Induction motor with short-circuited rotor (d-q model)#

The Induction motor with short-circuited rotor (d-q model) block is a model of an electric three-phase machine with short-circuited rotor.

Note

Stator windings are connected according to star circuit;
The mathematical model of the unit is described in the d-q coordinate system, which is oriented along the direction of the rotor flux vector and rotates 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 stator linkage flux to axis d, [Wb];

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

\(i_{sd}\) – projection of stator current to axis d, [А];

\(i_{sq}\) – projection of stator current to axis q, [А];

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

\(\omega_e\) – electric 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 rotor linkage flux to axis d, [Wb];

Note

Since d-q axis is oriented by rotor linkage flux, then \(\psi_{rq}\)=0, and full linkage flux of the rotor \(\psi_R=\psi_{rd}\)

\(i_{rd}\) – projection of rotor current to axis d, [А];

\(i_{rq}\) – projection of rotor current to axis q, [А];

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

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

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

where:

\(p\) – the number of pole pairs;

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

linkage flux equation (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 stator and rotor windings, [H];

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

Note

It is the sum of the stator 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 inductance of rotor, [H].

Note

It is the sum of the rotor own 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\(^2\)];

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

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

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

Note

This value is sent to the block from outside through the automation port and/or the mechanics port.

The moments presented in equation (11) are calculated as follows:

electromagnetic torque on 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 \(K_{fr}\) – friction factor.

1. Properties#

Property	Default Value	Description	Units of measurement
Number of pole pairs	2	-	-
Stator winding resistance	0.8	-	Ohm
Stator winding inductance	0.0004	-	H
Rotor winding resistance	0.2	-	Ohm
Rotor winding inductance	0.004	-	H
Stator to rotor mutual inductance	0.04	-	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 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 С	-	Electronics
Angular frequency	Output	Connects to 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