i_{sd}\cdot R_s-\omega_e \cdot \psi_{sq}\]

(2)#\[u_{sq}=\frac{d\psi_{sq}}{dt}+i_{sq}\cdot R_s+\omega_e \cdot \psi_{sd}\]

where:

\(\psi_{sd}\) – projection of the stator flux linkage onto the d-axis, [Wb];

\(\psi_{sq}\) – projection of the stator flux linkage onto the q-axis, [Wb];

\(i_{sd}\) – projection of the stator current onto the d-axis, [A];

\(i_{sq}\) – projection of the stator current onto the q-axis, [A];

\(R_s\) – active resistance of the stator winding, [Ohm];

\(\omega_e\) – electrical angular frequency, [rad/s].

rotor winding equations (3), (4):

(3)#\[u_{rd}=0=\frac{d\psi_{rd}}{dt}+i_{rd}\cdot R_r\]

(4)#\[u_{rq}=0=i_{rq}\cdot R_r + \Delta \omega \cdot \psi_{rd}\]

where:

\(\psi_{rd}\) – projection of the rotor flux linkage onto the d-axis, [Wb];

Note

Since the d-q axis is oriented along the rotor flux linkage, \(\psi_{rq}=0\), and the total rotor flux linkage \(\psi_R=\psi_{rd}\).

\(i_{rd}\) – projection of the rotor current onto the d-axis, [A];

\(i_{rq}\) – projection of the rotor current onto the q-axis, [A];

\(R_r\) – active resistance of the rotor winding, [Ohm];

\(\Delta \omega\) – slip angular frequency, [rad/s] (5):

(5)#\[\Delta \omega=\omega_e-p \cdot \omega_r\]

where:

\(p\) – number of pole pairs;

\(\omega_r\) – rotor angular velocity, [rad/s].

flux linkage equations (6),(7),(8):

(6)#\[\psi_{sd}=i_{sd}\cdot L_s+i_{rd}\cdot L_m;\]

(7)#\[\psi_{sq}=i_{sq}\cdot L_s+i_{rq}\cdot L_m;\]

(8)#\[\psi_{rd}=i_{rd}\cdot L_R+i_{sd}\cdot L_m;\]

where:

\(L_m\) – mutual inductance between the stator and rotor windings, [H];

\(L_S\) – total stator inductance, [H].

Note

This is the sum of the stator’s self-inductance (leakage inductance) and the mutual inductance between the stator and rotor windings (9):

(9)#\[L_S=L_{s\sigma}+L_m\]

\(L_R\) – total rotor inductance, [H].

Note

This is the sum of the rotor’s self-inductance (leakage inductance) and the mutual inductance between the stator and rotor windings (10):

(10)#\[L_R=L_{r\sigma}+L_m\]

mechanics (11):

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

where:

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

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

\(M_{тр}\) – friction torque, determined by bearing friction and the brush-collector assembly, [\(N\cdot m\)];

\(M_{сн}\) – load torque created by the load on the armature shaft, [\(N\cdot m\)];

Note

This value is supplied to the block externally via the automation port and/or mechanics port.

The torques presented in the equation (11) are calculated using the following formulas:

electromagnetic torque on the armature shaft (12):

(12)#\[M_{вр}=\frac{3}{2}\cdot p \cdot \frac{L_m}{L_R}\cdot \psi_{rd} \cdot i_{sq}\]

friction torque (13):

(13)#\[M_{тр}=\omega \cdot K_{тр}\]

where \(K_{тр}\) — friction coefficient.

The model implements a nonlinear dependency (14):

(14)#\[L_m=f(i_s)\]

where \(i_s\) — amplitude of the stator current vector, defined as (15):

(15)#\[i_s=\sqrt{i_{sd}^2+i_{sq}^2}\]

In the block, nonlinearity is specified as an array of numerical values (a table with numerical values for each nonlinearity point).

Note

Each point in the array represents a pair of values:
- stator current, [A];
- mutual inductance between the stator and rotor, [H].
Each point indicates the value of the mutual inductance between the stator and rotor at the current stator current value.

Important

The following requirements apply to the nonlinearity array in this block:

The first point must have the value \(i_s\)=0 [A];
There should be no points with the same \(i_s\) current value;
Each subsequent \(i_s\) current value must be greater than the previous one;
The number of \(i_s\) current values must equal the number of \(L_m\) mutual inductance values;
The number of points must be at least two;
Points must not contain negative values.

1. Properties#

Property	Default Value	Description	Units of measurement
Pole pairs	2.0	-	-
Active resistance of the stator winding	0.8	-	Ohm
Stator winding inductance (leakage inductance)	4.0E-4	-	H
Active resistance of the rotor winding	0.2	-	Ohm
Rotor winding inductance (leakage inductance)	0.004	-	H
The points of the nonlinearity array - the stator current	0.0;8.0;16.0;20.0;40.0;120.0	Positive values starting from 0	А
The points of the nonlinearity array - the mutual inductance	0.14;0.12;0.09;0.07;0.04;0.04	Number of values must equal the number of current values	H
Moment of inertia of the rotor	0.01	-	kg\(\cdot m^2\)
Rotor friction coefficient	0.01	-	N\(\cdot m\)/rad/s
Array of initial conditions [rotor angular frequency (rad/s), rotor rotation angle (rad)]	0.0;0.0;	-	[rad/s; rad]

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
External torque	Input	Torque value on the shaft from the load	N\(\cdot\)m	Automation
А	Input	Start of stator winding phase A	-	Electronics
В	Input	Start of stator winding phase B	-	Electronics
С	Input	Start of stator winding phase C	-	Electronics
Angular frequency	Output	Connects to angular velocity ports of blocks from the Mechanics (Rotational Motion) library	rad/s	Angular Velocity

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